High frequency backscattering by a solid cylinder with axis tilted relative to a nearby horizontal surface Daniel S. Plotnick and Philip L. MarstonKevin L. Williams and Aubrey L. EspañaOU
Citation: The Journal of the Acoustical Society of America 137, 470 (2015); doi: 10.1121/1.4904490 View online: http://dx.doi.org/10.1121/1.4904490 View Table of Contents: http://asa.scitation.org/toc/jas/137/1 Published by the Acoustical Society of America
Articles you may be interested in High frequency imaging and elastic effects for a solid cylinder with axis oblique relative to a nearby horizontal surface The Journal of the Acoustical Society of America 140, (2016); 10.1121/1.4961001
High frequency backscattering by a solid cylinder with axis tilted relative to a nearby horizontal surface Daniel S. Plotnicka) and Philip L. Marston Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814
~a Kevin L. Williams and Aubrey L. Espan Applied Physics Laboratory, Acoustics Department, University of Washington, Seattle, Washington 98105
(Received 9 April 2014; revised 16 September 2014; accepted 4 November 2014) The backscattering spectrum versus azimuthal angle, also called the “acoustic color” or “acoustic template,” of solid cylinders located in the free water column have been previously studied. For cylinders lying proud on horizontal sand sediment, there has been progress in understanding the backscattering spectrum as a function of grazing angle and the viewing angle relative to the cylinder’s axis. Significant changes in the proud backscattering spectrum versus the freefield case are associated with the interference of several multipaths involving the target and the surface. If the cylinder’s axis has a vertical tilt such that one end is partially buried in the sand, the multipath structure is changed, thus modifying the resulting spectrum. Some of the changes in the template can be approximately modeled using a combination of geometrical and physical acoustics. The resulting analysis gives a simple approximation relating certain changes in the template with the vertical tilt of the cylinder. This includes a splitting in the azimuthal angle at which broadside multipath features are observed. A similar approximation also applies to a metallic cylinder adjacent to a flat free C 2015 Acoustical Society of America. surface and was confirmed in tank experiments. V [http://dx.doi.org/10.1121/1.4904490] [OU]
Pages: 470–480
I. INTRODUCTION
Efforts have been made in recent years to understand the acoustic backscattering from objects in the ocean. Acoustic energy may be backscattered by direct acoustic reflections, referred to as specular reflections, and elastic scattering mechanisms. The interference between specular and elastic scattering creates an acoustic spectrum that can provide valuable information about that object’s shape, composition, and orientation. This information can also give insight into the basic elastic scattering mechanisms for different shapes such as solid cylinders and cylindrical shells. In acoustic backscattering the sound source and receiver are collocated. The full spectrum versus the object’s azimuthal angle will be referred to as the “acoustic color” or “acoustic template.” The acoustic color of an object depends not only on its shape and material, but also on its nearby environment. In particular, scattering by objects in contact with the sediment/water interface (often called a “proud” configuration) has received attention.1–4 The acoustic color for a solid cylinder placed in a horizontal proud configuration was previously examined in Refs. 1 and 2. In practice, it is likely that an object will be tilted vertically and partially buried in the seafloor rather than lying perfectly horizontal in a proud configuration. Figure 1 shows just such a tilted configuration for a cylinder. Figure 2 shows the results of an experiment performed on a tilted, partially buried solid cylinder at the Naval Surface Warfare Center, Panama City (NSWC-PCD) during PONDEX09. This cylinder had dimensions a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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60.96 cm 30.48 cm, and for brevity will be referred to as the 2 foot 1 foot cylinder. Tilting the cylinder axis causes changes to the acoustic template. We will show that several of the most prominent changes are understandable using geometric methods. This paper will examine the changes to the acoustic template of a solid cylinder as vertical tilt angle is varied. The principal experimental observations and the resulting models used to predict certain features of those observations are presented in the body of this paper. The method used for calibration of the experiment and details of the models are reserved for the Appendixes; the inquisitive reader is encouraged to make use of these Appendixes, but they are not required for a basic understanding of the observed changes to the acoustic template. Figure 3 provides a 3-D schematic of the geometry used throughout this work. The target is taken to be in the far field of the acoustic source/receiver. The coordinate system f^ e 01 ; e^02 ; e^03 g defines the laboratory frame. The incident wave vector and the interface normal define the plane of incidence; the e^01 e^02 plane is parallel to the interface and the e^02 e^03 plane is perpendicular to the interface and parallel to the plane of incidence. The second set of coordinates defines the cylinder frame, f^ e 01 ; e^02 ; e^03 g; the e^02 direction is defined so as to lie along the cylinder axis. The grazing angle hg is measured between the incident wave vector and the local interface in the plane of incidence. The cylinder tilt angle ht is formed by the cylinder axis and the vector projection ~ p of the cylinder axis on the interface. The azimuthal angle H is measured between ~ p and the ^ e 01 direction; the sign convention for H is chosen such that when H is negative the end vertically furthest from the interface (vertically closest to the
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FIG. 1. (Left) Schematic of geometry for tilted cylinder. Acoustic beam at grazing angle hg insonifies a proud or partially buried cylinder lying at a tilt angle ht . (Right) The experimental setup used in the small scale laboratory experiment. Two support lines run from the corners of the cylinder to a rotating stage (not pictured). The cylinder is tilted vertically to an arbitrary angle with the upper corner placed near the air-water interface. It is insonified from below. Note that tilting the cylinder shifts its center of mass position relative to the axis of rotation; however, this shift does not affect the acoustic template in the far field.
S/R) is pointed toward the S/R, for positive azimuths the opposite. An experiment on a solid cylinder performed in a freshwater tank is discussed in Sec. II. This section will briefly highlight the observed changes that occur at end-on and broadside incidence. Section III will discuss the backscattering structure at broadside incidence for both the tilted and untilted cases. The scattering structure at end-on will be examined in Sec. IV. Finally, a physical acoustics model for the broadside structure will be introduced and a comparison to a finite element (FE) model will be made in Sec. V. Appendix A addresses the calibration method used in the freshwater tank experiments; Appendix B discusses the simple geometric model discussed in Sec. III; and Appendix C details the physical acoustics model discussed in Sec. V. II. SCALED EXPERIMENT
In order to examine the effect of vertical tilt of the cylinder on the acoustic template a series of backscattering experiments were performed at the water tank facility at Washington State University (WSU). The goal was to provide backscattering data from the 180 of azimuth required to characterize the target. Performing such scans on proud or partially buried targets in sediment is difficult because of practical experimental concerns; these concerns include but are not limited to configuration details (e.g., changes in burial depth, exact tilt angle, and flatness of the local interface) and complications/constraints imposed by the scanning system used (e.g., see Ref. 4). Instead, a solid aluminum cylinder
was suspended from two corners near the air-water interface of a water tank and tilted to the desired angle ht relative to that interface. Vertical tilt was controlled by lengthening or shortening the suspending lines on either corner (see Fig. 1). The cylinder had dimensions 50.80 mm 25.40 mm and for brevity will be referred to as the 2 in. 1 in. cylinder. To avoid undesired wetting effects, the cylinder was fully submerged with the upper corner placed as close to the interface as possible without deforming the free surface. The experiment was performed with a collocated source/receiver transducer. A fixed, wide band transducer insonified the cylinder as it was spun about a vertical axis using a rotating stepper motor system. Backscattering data was collected every 1 of azimuth. The transducer was tilted vertically to a grazing of hg ¼ 19:5 relative to the interface so that the cylinder lay within the beam main lobe. A short duration, wide bandwidth signal (50–450 kHz) was used. The transducer slant range of 2.52 m was sufficient for far field measurements. The primary goal of this experiment was to demonstrate the geometric relationship between cylinder orientation and certain azimuth dependent features; the salient geometry is unaffected by either changes in the interface reflection coefficient or by partial burial. The use of an air-water interface as a stand in for the water-sediment interface does modify this reflection coefficient, and as a consequence of the complete submersion of the cylinder effects due to partial exposure and the possibility of contributions from refracted waves in the sediment could not be studied. Regardless, the air-water interface remains a useful tool for studying boundary effects (e.g.,
FIG. 2. (Color online) (Left) Results of a backscattering experiment performed on a tilted, partially buried 2 ft 1 ft solid aluminum cylinder performed at NSWC-PCD during PONDEX09 where 0 is broadside. The tilt of the cylinder leads to an azimuthal splitting of the broadside multipaths shown in Fig. 5. The splitting, discussed in Sec. III and Fig. 4, is clearly visible between path 1 and paths 2,3. Path 4 is blocked by the partial burial of the cylinder. (Right) A diver near the tilted cylinder, compare with Fig. 1. J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
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FIG. 3. Schematic of coordinate system used in this paper. Acoustic beam from a collocated source/receiver (S/R) at grazing angle hg insonifies a cylinder sitting above an interface at a tilt angle ht . The coordinate set 0 0 0 e 1 ; e^2 ; e^3 g the cylinder f^ e 01 ; e^02 ; e^03 g defines the laboratory frame and f^ frame. The sign convention for the azimuthal angle, H, is chosen such that when H is negative, the end vertically furthest from the interface (vertically closest to the S/R) is pointed toward the S/R; this will be called the “upper” end. In the orientation here, the end vertically closest to the interface, the “lower” end, is pointed toward the S/R and thus the cylinder is at a positive azimuth.
see Ref. 2). Readers interested in an additional discussion of these issues are referred to Appendix C. The acoustic template was created from the spectrum of the backscattered data at each azimuth. Calibration to absolute target strength was performed using the method outlined in Appendix A. Results for four different tilt angles are shown in Fig. 4. The azimuthal angle H ¼ 0 corresponds to
FIG. 5. Side view. Example of multipath structure and notation for an untilted, proud cylinder. This corresponds to a slice along the e^02 e^03 plane for H ¼ 0 and ht ¼ 0 (see Fig. 3). Path 1 is a reversible path involving direct reflection from the cylinder. Paths 2 and 3 are reciprocal and involve a single reflection from the interface. Path 4 is a reversible path and includes two interface reflections.
broadside incidence, H ¼ 90 indicates the cylinder end farthest from the interface and nearest the source/receiver, and for H ¼ þ90 the opposite. Frequencies in Fig. 4 are given in ka where the wave number k ¼ 2pf =c is determined by the frequency f and soundspeed (c ¼ 1484 m/s), while a ¼ 12:70 mm is the cylinder radius. In the first case, ht ¼ 0 , there exists a 90 symmetry as seen in Fig. 4(a). The acoustic color is dominated by the bright features at broadside and at end-on. The broadside feature is a superposition of multiple ray paths involving reflections from the interface and will be addressed in more detail in Sec. III. Figure 5 shows this multipath structure in
FIG. 4. (Color online) Results from Washington State University experiment in calibrated target strength (dB), see Appendix A, for the 2 in. 1 in. cylinder at various tilt angles ht . The beam grazing angle hg ¼ 19:5 . Azimuthal angles H ¼ 90 ; 0 ; þ90 , respectively, correspond to the end furthest from the interface (nearest the source), broadside, and the end closest to the interface (farthest from the source). The predicted broadside splitting angle /, calculated from Eq. (1), is given. (a) ht ¼ 0 , / ¼ 0 , (b) ht ¼ 10 , / ¼ 3:6 , (c) ht ¼ 20 , / ¼ 7:4 , (d) ht ¼ 34 , / ¼ 13:8 . The data shows close agreement with these predicted values. 472
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the untilted case. End-on enhancements correspond to a corner reflector made by the interface and the cylinder end. Scattering at end-on will be further examined in Sec. IV. At intermediate azimuths, backscattering is dominated by elastic features, such as surface waves circumnavigating the cylinder,5 running along the meridian,6 or reverberating across the cylinder end.7 As the cylinder is tilted to ht ¼ 10 , Fig. 4(b), a number of modifications to the acoustic color occur. Notably, the broadside feature has split so that enhancements occur at three different azimuths near H ¼ 0 . This splitting is the primary feature of interest in this paper. In addition, the elastic structures at intermediate azimuths also split as the symmetry is broken and additional coupling angles become available. Finally, the acoustic template lacks the previously bright features at either end (H ¼ 690 ). In the third case, Fig. 4(c), the tilt angle of ht ¼ 20 is approximately equal to the grazing angle hg of the incident beam. This contributes to increased backscatter at both ends; from direct reflection from the cylinder end at H ¼ 90 and a reversible path involving two interface reflections at H ¼ þ90 . The angular splitting of the broadside feature also increases. Finally, at the tilt angle ht ¼ 34 , Fig. 4(d), there is minimal backscatter at the H ¼ 90 cylinder end. Multiple scattering (see Sec. IV) leads to enhancements at the H ¼þ90 end. The broadside splitting has continued to increase as tilt is increased. III. BROADSIDE MULTIPATH STRUCTURE
Solid elastic finite cylinders in the absence of other boundaries (free field) have been previously examined.1,2,8,9 For the solid aluminum cylinder contributions to broadside backscattering by the specular reflection and by a circumferentially traveling leaky Rayleigh wave are seen. The interference of these two mechanisms dominate the acoustic spectrum at broadside. The inclusion of a flat reflecting interface allows several additional backscattering ray paths at broadside, as in Fig. 5. Path 1 is a reversible direct path that does not require the interface. Paths 2 and 3 are reciprocal paths; they each involve a single reflection from the interface and from the cylinder, with the order of interactions being the difference between the two. Path 2 is the path that reflects first from the cylinder while path 3 is the path that initially reflects from the interface. Path 4 is reversible and involves two interface reflections. In the untilted case all three paths occur at H ¼ 0 . Interference between the paths, caused by different propagation lengths and the reflection coefficient of the interface, determines the dominant features in the broadside backscattered spectrum. For an overview of scattering from solid cylinders laying horizontal on a sand sediment see Ref. 1. The reversible paths 1 and 4 are monostatic paths while paths 2 and 3 are bistatic paths. When experiments are performed on a cylinder with its axis is tilted vertically relative to the interface, an angular splitting in the multipath structure was observed. The paths instead occur at H ¼ ½0; 6/, where / will be referred to as J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
the splitting angle. A geometrical analysis, given in Appendix B, shows that / ¼ sin1 ðtan hg tan ht Þ:
(1)
Figure 4 gives the predicted values for / at each tilt angle ht used in the experiment. Path 1 moves to a positive azimuth (with the cylinder end closest to the interface pointed toward the S/R) while path 4 moves to a negative azimuth. Paths 2 and 3 remain at 0 azimuth, a consequence of the right angle (corner) reflector formed by the interface and the cylinder regardless of tilt. Predicted values for this splitting angle show close agreement with the experimental results. IV. SCATTERING AT END-ON
For solid cylinders in the freefield, peak backscatter occurs when the incident beam is normal to the cylinder end (parallel with the axis). The inclusion of a reflecting boundary at a right angle to the cylinder end forms a corner reflector; increased backscatter at high frequency will occur at both end-on azimuths regardless of the beam grazing angle. This is shown for the untilted case in Fig. 4(a). Tilting the cylinder breaks the symmetry required for corner reflection; comparison of the untilted case with the results for ht ¼ 10 , Fig. 4(b), shows that small tilt angles can drastically reduce the amount of sound backscattered at either end. However, enhancements were found to occur at end-on for certain combinations of beam grazing angle and cylinder tilt angle. As previously discussed in Sec. II one example is the case shown in Fig. 4(c), where the grazing and tilt angles are nearly equal and thus both ends strongly backscatter. Special attention should be paid to the observations at end-on shown in Fig. 4(d); strong backscattering occurs at one end but not the other. Geometric analysis, Fig. 6, provides a plausible mechanism; multiple scattering where the incident ray interacts twice with the flat end of the cylinder before returning to the receiver. This is equivalent to multiple scattering from the image cylinder formed by the interface.10 The condition for this particular multiple scattering mechanism is
FIG. 6. Multiple scattering geometry at cylinder end, corresponding to a slice along the e^02 e^03 plane for H ¼ 90 (see Fig. 3). An incident beam at grazing angle hg reflects from the end of a cylinder tilted at ht and is then normally incident on the interface, which reverses the raypath. The enhancement due to this mechanism is visible in Fig. 8 where hg ¼ 19:5 and ht ¼ 34 . Plotnick et al.: Proud tilted cylinder backscattering
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p m/w hg ¼ : 2
(3)
Examination shows that these paths may only be observed under the condition p=2m /w p=m. Similarly, a reversible raypath reflecting initially from the infinite edge [Fig. 7(b)] requires, FIG. 7. The wedge formed by the cylinder end and the interface allows multiple scattering mechanisms (multiple interactions with the target); the dashed lines are reversible acoustic raypaths. For the tilted cylinder case the wedge angle is /w ¼ p=2 ht and the grazing angle for the incident beam is hg . Both the raypath in (a) and in (b) involve three distinct reflection points (m ¼ 3); (a) has an initial reflection from the finite edge (cylinder) and obeys Eq. (3) while the raypath in (b) initially reflects from the infinite edge (interface) and obeys Eq. (4). An m ¼ 2 mechanism is shown in Fig. 6.
p 2ht þ hg ¼ ; 2
(2)
and observed in the scaled WSU experiment using ht ¼ 34 and hg ¼ 19:5 , Fig. 4(d). At large vertical tilt angles, additional multiple scattering raypaths become possible. Figure 7 shows two examples of these multiple scattering mechanisms. These occur when the raypath is reversible; the last segment of the raypath must be normal to either the interface or the cylinder end. For simplicity, consider the cylinder end and interface to form a 2-D wedge with one infinite edge representing the interface and a finite edge representing the cylinder end. The angle of the wedge is /w ¼ p=2 ht . The first ray reflection entering the wedge may be off of either of these edges. The integer index m counts the number of distinct reflections within the wedge, the case shown in Fig. 6 has m ¼ 2 while both cases shown in Fig. 7 are m ¼ 3 raypaths. Reversible raypaths with their initial reflection from the finite edge, Fig. 7(a), meet the grazing angle condition,
p ðm 1Þ/w þ hg ¼ : 2
(4)
These paths are possible only when /w p=2ðm 1Þ. As discussed below, identifying situations where multiple scattering may occur is important when comparing experimental data to certain numerical or physical models. V. NUMERICAL MODELS
Finite element methods (FEM) allow sound scattering to be examined in complex target and environment configurations and then compared to experimental results. The results will be referred to as a finite element (FE) model. FEM can be computationally expensive, so techniques have been developed that speed processing time at the cost of imposing symmetry based approximations.11,12 These approximations may cause features such as the multiple scattering at end-on to be missing in the resulting predicted acoustic color. One such example is given in Fig. 8, which compares experimental data to results computed using a hybrid FE/propagation model. This particular model uses a Helmholtz integral to handle the propagation of sound to/from the cylinder and FEM to handle the interaction of sound with the cylinder.13 While the broadside splitting is captured by the model, it significantly underestimates the scattering at the H ¼ 90 end. The FE/propagation model does not include multiple scattering as was also the case for a similar hybrid 2D/3D code
FIG. 8. (Color online) (bottom) Acoustic template from WSU experiment for ht ¼ 34 re-plotted from Fig. 4(d) with a dynamic range of 40 dB. Frequency and amplitude have been scaled to the target strength of a 2 ft 1 ft cylinder. (top) Results of a simplified FE/propagation model for a 2 ft 1 ft cylinder in the same geometry and plotted with the same dynamic range. Note that the model underestimates the backscattering at H ¼ þ90 (black arrows) and near 40 (white arrows). This is due to approximations in the model that do not account for multiple scattering. The mechanism responsible for the enhancement at þ90 is shown in Fig. 6.
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FIG. 9. (Color online) Numerical models of near broadside scattering from the 2 ft 1 ft solid aluminum cylinder for the case hg ¼ ht ¼ 20 . Results are in target strength (dB). The splitting angle between the three broadside features agrees with that predicted by Eq. (1). (a) Finite element model of the tilted cylinder. (b) Physical acoustics model of the cylinder as derived in Appendix C.
used in Ref. 1. As discussed in Sec. IV, multiple scattering at end on is the likely cause of the scattering enhancement, specifically the mechanism shown in Fig. 6. The model shows general agreement with experimental data at intermediate angles although it also underestimates scattering near H ¼ 40 due to a multiple scattering feature. A physical acoustics model was developed that includes the effects of tilt, finite cylinder length, and the elastic circumnavigating Rayleigh wave. The reflection coefficient of the air-water interface is taken to be Rs ¼ 1 and an angular spreading factor is introduced. Figure 9 compares the physical acoustics model and FE model near broadside for hg ¼ ht ¼ 20 . The dominant features show general agreement. Details of this model may be seen in Appendix C.
VI. SUMMARY AND DISCUSSION
The acoustic template of an object is a useful tool for understanding the physical mechanisms for scattering as well as for target classification in the field. The inclusion of a flat reflecting boundary near the object gives rise to a multipath structure that strongly modifies the template. These paths in turn allow for backscattering enhancements at additional azimuths as compared to the free field case. The backscattering peak at broadside becomes split in azimuth for cylinders tilted vertically relative to the interface. This splitting can be understood by geometrically rotating incident acoustic raypaths into the cylinder coordinate system and then applying certain constraints. The results predict a triplet of broadside paths separated by a splitting angle /, a finding confirmed by experiment. When the vertical tilt angle is small the splitting angle increases with tilt. Equation (1) provides a relationship between the splitting angle and the tilt and grazing angles. Backscattering enhancements at end-on are also modified depending on grazing and tilt angles. Certain combinations allow increased target response; this was seen in the scaled experiment. J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
This discussion applied to a solid cylinder; however, the geometric analysis is also applicable to tilted cylindrical shells and may have application to complex shapes. The splitting phenomenon reported here may be useful for inferring the target orientation from scattering data. Potential future work on this subject includes modifications to the finite element modeling to include multiple scattering, an examination of elastic wave coupling conditions, and discussion of the multipath structure as seen in sonar images constructed from the data in this paper. ACKNOWLEDGMENTS
Work supported by ONR. APPENDIX A: ABSOLUTE TARGET STRENGTH CALIBRATION
This appendix outlines the process used to convert experimental data to absolute target strength. For a sufficiently large solid sphere the specular backscattering strength is independent of frequency and angle and is thus an ideal calibration target.14 A solid sphere was placed in front of the source/receiver and insonified with the same signal used in the cylinder experiment. The sphere is large enough that the initial specular echo is completely distinguishable from later elastic echoes. The spectrum Vsph ðxÞ of the specular response is then used to calibrate the cylinder data using " # asph Vcyl ðxÞ A ; (A1) TS ¼ 20 log10 aref Vsph ðxÞ 2 where Vcyl ðxÞ is the cylinder spectrum, aref ¼ 1 m is a standard reference radius, asph is the radius of the sphere used, and the factor A is given by A ¼ Rsph
rcyl rsph
2
:
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(A2)
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Here measurement distances to the sphere and cylinder are, respectively, rsph and rcyl . This accounts for spherical spreading and the reflection coefficient Rsph of the sphere in water calculated in the limit kasph 1. Comparison between cylinders of different sizes is possible by scaling the frequency and target strength of the resulting acoustic color; let a1 and a2 be the radii of two cylinders with identical length to diameter ratios, the frequency f2 is the scaled version of f1 , and the target strength TS2 the scaled version of TS1 . Frequency data obeys a linear scaling rule; f2 ¼ f1 a1 =a2 . Target strength may be converted using; TS2 ¼ TS1 þ 20 log10 ða2 =a1 Þ. These conversions have been applied to the experimental data taken on the 1 in. 2 in. cylinder to allow comparison to the 1 ft 2 ft cylinder used in the FEM model in Fig. 8. APPENDIX B: DERIVATION OF BROADSIDE MULTIPATH ANGLES
This appendix outlines the derivation of the azimuthal angles of broadside multipath rays that leads to the splitting in Eq. (1). A plane wave is incident on a cylinder located at the origin with arbitrary orientation. We define two sets of Cartesian coordinates centered on the cylinder. One set, f^ e 01 ; e^02 ; e^03 g, defines the laboratory frame, where the superscript indicates vectors measured in the lab frame. The second set of coordinates defines the cylinder frame, f^ e 01 ; e^02 ; e^03 g; which is locked to the cylinder and allowed to rotate freely with respect to the lab frame (see Fig. 3). Primed vectors are measured in the cylinder frame. In the laboratory frame, the direction of the incident wave vector, 0 0 k~ ¼ kk^ , is 0 1 0 k^1 B C 0 B C k^ ¼ B k^02 C: (B1) @ A 0 k^ 3
The wave vector is then transformed from the lab frame to 0 the cylinder frame k^ using a series of rotations via the Euler angles a; b, and c. Figure 10 includes the geometry and coordinate system used15 for this operation; the cylinder and lab frame coordinate systems and source/receiver geometry
are the same as used in Fig. 3. In this formulation the line of nodes corresponds to the vector projection ~ p of the cylinder axis on the interface in Fig. 3. The general form of the rotation is 0 0 k^ ¼ Rða; b; cÞk^ :
(B2)
The rotation matrix is given by16 2 3 cccbca scsa cccbsa þ scca ccsb 6 7 7 Rða; b; cÞ ¼ 6 4 sccbca ccsa sccbsa þ ccca scsb 5; sbca sbsa cb (B3) where for brevity cx cosðxÞ and sx sinðxÞ. For the formulation in this paper, a ¼ 90 þ H, b ¼ 90 , and c ¼ ht . The path 1 ray does not involve any interactions with the interface. The incident wave vector, Fig. 10(b), is given by 0 1 0 0 (B4) k^1 ¼ @ cos hg A: sin hg Similarly for path 4 the wave vector is given by 0 1 0 0 B C k^4 ¼ @ cos hg A: sin hg
(B5)
Substituting either of these wave vectors and our above definitions for a, b, and c into (B2) places that wave vector into cylinder coordinates: 0 1 þcos hg sin ht sin H6sin hg cos ht 0 B C (B6) k^1; 4 ¼ @ þcos hg cos ht sin H7 sin hg sin ht A: cos hg cos H Paths 1 and 4 are reversible; the incident ray must be parallel to the cylinder surface normal at the point of reflection. For the cylinder, excluding the ends, the surface normal is always perpendicular to the cylinder axis and thus the incident wave
FIG. 10. (a) Euler angle geometry. Three successive rotations allow a conversion from lab to cylinder coordinates. The Euler angle a ¼ 90 þ H (the azimuthal angle), b ¼ 90 , and c ¼ ht (the tilt angle). See also Fig. 3. (b) Geometry for incident sound beam in the lab frame with grazing angle hg .
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must have no projection along the cylinder axis, the e^02 direction in Fig. 10(a). The azimuthal angles H1; 4 are thus the solutions to cos hg cos ht sin H1; 4 7sin hg sin ht ¼ 0 and H1; 4 ¼ 6sin1 ðtan hg tan ht Þ:
(B7)
For brevity the splitting angle is defined as / sin1 ðtan hg tan ht Þ in Eq. (1). Paths 2 and 3 are not reversible paths but are a recipro0þ cal pair. The incident wave vector k^2; 3 and reflected wave 0 vector k^2; 3 are 0 1 0 1 0 0 0þ 0 (B8) k^2; 3 ¼ @ cos hg A; k^2; 3 ¼ @ cos hg A; 7 sin hg 7 sin hg 0 where the sign dependence in the k^z component accounts for path 2 and 3, respectively. Examination in the cylinder frame 0þ 0 shows that upon reflection from the cylinder k^x ¼ k^x . After converting both incident and reflected wave vectors into cylinder coordinates, the azimuthal angle H2; 3 is the solution to
cos hg sin ht sin H þ sin hg cos ht ¼ þcos hg sin ht sin H þ sin hg cos ht :
(B9)
Thus, H2; 3 ¼ 0:
FIG. 11. (Not to scale) Geometry used in physical acoustics model for farfield scattering (see Appendix C). The coordinate system f x ; yg is used to perform the necessary integrations. The origin is taken to be the center of the cylinder and midway along its axis. The cylinder has length L, radius a. The distance D L; a. (a) Top view, source/receiver (S/R) is located along the scan line at position x at a farfield distance D from the cylinder axis, thus making azimuthal angle H with respect to the cylinder broadside. (b) Side view, cylinder viewed from the right side of (a). Acoustic beam incident on cylinder from S/R at a height of D tan hg measured from the scanning line to the interface. The vertical distance hð x Þ is the vertical distance measured from the scanning line to the cylinder axis. (c) Front view, cylinder tilted at ht . The tilt angle, radius, and length determine the height of origin above the interface; height of the cylinder axis measured to the origin is a function of x and tilt angle.
(B10)
Regardless of tilt and grazing angle, the reciprocal path 2,3 remains at 0 azimuth. This is understandable as a corner (right angle) reflector; there will always exist a locus of points along the meridian of the cylinder whose surface normal is perpendicular to the interface normal. This may be visualized using paths 2 and 3 in Fig. 5. The end result is a triplet of backscattering peaks occurring at H ¼ 0; 6/. APPENDIX C: PHYSICAL ACOUSTICS MODEL
corresponding to the raypaths shown in Fig. 5. The form functions fS; p and fl; p are in a hybrid 2D/3D form to aid in their evaluation. The leading term in (C2) has been extracted from the full 3D versions of these form functions. Combining specular and elastic contributions in this situation extends Stanton’s related approach.8 The prefactor in (C2) is explained below. Some care must be taken in the interpretation of these hybrid form factors. Section 2 of Ref. 17 has been utilized to convert the required integrals into a more compact form. The scattering form function f for a rigid cylinder of length L may be used to extract the geometric dependence: ik XY; pa ð L=2 X¼ exp½2ikrðx; xÞd x;
f
This appendix will examine a physical acoustics model which includes the scattering geometry and elastic mechanisms for the tilted cylinder. The geometry used to develop the physical acoustics model is shown in Fig. 11. A plane wave of amplitude pi is incident on the cylinder. The far field scattered pressure is the real part of17 afcyl eiðkrxtÞ ; 2 r
(C1)
where fcyl is the backscattering form function of a cylinder of radius a. For large values of ka the finite cylinder form function may be expressed as a sum over available raypaths p, 2 1=2 X ip=4 2 kaL (C2) fcyl e ðfS; p þ fl; p Þ: a 4p p; l Here, the subscript S denotes the specular contribution and the index l refers to the lth leaky wave elastic contribution. In this treatment, only the circumferential Rayleigh waves l ¼ R will be included. The path index p runs from 1 to 4, J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
(C4)
L=2
Y¼
ða a
pscat ¼ pi
(C3)
1=2 2 pa exp ik y =a d y eip=4 ; k
(C5)
where the stationary phase approximation was used in (C5). For the plane of integration considered here the backscattering obliquity factor used in Eq. (73) of Ref. 17 is taken as unity for all raypaths. The integral in (C5) depends on the curvature of the relevant reflected wavefronts, as explained in Ref. 17. For paths 1 and 4 the curvature is given by geometrical considerations explained in Ref. 17, which yield the results in (C5). For paths 2 and 3 the off-meridional dependence of the curvature is small for the parameters considered in Fig. 9(b) and that dependence has been neglected in the approximation in (C5). The ik=pa and Y terms have been combined to give the prefactor in (C2). The remaining integral is dependent on the distance rðx; xÞ between the source/ Plotnick et al.: Proud tilted cylinder backscattering
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receiver at a location along the scanning line x, and each point along the cylinder axis x. Thus, each specular and leaky wave form function requires evaluating X for that path. The multipath geometries may be displayed using the method of images. In the rigid cylinder, single path, zero tilt limit the total form function goes over to that predicted in Refs. 17 and 18. 1. Specular contributions
The reflection coefficient Ral ð90 Þ will be discussed below, and the 1=L factor is due to the finite cylinder factor in (C2). The distance rðx; xÞ from a point x in the scanning line to a point x on the cylinder axis is 2 1=2
xÞ rðx; xÞ ¼ ½D þ ðx x cos ht Þ þ h1 ð
;
(C7)
where the horizontal farfield distance from the scanning line x Þ is the vertical distance to the cylinder axis is D and h1 ð from the scanning line to the axis. Here x Þ ¼ D tanðhg Þ ½ðL=2Þ sin ht þ a cos ht x sin ht : h1 ð (C8) In the farfield limit17 only terms linear in x are kept. Evaluating the integral and taking the limit D L; a it can be shown that fS; 1 Ral ð90 Þe2ik½Dwatan hg ððL=2Þsin ht þa cos ht Þ=w ! kL cos ht tan hg tan ht sin H ; sinc w where w ¼
(C12) Here, distances rþ ðx; xÞ and r ðx; xÞ are measured from the scanning line to the reflection point along the cylinder meridian and denote rays from the physical source and an image source, respectively,
(C6)
L=2
2
L=2
r6 ðx; xÞ ¼ ½ðD aÞ2 þ ðx x cos ht Þ2 þ h6 ð x Þ2 1=2 ; (C13)
For path 1 the specular form function fS; 1 is ð L=2 i2ka ei2krðx; xÞ d x =L: fS; 1 ¼ Ral ð90 Þe
2
fS; 2þ3
ð L=2 ¼ 2Ral ð90 hg ÞRs ðhg Þ eik½rþ ðx; xÞþr ðx; xÞ d x =L:
(C9)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ tan2 hg þ sin2 H. The leading term gives
the phase of the specular reflection while the directional sinc term is centered on tan hg tan ht sin H ¼ 0, as expected from the results in Appendix B. Similarly, Path 4 requires an integral ð L=2 2 i2ka ei2krðx; xÞ d x =L; (C10) fS; 4 ¼ Ral ð90 ÞRs ðhg Þe
where h6 ð x Þ ¼ D tanðhg Þ6½L=2 sin ht þ a cos ht þ x sin ht . The integration leads to a sinc function with argument kL cos ht sin H=w. 2. Elastic contributions
This paragraph will address the contribution from leaky traveling waves. Launching and detachment of the lth surface wave occurs via a velocity matching condition where cw is the speed of sound in the medium, cl is the leaky wave velocity, and hl is the angle between the launching ray and local surface normal, sin hl ¼ cw =cl :
While this condition allows for several different categories of elastic waves on the cylinder, attention will be restricted to circumferentially traveling Rayleigh waves l ¼ R. Geometric evaluation of the Rayleigh wave component of the form factor for each path gives analogous results to those for specular scattering in Appendix C 1. The aluminum reflection coefficient Ral is replaced with a product of the complex coupling coefficient GR and complex factors that account for radiation damping and propagation. The coupling coefficient is approximated as19,20 GR 8pbR ðpkaÞ1=2 expðip=4Þ;
PR ðhp Þ ¼ GR
x Þ ¼ D tanðhg Þ þ ½ðL=2Þ sin ht þ a cos ht x sin ht : h4 ð (C11) This results in a phase term that is shifted relative to path 1, and a directional sinc function centered on tan hg tan ht þsin H ¼ 0, again as expected from previous results. Reciprocity of paths 2 and 3 may be used to combine their form functions, requiring an integral 478
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(C15)
where bR is the radiation damping parameter for the leaky Rayleigh wave. Rayleigh waves couple onto the cylinder, propagate circumferentially in both clockwise and counterclockwise directions, and re-radiate to the receiver. The combined coupling and propagation for each type of path is19
L=2
where Rs ðhg Þ is the interface reflection coefficient discussed below. The evaluation is identical to path 1 except that the x Þ is to an image source and thus vertical distance h4 ð
(C14)
exp½2ðp hp ÞbR þ igR ðhp Þ ; 1 exp½2pbR þ i2pkacw =cR
(C16)
where gR ðhp Þ ¼ kað2p 2hp Þcw =cR 2ka cos hR . The term hp is a path dependent coupling angle that accounts for different coupling conditions for monostatic paths 1 and 4 and bistatic paths 2 and 3. The Rayleigh wave velocity cR and the radiation damping coefficient bR were evaluated using methods outlined in Ref. 21. Any effect of the close proximity of the interface on the local values of bR and cR have been neglected. For paths 1 and 4, the coupling angles are h1; 4 ¼ hR , where hR is the Rayleigh coupling angle meeting the condition in (C14). Unlike paths 1 and 4, however, the Plotnick et al.: Proud tilted cylinder backscattering
projection of the incident ray from reciprocal paths 2 and 3 onto the cylinder includes an axial component, so that these leaky waves are not purely circumferential. The affiliated helical wave aspects of the response are omitted from consideration because, for the situation emphasized here in Fig. 9, the associated complications to the theory, e.g., see Ref. 22, do not appear to be essential. Paths 2 and 3 are bistatic, and the clockwise (CW) and counterclockwise (CCW) traveling waves have different coupling conditions. For path 3, which reflects first from the interface and then couples onto the cylinder, the associated coupling angles are h3; cw ¼ ðhR þ hg Þ and h3; ccw ¼ ðhR hg Þ. Path 2 is analogous
except clockwise and counterclockwise are reversed so that h2; ccw ¼ ðhR þ hg Þ and h2; cw ¼ ðhR hg Þ. This asymmetry introduces the factor of 1=2 into the elastic contribution of paths 2 and 3 in (C17).
3. Total model
Spherical spreading is incorporated by expressing the scattered pressure in target strength, referenced to 1 m. Based upon (C1) and (C2) the target strength can be expressed as TS ¼ 20 log10 jFj. For the physical acoustics model above,
1=2 n kaL2 e2ikDw P1 S1 Ral ð90 Þ þ PR ðhR Þe2ika þ 2Rs hg P23 S23 F¼ 4p
2ika 1 2ika cos hg 2 Ral 90 hg þ PR hR þ hg þ PR hR hg e þ Rs hg P4 S4 Ral ð90 Þ þ PR ðhR Þ e : 2
The above expression has been separated into phase and coupling factors P, directional coefficients S, and effective reflection coefficients, where P1 ¼ exp f2ik½a ðL=2 sin ht þ a cos ht Þ tan hg =wg; (C18a) P23 ¼ exp f2ik½a cos hg g;
(C18b)
P4 ¼ exp f2ik½a þ ðL=2 sin ht þ a cos ht Þ tan hg =wg; (C18c) S1 ¼ sincfkL cos ht ðtan ht tan hg sin HÞ=wg;
(C18d)
S23 ¼ sincfkL cos ht sin H=wg;
(C18e)
S4 ¼ sincfkL cos ht ðtan ht tan hg þ sin HÞ=wg;
(C18f)
and PR ðhp Þ is given in (C16). As before w ¼ ð1 þ tan2 hg þ sin2 HÞ1=2 . Figure 9(b) shows the resulting acoustic color predicted by the above model for the 2 ft 1 ft aluminum cylinder. This model captures several of the dominant features in the resulting acoustic color that are seen in both the FE model [Fig. 9(a)] and the scaled 2 in. 1 in. experimental results [Fig. 4(c)] at a comparable tilt and grazing angle. Reflection coefficients for aluminum, Ral , at both normal and grazing incidence were obtained using the method outlined in Ref. 23, Appendix A. The experiment presented in this paper utilized a free surface, and thus, the interface reflection coefficient was taken to be Rs ¼ 1. For real world cases, such as scattering from a sediment bottom, the complex reflection coefficient for that material and grazing angle must be substituted. Several approximations were utilized in this model, including a far field assumption. The suppression of the obliquity factor in (C2) and the helical nature of the elastic J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
(C17)
waves for paths 2 and 3 means that the path 2 and 3 terms are most sensitive to these approximations. Fully treated helical rays22 and meridional rays5–7 have not been included in this model, nor has multiple scattering in which a wave may decouple from the cylinder, reflect off of the interface, and couple back on. Nevertheless, this physical acoustics model captures the dominant features near broadside. 1
K. L. Williams, S. G. Kargl, E. I. Thorsos, D. S. Burnett, J. L. Lopes, M. Zampolli, and P. L. Marston, “Acoustic scattering from a solid aluminum cylinder in contact with a sand sediment: Measurements, modeling, and interpretation,” J. Acoust. Soc. Am. 127, 3356–3371 (2010). 2 J. R. LaFollett, K. L. Williams, and P. L. Marston, “Boundary effects on backscattering by a solid aluminum cylinder: Experiment and finite element model comparisons (L),” J. Acoust. Soc. Am. 130, 669–672 (2011). 3 R. Lim, K. L. Williams, and E. I. Thorsos, “Acoustic scattering by a threedimensional elastic object near a rough surface,” J. Acoust. Soc. Am. 107, 1246–1262 (2000). 4 J. L. Kennedy, T. M. Marston, K. Lee, J. L. Lopes, and R. Lim, “A rail system for circular synthetic aperture sonar imaging and acoustic target strength measurements: Design/operation/preliminary results,” Rev. Sci. Instrum. 85, 014901 (2014). 5 P. L. Marston, “Approximate meridional leaky ray amplitudes for tilted cylinders: End-backscattering enhancements and comparisons with exact theory for infinite solid cylinders,” J. Acoust. Soc. Am. 102, 358–369 (1997). 6 K. Gipson and P. L. Marston, “Backscattering enhancements due to reflection of meridional leaky Rayleigh waves at the blunt truncation of a tilted solid cylinder in water: Observations and theory,” J. Acoust. Soc. Am. 106, 1673–1680 (1999). 7 K. Gipson and P. L. Marston, “Backscattering enhancements from Rayleigh waves on the flat face of a tilted solid cylinder in water,” J. Acoust. Soc. Am. 107, 112–117 (2000). 8 T. K. Stanton, “Sound scattering by cylinders of finite length. II. elastic cylinders,” J. Acoust. Soc. Am. 83, 64–67 (1988). 9 X.-L. Bao, “Echoes and helical surface waves on a finite elastic cylinder excited by sound pulses in water,” J. Acoust. Soc. Am. 94, 1461–1466 (1993). 10 G. C. Gaunaurd and H. Huang, “Acoustic scattering by a spherical body near a plane boundary,” J. Acoust. Soc. Am. 96, 2526–2536 (1994). 11 M. Zampolli, A. Tesei, F. B. Jensen, N. Malm, and J. B. Blottman, “A computationally efficient finite element model with perfectly matched Plotnick et al.: Proud tilted cylinder backscattering
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layers applied to scattering from axially symmetric objects,” J. Acoust. Soc. Am. 122, 1472–1485 (2007). 12 M. Zampolli, A. L. Espa~ na, K. L. Williams, S. G. Kargl, E. I. Thorsos, J. L. Lopes, J. L. Kennedy, and P. L. Marston, “Low- to mid-frequency scattering from elastic objects on a sand sea floor: Simulation of frequency and aspect angle dependant structural echoes,” J. Comp. Acoust. 20, 1240007 (2012). 13 M. Nijhof, A. L. Espa~ na, and K. L. Williams, “Efficient calculation of broadband acoustic scattering from a partially, obliquely buried cylinder,” in Proceedings of the 1st International Conference Exhibition Underwater Acoustics (2013), pp. 697–702. 14 L. R. Dragonette, S. K. Numrich, and L. J. Frank, “Calibration technique for acoustic scattering measurements,” J. Acoust. Soc. Am. 69, 1186–1189 (1981). 15 A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua (Dover, Mineola, NY, 2003), pp. 158–161. 16 J. D. Pendleton, “Euler angle geometry, helicity basis vectors, and the Wigner D-function addition theorem,” Am. J. Phys. 71, 1280–1291 (2003).
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P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992). 18 G. C. Gaunaurd, “Sonar cross sections of bodies partially insonified by finite sound beams,” IEEE J. Oceanic Eng. 10, 213–230 (1985). 19 N. H. Sun and P. L. Marston, “Ray synthesis of leaky Lamb wave contributions to backscattering from thick cylindrical shells,” J. Acoust. Soc. Am. 91, 1398–1402 (1992). 20 P. L. Marston, “Leaky waves on weakly curved scatterers. II. Convolution formulation for two-dimensional high-frequency scattering,” J. Acoust. Soc. Am. 97, 34–41 (1995). 21 € R. D. Doolittle, H. Uberall, and P. Ugin^ıcius, “Sound scattering by elastic cylinders,” J. Acoust. Soc. Am. 43, 1–14 (1968). 22 F. J. Blonigen and P. L. Marston, “Leaky helical flexural wave backscattering contributions from tilted cylindrical shells in water: Observations and modeling,” J. Acoust. Soc. Am. 112, 528–536 (2002). 23 K. L. Williams and P. L. Marston, “Mixed-mode acoustical glory scattering from a large elastic sphere: Model and experimental verification,” J. Acoust. Soc. Am. 76, 1555–1563 (1984).
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Citation: The Journal of the Acoustical Society of America 137, 470 (2015); doi: 10.1121/1.4904490 View online: http://dx.doi.org/10.1121/1.4904490 View Table of Contents: http://asa.scitation.org/toc/jas/137/1 Published by the Acoustical Society of America
Articles you may be interested in High frequency imaging and elastic effects for a solid cylinder with axis oblique relative to a nearby horizontal surface The Journal of the Acoustical Society of America 140, (2016); 10.1121/1.4961001
High frequency backscattering by a solid cylinder with axis tilted relative to a nearby horizontal surface Daniel S. Plotnicka) and Philip L. Marston Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814
~a Kevin L. Williams and Aubrey L. Espan Applied Physics Laboratory, Acoustics Department, University of Washington, Seattle, Washington 98105
(Received 9 April 2014; revised 16 September 2014; accepted 4 November 2014) The backscattering spectrum versus azimuthal angle, also called the “acoustic color” or “acoustic template,” of solid cylinders located in the free water column have been previously studied. For cylinders lying proud on horizontal sand sediment, there has been progress in understanding the backscattering spectrum as a function of grazing angle and the viewing angle relative to the cylinder’s axis. Significant changes in the proud backscattering spectrum versus the freefield case are associated with the interference of several multipaths involving the target and the surface. If the cylinder’s axis has a vertical tilt such that one end is partially buried in the sand, the multipath structure is changed, thus modifying the resulting spectrum. Some of the changes in the template can be approximately modeled using a combination of geometrical and physical acoustics. The resulting analysis gives a simple approximation relating certain changes in the template with the vertical tilt of the cylinder. This includes a splitting in the azimuthal angle at which broadside multipath features are observed. A similar approximation also applies to a metallic cylinder adjacent to a flat free C 2015 Acoustical Society of America. surface and was confirmed in tank experiments. V [http://dx.doi.org/10.1121/1.4904490] [OU]
Pages: 470–480
I. INTRODUCTION
Efforts have been made in recent years to understand the acoustic backscattering from objects in the ocean. Acoustic energy may be backscattered by direct acoustic reflections, referred to as specular reflections, and elastic scattering mechanisms. The interference between specular and elastic scattering creates an acoustic spectrum that can provide valuable information about that object’s shape, composition, and orientation. This information can also give insight into the basic elastic scattering mechanisms for different shapes such as solid cylinders and cylindrical shells. In acoustic backscattering the sound source and receiver are collocated. The full spectrum versus the object’s azimuthal angle will be referred to as the “acoustic color” or “acoustic template.” The acoustic color of an object depends not only on its shape and material, but also on its nearby environment. In particular, scattering by objects in contact with the sediment/water interface (often called a “proud” configuration) has received attention.1–4 The acoustic color for a solid cylinder placed in a horizontal proud configuration was previously examined in Refs. 1 and 2. In practice, it is likely that an object will be tilted vertically and partially buried in the seafloor rather than lying perfectly horizontal in a proud configuration. Figure 1 shows just such a tilted configuration for a cylinder. Figure 2 shows the results of an experiment performed on a tilted, partially buried solid cylinder at the Naval Surface Warfare Center, Panama City (NSWC-PCD) during PONDEX09. This cylinder had dimensions a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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60.96 cm 30.48 cm, and for brevity will be referred to as the 2 foot 1 foot cylinder. Tilting the cylinder axis causes changes to the acoustic template. We will show that several of the most prominent changes are understandable using geometric methods. This paper will examine the changes to the acoustic template of a solid cylinder as vertical tilt angle is varied. The principal experimental observations and the resulting models used to predict certain features of those observations are presented in the body of this paper. The method used for calibration of the experiment and details of the models are reserved for the Appendixes; the inquisitive reader is encouraged to make use of these Appendixes, but they are not required for a basic understanding of the observed changes to the acoustic template. Figure 3 provides a 3-D schematic of the geometry used throughout this work. The target is taken to be in the far field of the acoustic source/receiver. The coordinate system f^ e 01 ; e^02 ; e^03 g defines the laboratory frame. The incident wave vector and the interface normal define the plane of incidence; the e^01 e^02 plane is parallel to the interface and the e^02 e^03 plane is perpendicular to the interface and parallel to the plane of incidence. The second set of coordinates defines the cylinder frame, f^ e 01 ; e^02 ; e^03 g; the e^02 direction is defined so as to lie along the cylinder axis. The grazing angle hg is measured between the incident wave vector and the local interface in the plane of incidence. The cylinder tilt angle ht is formed by the cylinder axis and the vector projection ~ p of the cylinder axis on the interface. The azimuthal angle H is measured between ~ p and the ^ e 01 direction; the sign convention for H is chosen such that when H is negative the end vertically furthest from the interface (vertically closest to the
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C 2015 Acoustical Society of America V
FIG. 1. (Left) Schematic of geometry for tilted cylinder. Acoustic beam at grazing angle hg insonifies a proud or partially buried cylinder lying at a tilt angle ht . (Right) The experimental setup used in the small scale laboratory experiment. Two support lines run from the corners of the cylinder to a rotating stage (not pictured). The cylinder is tilted vertically to an arbitrary angle with the upper corner placed near the air-water interface. It is insonified from below. Note that tilting the cylinder shifts its center of mass position relative to the axis of rotation; however, this shift does not affect the acoustic template in the far field.
S/R) is pointed toward the S/R, for positive azimuths the opposite. An experiment on a solid cylinder performed in a freshwater tank is discussed in Sec. II. This section will briefly highlight the observed changes that occur at end-on and broadside incidence. Section III will discuss the backscattering structure at broadside incidence for both the tilted and untilted cases. The scattering structure at end-on will be examined in Sec. IV. Finally, a physical acoustics model for the broadside structure will be introduced and a comparison to a finite element (FE) model will be made in Sec. V. Appendix A addresses the calibration method used in the freshwater tank experiments; Appendix B discusses the simple geometric model discussed in Sec. III; and Appendix C details the physical acoustics model discussed in Sec. V. II. SCALED EXPERIMENT
In order to examine the effect of vertical tilt of the cylinder on the acoustic template a series of backscattering experiments were performed at the water tank facility at Washington State University (WSU). The goal was to provide backscattering data from the 180 of azimuth required to characterize the target. Performing such scans on proud or partially buried targets in sediment is difficult because of practical experimental concerns; these concerns include but are not limited to configuration details (e.g., changes in burial depth, exact tilt angle, and flatness of the local interface) and complications/constraints imposed by the scanning system used (e.g., see Ref. 4). Instead, a solid aluminum cylinder
was suspended from two corners near the air-water interface of a water tank and tilted to the desired angle ht relative to that interface. Vertical tilt was controlled by lengthening or shortening the suspending lines on either corner (see Fig. 1). The cylinder had dimensions 50.80 mm 25.40 mm and for brevity will be referred to as the 2 in. 1 in. cylinder. To avoid undesired wetting effects, the cylinder was fully submerged with the upper corner placed as close to the interface as possible without deforming the free surface. The experiment was performed with a collocated source/receiver transducer. A fixed, wide band transducer insonified the cylinder as it was spun about a vertical axis using a rotating stepper motor system. Backscattering data was collected every 1 of azimuth. The transducer was tilted vertically to a grazing of hg ¼ 19:5 relative to the interface so that the cylinder lay within the beam main lobe. A short duration, wide bandwidth signal (50–450 kHz) was used. The transducer slant range of 2.52 m was sufficient for far field measurements. The primary goal of this experiment was to demonstrate the geometric relationship between cylinder orientation and certain azimuth dependent features; the salient geometry is unaffected by either changes in the interface reflection coefficient or by partial burial. The use of an air-water interface as a stand in for the water-sediment interface does modify this reflection coefficient, and as a consequence of the complete submersion of the cylinder effects due to partial exposure and the possibility of contributions from refracted waves in the sediment could not be studied. Regardless, the air-water interface remains a useful tool for studying boundary effects (e.g.,
FIG. 2. (Color online) (Left) Results of a backscattering experiment performed on a tilted, partially buried 2 ft 1 ft solid aluminum cylinder performed at NSWC-PCD during PONDEX09 where 0 is broadside. The tilt of the cylinder leads to an azimuthal splitting of the broadside multipaths shown in Fig. 5. The splitting, discussed in Sec. III and Fig. 4, is clearly visible between path 1 and paths 2,3. Path 4 is blocked by the partial burial of the cylinder. (Right) A diver near the tilted cylinder, compare with Fig. 1. J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
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FIG. 3. Schematic of coordinate system used in this paper. Acoustic beam from a collocated source/receiver (S/R) at grazing angle hg insonifies a cylinder sitting above an interface at a tilt angle ht . The coordinate set 0 0 0 e 1 ; e^2 ; e^3 g the cylinder f^ e 01 ; e^02 ; e^03 g defines the laboratory frame and f^ frame. The sign convention for the azimuthal angle, H, is chosen such that when H is negative, the end vertically furthest from the interface (vertically closest to the S/R) is pointed toward the S/R; this will be called the “upper” end. In the orientation here, the end vertically closest to the interface, the “lower” end, is pointed toward the S/R and thus the cylinder is at a positive azimuth.
see Ref. 2). Readers interested in an additional discussion of these issues are referred to Appendix C. The acoustic template was created from the spectrum of the backscattered data at each azimuth. Calibration to absolute target strength was performed using the method outlined in Appendix A. Results for four different tilt angles are shown in Fig. 4. The azimuthal angle H ¼ 0 corresponds to
FIG. 5. Side view. Example of multipath structure and notation for an untilted, proud cylinder. This corresponds to a slice along the e^02 e^03 plane for H ¼ 0 and ht ¼ 0 (see Fig. 3). Path 1 is a reversible path involving direct reflection from the cylinder. Paths 2 and 3 are reciprocal and involve a single reflection from the interface. Path 4 is a reversible path and includes two interface reflections.
broadside incidence, H ¼ 90 indicates the cylinder end farthest from the interface and nearest the source/receiver, and for H ¼ þ90 the opposite. Frequencies in Fig. 4 are given in ka where the wave number k ¼ 2pf =c is determined by the frequency f and soundspeed (c ¼ 1484 m/s), while a ¼ 12:70 mm is the cylinder radius. In the first case, ht ¼ 0 , there exists a 90 symmetry as seen in Fig. 4(a). The acoustic color is dominated by the bright features at broadside and at end-on. The broadside feature is a superposition of multiple ray paths involving reflections from the interface and will be addressed in more detail in Sec. III. Figure 5 shows this multipath structure in
FIG. 4. (Color online) Results from Washington State University experiment in calibrated target strength (dB), see Appendix A, for the 2 in. 1 in. cylinder at various tilt angles ht . The beam grazing angle hg ¼ 19:5 . Azimuthal angles H ¼ 90 ; 0 ; þ90 , respectively, correspond to the end furthest from the interface (nearest the source), broadside, and the end closest to the interface (farthest from the source). The predicted broadside splitting angle /, calculated from Eq. (1), is given. (a) ht ¼ 0 , / ¼ 0 , (b) ht ¼ 10 , / ¼ 3:6 , (c) ht ¼ 20 , / ¼ 7:4 , (d) ht ¼ 34 , / ¼ 13:8 . The data shows close agreement with these predicted values. 472
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the untilted case. End-on enhancements correspond to a corner reflector made by the interface and the cylinder end. Scattering at end-on will be further examined in Sec. IV. At intermediate azimuths, backscattering is dominated by elastic features, such as surface waves circumnavigating the cylinder,5 running along the meridian,6 or reverberating across the cylinder end.7 As the cylinder is tilted to ht ¼ 10 , Fig. 4(b), a number of modifications to the acoustic color occur. Notably, the broadside feature has split so that enhancements occur at three different azimuths near H ¼ 0 . This splitting is the primary feature of interest in this paper. In addition, the elastic structures at intermediate azimuths also split as the symmetry is broken and additional coupling angles become available. Finally, the acoustic template lacks the previously bright features at either end (H ¼ 690 ). In the third case, Fig. 4(c), the tilt angle of ht ¼ 20 is approximately equal to the grazing angle hg of the incident beam. This contributes to increased backscatter at both ends; from direct reflection from the cylinder end at H ¼ 90 and a reversible path involving two interface reflections at H ¼ þ90 . The angular splitting of the broadside feature also increases. Finally, at the tilt angle ht ¼ 34 , Fig. 4(d), there is minimal backscatter at the H ¼ 90 cylinder end. Multiple scattering (see Sec. IV) leads to enhancements at the H ¼þ90 end. The broadside splitting has continued to increase as tilt is increased. III. BROADSIDE MULTIPATH STRUCTURE
Solid elastic finite cylinders in the absence of other boundaries (free field) have been previously examined.1,2,8,9 For the solid aluminum cylinder contributions to broadside backscattering by the specular reflection and by a circumferentially traveling leaky Rayleigh wave are seen. The interference of these two mechanisms dominate the acoustic spectrum at broadside. The inclusion of a flat reflecting interface allows several additional backscattering ray paths at broadside, as in Fig. 5. Path 1 is a reversible direct path that does not require the interface. Paths 2 and 3 are reciprocal paths; they each involve a single reflection from the interface and from the cylinder, with the order of interactions being the difference between the two. Path 2 is the path that reflects first from the cylinder while path 3 is the path that initially reflects from the interface. Path 4 is reversible and involves two interface reflections. In the untilted case all three paths occur at H ¼ 0 . Interference between the paths, caused by different propagation lengths and the reflection coefficient of the interface, determines the dominant features in the broadside backscattered spectrum. For an overview of scattering from solid cylinders laying horizontal on a sand sediment see Ref. 1. The reversible paths 1 and 4 are monostatic paths while paths 2 and 3 are bistatic paths. When experiments are performed on a cylinder with its axis is tilted vertically relative to the interface, an angular splitting in the multipath structure was observed. The paths instead occur at H ¼ ½0; 6/, where / will be referred to as J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
the splitting angle. A geometrical analysis, given in Appendix B, shows that / ¼ sin1 ðtan hg tan ht Þ:
(1)
Figure 4 gives the predicted values for / at each tilt angle ht used in the experiment. Path 1 moves to a positive azimuth (with the cylinder end closest to the interface pointed toward the S/R) while path 4 moves to a negative azimuth. Paths 2 and 3 remain at 0 azimuth, a consequence of the right angle (corner) reflector formed by the interface and the cylinder regardless of tilt. Predicted values for this splitting angle show close agreement with the experimental results. IV. SCATTERING AT END-ON
For solid cylinders in the freefield, peak backscatter occurs when the incident beam is normal to the cylinder end (parallel with the axis). The inclusion of a reflecting boundary at a right angle to the cylinder end forms a corner reflector; increased backscatter at high frequency will occur at both end-on azimuths regardless of the beam grazing angle. This is shown for the untilted case in Fig. 4(a). Tilting the cylinder breaks the symmetry required for corner reflection; comparison of the untilted case with the results for ht ¼ 10 , Fig. 4(b), shows that small tilt angles can drastically reduce the amount of sound backscattered at either end. However, enhancements were found to occur at end-on for certain combinations of beam grazing angle and cylinder tilt angle. As previously discussed in Sec. II one example is the case shown in Fig. 4(c), where the grazing and tilt angles are nearly equal and thus both ends strongly backscatter. Special attention should be paid to the observations at end-on shown in Fig. 4(d); strong backscattering occurs at one end but not the other. Geometric analysis, Fig. 6, provides a plausible mechanism; multiple scattering where the incident ray interacts twice with the flat end of the cylinder before returning to the receiver. This is equivalent to multiple scattering from the image cylinder formed by the interface.10 The condition for this particular multiple scattering mechanism is
FIG. 6. Multiple scattering geometry at cylinder end, corresponding to a slice along the e^02 e^03 plane for H ¼ 90 (see Fig. 3). An incident beam at grazing angle hg reflects from the end of a cylinder tilted at ht and is then normally incident on the interface, which reverses the raypath. The enhancement due to this mechanism is visible in Fig. 8 where hg ¼ 19:5 and ht ¼ 34 . Plotnick et al.: Proud tilted cylinder backscattering
473
p m/w hg ¼ : 2
(3)
Examination shows that these paths may only be observed under the condition p=2m /w p=m. Similarly, a reversible raypath reflecting initially from the infinite edge [Fig. 7(b)] requires, FIG. 7. The wedge formed by the cylinder end and the interface allows multiple scattering mechanisms (multiple interactions with the target); the dashed lines are reversible acoustic raypaths. For the tilted cylinder case the wedge angle is /w ¼ p=2 ht and the grazing angle for the incident beam is hg . Both the raypath in (a) and in (b) involve three distinct reflection points (m ¼ 3); (a) has an initial reflection from the finite edge (cylinder) and obeys Eq. (3) while the raypath in (b) initially reflects from the infinite edge (interface) and obeys Eq. (4). An m ¼ 2 mechanism is shown in Fig. 6.
p 2ht þ hg ¼ ; 2
(2)
and observed in the scaled WSU experiment using ht ¼ 34 and hg ¼ 19:5 , Fig. 4(d). At large vertical tilt angles, additional multiple scattering raypaths become possible. Figure 7 shows two examples of these multiple scattering mechanisms. These occur when the raypath is reversible; the last segment of the raypath must be normal to either the interface or the cylinder end. For simplicity, consider the cylinder end and interface to form a 2-D wedge with one infinite edge representing the interface and a finite edge representing the cylinder end. The angle of the wedge is /w ¼ p=2 ht . The first ray reflection entering the wedge may be off of either of these edges. The integer index m counts the number of distinct reflections within the wedge, the case shown in Fig. 6 has m ¼ 2 while both cases shown in Fig. 7 are m ¼ 3 raypaths. Reversible raypaths with their initial reflection from the finite edge, Fig. 7(a), meet the grazing angle condition,
p ðm 1Þ/w þ hg ¼ : 2
(4)
These paths are possible only when /w p=2ðm 1Þ. As discussed below, identifying situations where multiple scattering may occur is important when comparing experimental data to certain numerical or physical models. V. NUMERICAL MODELS
Finite element methods (FEM) allow sound scattering to be examined in complex target and environment configurations and then compared to experimental results. The results will be referred to as a finite element (FE) model. FEM can be computationally expensive, so techniques have been developed that speed processing time at the cost of imposing symmetry based approximations.11,12 These approximations may cause features such as the multiple scattering at end-on to be missing in the resulting predicted acoustic color. One such example is given in Fig. 8, which compares experimental data to results computed using a hybrid FE/propagation model. This particular model uses a Helmholtz integral to handle the propagation of sound to/from the cylinder and FEM to handle the interaction of sound with the cylinder.13 While the broadside splitting is captured by the model, it significantly underestimates the scattering at the H ¼ 90 end. The FE/propagation model does not include multiple scattering as was also the case for a similar hybrid 2D/3D code
FIG. 8. (Color online) (bottom) Acoustic template from WSU experiment for ht ¼ 34 re-plotted from Fig. 4(d) with a dynamic range of 40 dB. Frequency and amplitude have been scaled to the target strength of a 2 ft 1 ft cylinder. (top) Results of a simplified FE/propagation model for a 2 ft 1 ft cylinder in the same geometry and plotted with the same dynamic range. Note that the model underestimates the backscattering at H ¼ þ90 (black arrows) and near 40 (white arrows). This is due to approximations in the model that do not account for multiple scattering. The mechanism responsible for the enhancement at þ90 is shown in Fig. 6.
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FIG. 9. (Color online) Numerical models of near broadside scattering from the 2 ft 1 ft solid aluminum cylinder for the case hg ¼ ht ¼ 20 . Results are in target strength (dB). The splitting angle between the three broadside features agrees with that predicted by Eq. (1). (a) Finite element model of the tilted cylinder. (b) Physical acoustics model of the cylinder as derived in Appendix C.
used in Ref. 1. As discussed in Sec. IV, multiple scattering at end on is the likely cause of the scattering enhancement, specifically the mechanism shown in Fig. 6. The model shows general agreement with experimental data at intermediate angles although it also underestimates scattering near H ¼ 40 due to a multiple scattering feature. A physical acoustics model was developed that includes the effects of tilt, finite cylinder length, and the elastic circumnavigating Rayleigh wave. The reflection coefficient of the air-water interface is taken to be Rs ¼ 1 and an angular spreading factor is introduced. Figure 9 compares the physical acoustics model and FE model near broadside for hg ¼ ht ¼ 20 . The dominant features show general agreement. Details of this model may be seen in Appendix C.
VI. SUMMARY AND DISCUSSION
The acoustic template of an object is a useful tool for understanding the physical mechanisms for scattering as well as for target classification in the field. The inclusion of a flat reflecting boundary near the object gives rise to a multipath structure that strongly modifies the template. These paths in turn allow for backscattering enhancements at additional azimuths as compared to the free field case. The backscattering peak at broadside becomes split in azimuth for cylinders tilted vertically relative to the interface. This splitting can be understood by geometrically rotating incident acoustic raypaths into the cylinder coordinate system and then applying certain constraints. The results predict a triplet of broadside paths separated by a splitting angle /, a finding confirmed by experiment. When the vertical tilt angle is small the splitting angle increases with tilt. Equation (1) provides a relationship between the splitting angle and the tilt and grazing angles. Backscattering enhancements at end-on are also modified depending on grazing and tilt angles. Certain combinations allow increased target response; this was seen in the scaled experiment. J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
This discussion applied to a solid cylinder; however, the geometric analysis is also applicable to tilted cylindrical shells and may have application to complex shapes. The splitting phenomenon reported here may be useful for inferring the target orientation from scattering data. Potential future work on this subject includes modifications to the finite element modeling to include multiple scattering, an examination of elastic wave coupling conditions, and discussion of the multipath structure as seen in sonar images constructed from the data in this paper. ACKNOWLEDGMENTS
Work supported by ONR. APPENDIX A: ABSOLUTE TARGET STRENGTH CALIBRATION
This appendix outlines the process used to convert experimental data to absolute target strength. For a sufficiently large solid sphere the specular backscattering strength is independent of frequency and angle and is thus an ideal calibration target.14 A solid sphere was placed in front of the source/receiver and insonified with the same signal used in the cylinder experiment. The sphere is large enough that the initial specular echo is completely distinguishable from later elastic echoes. The spectrum Vsph ðxÞ of the specular response is then used to calibrate the cylinder data using " # asph Vcyl ðxÞ A ; (A1) TS ¼ 20 log10 aref Vsph ðxÞ 2 where Vcyl ðxÞ is the cylinder spectrum, aref ¼ 1 m is a standard reference radius, asph is the radius of the sphere used, and the factor A is given by A ¼ Rsph
rcyl rsph
2
:
Plotnick et al.: Proud tilted cylinder backscattering
(A2)
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Here measurement distances to the sphere and cylinder are, respectively, rsph and rcyl . This accounts for spherical spreading and the reflection coefficient Rsph of the sphere in water calculated in the limit kasph 1. Comparison between cylinders of different sizes is possible by scaling the frequency and target strength of the resulting acoustic color; let a1 and a2 be the radii of two cylinders with identical length to diameter ratios, the frequency f2 is the scaled version of f1 , and the target strength TS2 the scaled version of TS1 . Frequency data obeys a linear scaling rule; f2 ¼ f1 a1 =a2 . Target strength may be converted using; TS2 ¼ TS1 þ 20 log10 ða2 =a1 Þ. These conversions have been applied to the experimental data taken on the 1 in. 2 in. cylinder to allow comparison to the 1 ft 2 ft cylinder used in the FEM model in Fig. 8. APPENDIX B: DERIVATION OF BROADSIDE MULTIPATH ANGLES
This appendix outlines the derivation of the azimuthal angles of broadside multipath rays that leads to the splitting in Eq. (1). A plane wave is incident on a cylinder located at the origin with arbitrary orientation. We define two sets of Cartesian coordinates centered on the cylinder. One set, f^ e 01 ; e^02 ; e^03 g, defines the laboratory frame, where the superscript indicates vectors measured in the lab frame. The second set of coordinates defines the cylinder frame, f^ e 01 ; e^02 ; e^03 g; which is locked to the cylinder and allowed to rotate freely with respect to the lab frame (see Fig. 3). Primed vectors are measured in the cylinder frame. In the laboratory frame, the direction of the incident wave vector, 0 0 k~ ¼ kk^ , is 0 1 0 k^1 B C 0 B C k^ ¼ B k^02 C: (B1) @ A 0 k^ 3
The wave vector is then transformed from the lab frame to 0 the cylinder frame k^ using a series of rotations via the Euler angles a; b, and c. Figure 10 includes the geometry and coordinate system used15 for this operation; the cylinder and lab frame coordinate systems and source/receiver geometry
are the same as used in Fig. 3. In this formulation the line of nodes corresponds to the vector projection ~ p of the cylinder axis on the interface in Fig. 3. The general form of the rotation is 0 0 k^ ¼ Rða; b; cÞk^ :
(B2)
The rotation matrix is given by16 2 3 cccbca scsa cccbsa þ scca ccsb 6 7 7 Rða; b; cÞ ¼ 6 4 sccbca ccsa sccbsa þ ccca scsb 5; sbca sbsa cb (B3) where for brevity cx cosðxÞ and sx sinðxÞ. For the formulation in this paper, a ¼ 90 þ H, b ¼ 90 , and c ¼ ht . The path 1 ray does not involve any interactions with the interface. The incident wave vector, Fig. 10(b), is given by 0 1 0 0 (B4) k^1 ¼ @ cos hg A: sin hg Similarly for path 4 the wave vector is given by 0 1 0 0 B C k^4 ¼ @ cos hg A: sin hg
(B5)
Substituting either of these wave vectors and our above definitions for a, b, and c into (B2) places that wave vector into cylinder coordinates: 0 1 þcos hg sin ht sin H6sin hg cos ht 0 B C (B6) k^1; 4 ¼ @ þcos hg cos ht sin H7 sin hg sin ht A: cos hg cos H Paths 1 and 4 are reversible; the incident ray must be parallel to the cylinder surface normal at the point of reflection. For the cylinder, excluding the ends, the surface normal is always perpendicular to the cylinder axis and thus the incident wave
FIG. 10. (a) Euler angle geometry. Three successive rotations allow a conversion from lab to cylinder coordinates. The Euler angle a ¼ 90 þ H (the azimuthal angle), b ¼ 90 , and c ¼ ht (the tilt angle). See also Fig. 3. (b) Geometry for incident sound beam in the lab frame with grazing angle hg .
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must have no projection along the cylinder axis, the e^02 direction in Fig. 10(a). The azimuthal angles H1; 4 are thus the solutions to cos hg cos ht sin H1; 4 7sin hg sin ht ¼ 0 and H1; 4 ¼ 6sin1 ðtan hg tan ht Þ:
(B7)
For brevity the splitting angle is defined as / sin1 ðtan hg tan ht Þ in Eq. (1). Paths 2 and 3 are not reversible paths but are a recipro0þ cal pair. The incident wave vector k^2; 3 and reflected wave 0 vector k^2; 3 are 0 1 0 1 0 0 0þ 0 (B8) k^2; 3 ¼ @ cos hg A; k^2; 3 ¼ @ cos hg A; 7 sin hg 7 sin hg 0 where the sign dependence in the k^z component accounts for path 2 and 3, respectively. Examination in the cylinder frame 0þ 0 shows that upon reflection from the cylinder k^x ¼ k^x . After converting both incident and reflected wave vectors into cylinder coordinates, the azimuthal angle H2; 3 is the solution to
cos hg sin ht sin H þ sin hg cos ht ¼ þcos hg sin ht sin H þ sin hg cos ht :
(B9)
Thus, H2; 3 ¼ 0:
FIG. 11. (Not to scale) Geometry used in physical acoustics model for farfield scattering (see Appendix C). The coordinate system f x ; yg is used to perform the necessary integrations. The origin is taken to be the center of the cylinder and midway along its axis. The cylinder has length L, radius a. The distance D L; a. (a) Top view, source/receiver (S/R) is located along the scan line at position x at a farfield distance D from the cylinder axis, thus making azimuthal angle H with respect to the cylinder broadside. (b) Side view, cylinder viewed from the right side of (a). Acoustic beam incident on cylinder from S/R at a height of D tan hg measured from the scanning line to the interface. The vertical distance hð x Þ is the vertical distance measured from the scanning line to the cylinder axis. (c) Front view, cylinder tilted at ht . The tilt angle, radius, and length determine the height of origin above the interface; height of the cylinder axis measured to the origin is a function of x and tilt angle.
(B10)
Regardless of tilt and grazing angle, the reciprocal path 2,3 remains at 0 azimuth. This is understandable as a corner (right angle) reflector; there will always exist a locus of points along the meridian of the cylinder whose surface normal is perpendicular to the interface normal. This may be visualized using paths 2 and 3 in Fig. 5. The end result is a triplet of backscattering peaks occurring at H ¼ 0; 6/. APPENDIX C: PHYSICAL ACOUSTICS MODEL
corresponding to the raypaths shown in Fig. 5. The form functions fS; p and fl; p are in a hybrid 2D/3D form to aid in their evaluation. The leading term in (C2) has been extracted from the full 3D versions of these form functions. Combining specular and elastic contributions in this situation extends Stanton’s related approach.8 The prefactor in (C2) is explained below. Some care must be taken in the interpretation of these hybrid form factors. Section 2 of Ref. 17 has been utilized to convert the required integrals into a more compact form. The scattering form function f for a rigid cylinder of length L may be used to extract the geometric dependence: ik XY; pa ð L=2 X¼ exp½2ikrðx; xÞd x;
f
This appendix will examine a physical acoustics model which includes the scattering geometry and elastic mechanisms for the tilted cylinder. The geometry used to develop the physical acoustics model is shown in Fig. 11. A plane wave of amplitude pi is incident on the cylinder. The far field scattered pressure is the real part of17 afcyl eiðkrxtÞ ; 2 r
(C1)
where fcyl is the backscattering form function of a cylinder of radius a. For large values of ka the finite cylinder form function may be expressed as a sum over available raypaths p, 2 1=2 X ip=4 2 kaL (C2) fcyl e ðfS; p þ fl; p Þ: a 4p p; l Here, the subscript S denotes the specular contribution and the index l refers to the lth leaky wave elastic contribution. In this treatment, only the circumferential Rayleigh waves l ¼ R will be included. The path index p runs from 1 to 4, J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
(C4)
L=2
Y¼
ða a
pscat ¼ pi
(C3)
1=2 2 pa exp ik y =a d y eip=4 ; k
(C5)
where the stationary phase approximation was used in (C5). For the plane of integration considered here the backscattering obliquity factor used in Eq. (73) of Ref. 17 is taken as unity for all raypaths. The integral in (C5) depends on the curvature of the relevant reflected wavefronts, as explained in Ref. 17. For paths 1 and 4 the curvature is given by geometrical considerations explained in Ref. 17, which yield the results in (C5). For paths 2 and 3 the off-meridional dependence of the curvature is small for the parameters considered in Fig. 9(b) and that dependence has been neglected in the approximation in (C5). The ik=pa and Y terms have been combined to give the prefactor in (C2). The remaining integral is dependent on the distance rðx; xÞ between the source/ Plotnick et al.: Proud tilted cylinder backscattering
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receiver at a location along the scanning line x, and each point along the cylinder axis x. Thus, each specular and leaky wave form function requires evaluating X for that path. The multipath geometries may be displayed using the method of images. In the rigid cylinder, single path, zero tilt limit the total form function goes over to that predicted in Refs. 17 and 18. 1. Specular contributions
The reflection coefficient Ral ð90 Þ will be discussed below, and the 1=L factor is due to the finite cylinder factor in (C2). The distance rðx; xÞ from a point x in the scanning line to a point x on the cylinder axis is 2 1=2
xÞ rðx; xÞ ¼ ½D þ ðx x cos ht Þ þ h1 ð
;
(C7)
where the horizontal farfield distance from the scanning line x Þ is the vertical distance to the cylinder axis is D and h1 ð from the scanning line to the axis. Here x Þ ¼ D tanðhg Þ ½ðL=2Þ sin ht þ a cos ht x sin ht : h1 ð (C8) In the farfield limit17 only terms linear in x are kept. Evaluating the integral and taking the limit D L; a it can be shown that fS; 1 Ral ð90 Þe2ik½Dwatan hg ððL=2Þsin ht þa cos ht Þ=w ! kL cos ht tan hg tan ht sin H ; sinc w where w ¼
(C12) Here, distances rþ ðx; xÞ and r ðx; xÞ are measured from the scanning line to the reflection point along the cylinder meridian and denote rays from the physical source and an image source, respectively,
(C6)
L=2
2
L=2
r6 ðx; xÞ ¼ ½ðD aÞ2 þ ðx x cos ht Þ2 þ h6 ð x Þ2 1=2 ; (C13)
For path 1 the specular form function fS; 1 is ð L=2 i2ka ei2krðx; xÞ d x =L: fS; 1 ¼ Ral ð90 Þe
2
fS; 2þ3
ð L=2 ¼ 2Ral ð90 hg ÞRs ðhg Þ eik½rþ ðx; xÞþr ðx; xÞ d x =L:
(C9)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ tan2 hg þ sin2 H. The leading term gives
the phase of the specular reflection while the directional sinc term is centered on tan hg tan ht sin H ¼ 0, as expected from the results in Appendix B. Similarly, Path 4 requires an integral ð L=2 2 i2ka ei2krðx; xÞ d x =L; (C10) fS; 4 ¼ Ral ð90 ÞRs ðhg Þe
where h6 ð x Þ ¼ D tanðhg Þ6½L=2 sin ht þ a cos ht þ x sin ht . The integration leads to a sinc function with argument kL cos ht sin H=w. 2. Elastic contributions
This paragraph will address the contribution from leaky traveling waves. Launching and detachment of the lth surface wave occurs via a velocity matching condition where cw is the speed of sound in the medium, cl is the leaky wave velocity, and hl is the angle between the launching ray and local surface normal, sin hl ¼ cw =cl :
While this condition allows for several different categories of elastic waves on the cylinder, attention will be restricted to circumferentially traveling Rayleigh waves l ¼ R. Geometric evaluation of the Rayleigh wave component of the form factor for each path gives analogous results to those for specular scattering in Appendix C 1. The aluminum reflection coefficient Ral is replaced with a product of the complex coupling coefficient GR and complex factors that account for radiation damping and propagation. The coupling coefficient is approximated as19,20 GR 8pbR ðpkaÞ1=2 expðip=4Þ;
PR ðhp Þ ¼ GR
x Þ ¼ D tanðhg Þ þ ½ðL=2Þ sin ht þ a cos ht x sin ht : h4 ð (C11) This results in a phase term that is shifted relative to path 1, and a directional sinc function centered on tan hg tan ht þsin H ¼ 0, again as expected from previous results. Reciprocity of paths 2 and 3 may be used to combine their form functions, requiring an integral 478
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(C15)
where bR is the radiation damping parameter for the leaky Rayleigh wave. Rayleigh waves couple onto the cylinder, propagate circumferentially in both clockwise and counterclockwise directions, and re-radiate to the receiver. The combined coupling and propagation for each type of path is19
L=2
where Rs ðhg Þ is the interface reflection coefficient discussed below. The evaluation is identical to path 1 except that the x Þ is to an image source and thus vertical distance h4 ð
(C14)
exp½2ðp hp ÞbR þ igR ðhp Þ ; 1 exp½2pbR þ i2pkacw =cR
(C16)
where gR ðhp Þ ¼ kað2p 2hp Þcw =cR 2ka cos hR . The term hp is a path dependent coupling angle that accounts for different coupling conditions for monostatic paths 1 and 4 and bistatic paths 2 and 3. The Rayleigh wave velocity cR and the radiation damping coefficient bR were evaluated using methods outlined in Ref. 21. Any effect of the close proximity of the interface on the local values of bR and cR have been neglected. For paths 1 and 4, the coupling angles are h1; 4 ¼ hR , where hR is the Rayleigh coupling angle meeting the condition in (C14). Unlike paths 1 and 4, however, the Plotnick et al.: Proud tilted cylinder backscattering
projection of the incident ray from reciprocal paths 2 and 3 onto the cylinder includes an axial component, so that these leaky waves are not purely circumferential. The affiliated helical wave aspects of the response are omitted from consideration because, for the situation emphasized here in Fig. 9, the associated complications to the theory, e.g., see Ref. 22, do not appear to be essential. Paths 2 and 3 are bistatic, and the clockwise (CW) and counterclockwise (CCW) traveling waves have different coupling conditions. For path 3, which reflects first from the interface and then couples onto the cylinder, the associated coupling angles are h3; cw ¼ ðhR þ hg Þ and h3; ccw ¼ ðhR hg Þ. Path 2 is analogous
except clockwise and counterclockwise are reversed so that h2; ccw ¼ ðhR þ hg Þ and h2; cw ¼ ðhR hg Þ. This asymmetry introduces the factor of 1=2 into the elastic contribution of paths 2 and 3 in (C17).
3. Total model
Spherical spreading is incorporated by expressing the scattered pressure in target strength, referenced to 1 m. Based upon (C1) and (C2) the target strength can be expressed as TS ¼ 20 log10 jFj. For the physical acoustics model above,
1=2 n kaL2 e2ikDw P1 S1 Ral ð90 Þ þ PR ðhR Þe2ika þ 2Rs hg P23 S23 F¼ 4p
2ika 1 2ika cos hg 2 Ral 90 hg þ PR hR þ hg þ PR hR hg e þ Rs hg P4 S4 Ral ð90 Þ þ PR ðhR Þ e : 2
The above expression has been separated into phase and coupling factors P, directional coefficients S, and effective reflection coefficients, where P1 ¼ exp f2ik½a ðL=2 sin ht þ a cos ht Þ tan hg =wg; (C18a) P23 ¼ exp f2ik½a cos hg g;
(C18b)
P4 ¼ exp f2ik½a þ ðL=2 sin ht þ a cos ht Þ tan hg =wg; (C18c) S1 ¼ sincfkL cos ht ðtan ht tan hg sin HÞ=wg;
(C18d)
S23 ¼ sincfkL cos ht sin H=wg;
(C18e)
S4 ¼ sincfkL cos ht ðtan ht tan hg þ sin HÞ=wg;
(C18f)
and PR ðhp Þ is given in (C16). As before w ¼ ð1 þ tan2 hg þ sin2 HÞ1=2 . Figure 9(b) shows the resulting acoustic color predicted by the above model for the 2 ft 1 ft aluminum cylinder. This model captures several of the dominant features in the resulting acoustic color that are seen in both the FE model [Fig. 9(a)] and the scaled 2 in. 1 in. experimental results [Fig. 4(c)] at a comparable tilt and grazing angle. Reflection coefficients for aluminum, Ral , at both normal and grazing incidence were obtained using the method outlined in Ref. 23, Appendix A. The experiment presented in this paper utilized a free surface, and thus, the interface reflection coefficient was taken to be Rs ¼ 1. For real world cases, such as scattering from a sediment bottom, the complex reflection coefficient for that material and grazing angle must be substituted. Several approximations were utilized in this model, including a far field assumption. The suppression of the obliquity factor in (C2) and the helical nature of the elastic J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015
(C17)
waves for paths 2 and 3 means that the path 2 and 3 terms are most sensitive to these approximations. Fully treated helical rays22 and meridional rays5–7 have not been included in this model, nor has multiple scattering in which a wave may decouple from the cylinder, reflect off of the interface, and couple back on. Nevertheless, this physical acoustics model captures the dominant features near broadside. 1
K. L. Williams, S. G. Kargl, E. I. Thorsos, D. S. Burnett, J. L. Lopes, M. Zampolli, and P. L. Marston, “Acoustic scattering from a solid aluminum cylinder in contact with a sand sediment: Measurements, modeling, and interpretation,” J. Acoust. Soc. Am. 127, 3356–3371 (2010). 2 J. R. LaFollett, K. L. Williams, and P. L. Marston, “Boundary effects on backscattering by a solid aluminum cylinder: Experiment and finite element model comparisons (L),” J. Acoust. Soc. Am. 130, 669–672 (2011). 3 R. Lim, K. L. Williams, and E. I. Thorsos, “Acoustic scattering by a threedimensional elastic object near a rough surface,” J. Acoust. Soc. Am. 107, 1246–1262 (2000). 4 J. L. Kennedy, T. M. Marston, K. Lee, J. L. Lopes, and R. Lim, “A rail system for circular synthetic aperture sonar imaging and acoustic target strength measurements: Design/operation/preliminary results,” Rev. Sci. Instrum. 85, 014901 (2014). 5 P. L. Marston, “Approximate meridional leaky ray amplitudes for tilted cylinders: End-backscattering enhancements and comparisons with exact theory for infinite solid cylinders,” J. Acoust. Soc. Am. 102, 358–369 (1997). 6 K. Gipson and P. L. Marston, “Backscattering enhancements due to reflection of meridional leaky Rayleigh waves at the blunt truncation of a tilted solid cylinder in water: Observations and theory,” J. Acoust. Soc. Am. 106, 1673–1680 (1999). 7 K. Gipson and P. L. Marston, “Backscattering enhancements from Rayleigh waves on the flat face of a tilted solid cylinder in water,” J. Acoust. Soc. Am. 107, 112–117 (2000). 8 T. K. Stanton, “Sound scattering by cylinders of finite length. II. elastic cylinders,” J. Acoust. Soc. Am. 83, 64–67 (1988). 9 X.-L. Bao, “Echoes and helical surface waves on a finite elastic cylinder excited by sound pulses in water,” J. Acoust. Soc. Am. 94, 1461–1466 (1993). 10 G. C. Gaunaurd and H. Huang, “Acoustic scattering by a spherical body near a plane boundary,” J. Acoust. Soc. Am. 96, 2526–2536 (1994). 11 M. Zampolli, A. Tesei, F. B. Jensen, N. Malm, and J. B. Blottman, “A computationally efficient finite element model with perfectly matched Plotnick et al.: Proud tilted cylinder backscattering
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layers applied to scattering from axially symmetric objects,” J. Acoust. Soc. Am. 122, 1472–1485 (2007). 12 M. Zampolli, A. L. Espa~ na, K. L. Williams, S. G. Kargl, E. I. Thorsos, J. L. Lopes, J. L. Kennedy, and P. L. Marston, “Low- to mid-frequency scattering from elastic objects on a sand sea floor: Simulation of frequency and aspect angle dependant structural echoes,” J. Comp. Acoust. 20, 1240007 (2012). 13 M. Nijhof, A. L. Espa~ na, and K. L. Williams, “Efficient calculation of broadband acoustic scattering from a partially, obliquely buried cylinder,” in Proceedings of the 1st International Conference Exhibition Underwater Acoustics (2013), pp. 697–702. 14 L. R. Dragonette, S. K. Numrich, and L. J. Frank, “Calibration technique for acoustic scattering measurements,” J. Acoust. Soc. Am. 69, 1186–1189 (1981). 15 A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua (Dover, Mineola, NY, 2003), pp. 158–161. 16 J. D. Pendleton, “Euler angle geometry, helicity basis vectors, and the Wigner D-function addition theorem,” Am. J. Phys. 71, 1280–1291 (2003).
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Plotnick et al.: Proud tilted cylinder backscattering
