Shear waves in inhomogeneous, compressible fluids in a gravity field Oleg A. Godina) CIRES, University of Colorado and NOAA Earth System Research Laboratory, Physical Sciences Division, DSRC, Mail Code R/PSD99, 325 Broadway, Boulder, Colorado 80305-3328

(Received 18 September 2013; revised 26 December 2013; accepted 15 January 2014) While elastic solids support compressional and shear waves, waves in ideal compressible fluids are usually thought of as compressional waves. Here, a class of acoustic-gravity waves is studied in which the dilatation is identically zero, and the pressure and density remain constant in each fluid particle. These shear waves are described by an exact analytic solution of linearized hydrodynamics equations in inhomogeneous, quiescent, inviscid, compressible fluids with piecewise continuous parameters in a uniform gravity field. It is demonstrated that the shear acoustic-gravity waves also can be supported by moving fluids as well as quiescent, viscous fluids with and without thermal conductivity. Excitation of a shear-wave normal mode by a point source and the normal mode distortion in realistic environmental models are considered. The shear acoustic-gravity waves are likely to play a significant role in coupling wave processes in the ocean and atmosphere. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4863655] V PACS number(s): 43.28.Dm, 43.20.Bi, 43.20.Mv [RMW]

I. INTRODUCTION

Understanding and quantifying the interaction between wave processes in the ocean and atmosphere are critical in a multitude of problems ranging from climate modeling1–4 to ionospheric and thermospheric manifestations5–12 of underwater earthquakes and from satellite detection and early warning of tsunamis13–16 to localization and characterization of underwater explosions for the purposes of monitoring compliance with the Comprehensive Nuclear Test Ban Treaty.17–20 An investigation21 of the transparency of gas–liquid interfaces to acoustic-gravity waves generated by a compact source within the liquid revealed the existence of two narrow frequency bands, where the power flux through the interface increases by several orders of magnitude compared to its values outside these frequency bands. An unusual type of surface acoustic-gravity wave was found21 to be responsible for the emergence of one of these “windows of transparency.” The surface wave has the dispersion equation x2 ¼ gk, which does not contain any parameters of the liquid or the gas. Here g is acceleration due to gravity, x and k are wave frequency and wavenumber. An idealized environmental model was considered in Ref. 21, where unbounded fluid consists of gas and liquid half-spaces, each with a constant sound speed and an exponential density profile. One limiting case of this model is the textbook problem, where an incompressible liquid of constant density occupies a half-space with a free surface. Surface waves in this problem are known as the surface gravity waves in deep water and have the dispersion relation x2 ¼ gk.22,23 When the fluid consists of two incompressible, homogeneous fluid half-spaces with densities qþ and q above and below a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

J. Acoust. Soc. Am. 135 (3), March 2014

Pages: 1071–1082

the interface and qþ > q, the dispersion equation of the surface waves becomes22 x2 ¼ gk(qþ  q)/(qþ þ q); there are no surface waves with the dispersion equation x2 ¼ gk. In both textbook problems, the amplitude of the vertical displacement of fluid particles in continuous surface waves decreases exponentially with distance from the interface22,23 (Fig. 1). When fluids in both half-spaces are compressible, then, in addition to a surface wave that reduces to the textbook solutions in the incompressible limit, a different kind of surface wave can exist, where the particle displacement amplitude increases exponentially throughout the fluid (Fig. 1).21 This surface wave has the dispersion equation x2 ¼ gk and is a shear wave in the sense that the dilatation of the wave motion is identically zero. Hence, every moving fluid parcel preserves its volume and density. Wave motion, in which the volume of each fluid parcel remains constant, also can be called incompressible. It was recently found that incompressible wave motion is supported by three-dimensionally inhomogeneous, quiescent, compressible fluids occupying either unbounded domains or domains with horizontal pressure-release surfaces and sloping rigid boundaries.24 Exact, analytic solutions of linearized equations of hydrodynamics, which describe the incompressible wave motion, apply to rotating and non-rotating ideal, compressible fluids with arbitrary density profiles24 and lead to a simple dispersion equation x2 ¼ gk in the absence of sloping boundaries. For stratified, non-rotating media, similar results were previously reported by Lamb22 for incompressible and by Whitney25 and Jones26 for compressible fluids. In this paper, we focus on non-rotating fluids with horizontal boundaries and extend the analysis of Ref. 24 to more realistic environmental models by considering shear acoustic-gravity waves in moving ideal fluids as well as in viscous fluids with and without thermal conductivity. In addition to free waves, we study excitation of the shear

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C 2014 Acoustical Society of America V

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FIG. 1. (Color online) Surface waves in three canonical problems. Variation with height (depth) of the amplitude of the vertical displacement w3 of fluid particles is shown for surface waves in homogeneous incompressible fluid with a free surface (Refs. 22 and 23) (a), at the interface of two homogeneous incompressible fluids (Ref. 22) (b), and at the interface of two compressible fluids with exponential density profiles (Ref. 21) (c). Particle displacement does not necessarily decrease away from the interface in compressible fluids.

waves by a point source. Moreover, we quantify the distortion of the shear waves caused by deviations of the environmental parameters from the conditions necessary for existence of the pure shear waves. The paper is organized as follows. In Sec. II, equations of motion and boundary conditions for linear acousticgravity waves are formulated (Sec. II A) and exact analytic solutions of these equations, which represent free shear waves, are obtained in ideal fluids (Secs. II B and II C) and in viscous, heat-conducting fluids (Sec. II D). Excitation of the shear waves by a point source and conditions of existence of shear-wave normal modes in layered waveguides are studied in Sec. III. More general environmental models, where no pure shear waves exist, are considered in Sec. IV. Section V summarizes our findings. II. EXACT SOLUTIONS FOR FREE WAVES A. Governing equations

Consider continuous linear waves of frequency x in a fluid with background (i.e., unperturbed by waves) pressure p0, density q0, sound speed c and flow velocity v0 in a uniform gravity field with acceleration g. The fluid is stationary (i.e., its parameters are independent of time t) in the absence of waves. Time dependence exp(ixt) of the wave field is implied and suppressed. We assume that either the background state of the fluid is stable or the instabilities, if any, develop on time scales much larger than periods of the wave considered. Linearization of the Euler, continuity, and state equations with respect to wave amplitude leads to the following set of equations27,28 governing wave fields: rp þ q0

d2 w rp0 þ ðw  rÞrp0  ð p þ w  rp0 Þ ¼ 0; q 0 c2 dt2 (1)

r  w þ ð p þ w  rp0 Þ=q0 c2 ¼ 0;

(2)

where p and w are the pressure perturbation and oscillatory displacement of fluid particles due to the wave and 1072

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d=dt ¼ ix þ v0  r is the convective time derivative. Wave-induced fluid velocity perturbation v is related to the oscillatory displacement by the equation27,28 v ¼ dw=dt  ðw  rÞv0 : Equations (1) and (2) allow for changes in the fluid’s composition, including air humidity, but do not account for phase transitions such as water condensation (and associated heat release) or evaporation. Perturbations in the gravity field due to wave-induced redistribution of the fluid’s mass are disregarded in Eqs. (1) and (2). This is usually referred to as Cowling approximation.29 No specific form of the equation of state is assumed in the derivation of Eqs. (1) and (2).27,28 Remarkably, of all the thermodynamic partial derivatives entering the linearized equation of state, only one characteristic of the fluid, the sound speed, is present in Eqs. (1) and (2). The equations have been derived27,28 assuming that wave propagation is an adiabatic thermodynamic process and disregarding irreversible processes associated with viscosity, thermal conductivity, and diffusion of admixtures such as salt in seawater and water vapor in atmospheric air. However, some mechanisms of wave dissipation can be described phenomenologically by ascribing frequency-dependent, complex values to the sound speed and/or density. Linearized equations of motion (1) and (2) are to be supplemented by linearized boundary conditions. On fluid–fluid interfaces, the normal component of the oscillatory displacement w and the quantity p~ ¼ p þ w  rp0

(3)

remain continuous.27,28 p~ has the meaning of the Lagrangian pressure perturbation, i.e., wave-induced pressure perturbation in a moving fluid particle22,28 as opposed to the (Eulerian) pressure perturbation p at a fixed point in space. Note that p is discontinuous at the interfaces where q0 is discontinuous. On a free surface (or any surface where pressure remains constant) the boundary condition is p~ ¼ 0; on a rigid surface, the normal component of w is zero.27,28

B. Shear waves in inviscid fluids

Introduce a Cartesian coordinate system with horizontal coordinates x and y and vertical coordinate z increasing upward (Fig. 2). Then g ¼ (0, 0, g). Let the background flow be horizontal: v0 ¼ (v01, v02, 0) and independent of the horizontal coordinates. Then the Euler equation for the background state gives rp0 ¼ q0 g:

(4)

By applying the differential operator curl to both sides of Eq. (4) and recalling that g ¼ const, we find that g  rq0 ¼ 0 and, hence, q0 ¼ q0(z). The sound speed can depend on all three spatial coordinates but it follows from the equation v0  rq0 ¼ 0 that v0  rc ¼ 0: We are interested in a particular class of wave motion where pressure remains constant in each moving fluid particle: Oleg A. Godin: Shear acoustic-gravity waves

The solution defined by Eq. (9) for the displacement and Eqs. (5) and (6) for pressure is valid in unbounded fluid as well as in a volume with pressure-release boundaries of arbitrary geometry. It also remains valid in the presence of fluid–fluid interfaces. Indeed, sound speed does not affect the solution; in the absence of waves, density discontinuities can occur only on horizontal planes, and Eqs. (5) and (8) ensure that the boundary conditions of p~ and w3 continuity are met at such interfaces. Note that the displacement field [Eq. (8)] in the shear wave remains continuous everywhere, while the Eulerian pressure [Eq. (6)] is discontinuous at interfaces between fluids with distinct densities (Fig. 3). In moving media, we look for solutions to simultaneous Eqs. (6) and (7) assuming the harmonic dependence expðik  rÞ; k ¼ ðk1 ; k2 ; 0Þ on horizontal coordinates. Solutions to Eq. (7) exist only when the quantity b ¼ 1  k  v0 =x

(10)

is constant and have the form wh ¼ ix2 b2 gkw3 ;   w3 ðrÞ ¼ W0 exp ik  r þ x2 b2 z=g ; W0 ¼ const: FIG. 2. (Color online) Geometry of the problem. Background density q0 and velocity v0 ¼ (v01, v02, 0) of the horizontal background flow are functions of the vertical coordinate z and may be discontinuous at interfaces within the fluid (a). In waves with harmonic dependence on horizontal coordinates x and y, k ¼ (k1, k2, 0) is the horizontal wave vector (b).

p~  0:

(5)

Then, according to Eqs. (2) and (3), p ¼ q0 gw3 ;

r  w ¼ 0;

(6)

and dilatation is identically zero; r  v ¼ 0: Here and below, we use the following notation for horizontal and vertical components of the oscillatory displacement: w ¼ (w1, w2, w3), wh ¼ (w1, w2, 0). Since every fluid particle maintains its volume, only shear deformations take place, and the wave motion can be referred to as an incompressible24 one or as a shear wave, as opposed to compressional waves. Substitution of Eqs. (4) and (6) into Eq. (1) gives d2 w=dt2 ¼ grw3 :

(7)

In the absence of background flow, d=dt ¼ ix, and from Eq. (7) we find   wh ¼ x2 grh w3 ; w3 ðrÞ ¼ W ðx; yÞexp x2 z=g ; (8) where rh ¼ ð@=@x; @=@y; 0Þ. Then, from Eqs. (6) and (8) it follows that W is an arbitrary solution of a two-dimensional (2-D) Helmholtz equation: @ 2 W=@x2 þ @ 2 W=@y2 þ x4 g2 W ¼ 0:

Substitution of Eq. (11) into Eq. (6) gives the dispersion equation of the shear wave: ðx  k  v0 Þ2 ¼ gk:

(12)

In a uniform flow, displacement of a fluid parcel is a sum of its displacement with the background flow and the oscillatory displacement w.27,28 According to Eqs. (11) and (12), jwh j ¼ jw3 j, the phase shift between the horizontal and vertical oscillatory displacements is p/2, and trajectories of fluid parcels are circles in the reference frame following the background flow. Each circle lies in a vertical plane parallel to the wave vector k. Radii of the circular trajectories increase exponentially with height z. b [Eq. (10)] has the meaning of a Doppler factor relating frequencies of the wave in the original reference frame and the reference frame moving with the local background flow. b is constant and the shear wave exists in moving fluid with continuously varying flow velocity when either v0 ¼ const or there exists a horizontal direction, the projection of v0 on which is independent of z. The latter condition is always met, in particular, for unidirectional flows. As in quiescent fluids, the shear wave, which is described by Eqs. (6), (11), and (12), satisfies boundary conditions at arbitrary pressure release boundaries and is unaffected by the presence of discontinuities in c(r) and q0(z). The solution [Eqs. (6), (11), and (12)] is unaffected by the component of background flow velocity that is orthogonal to the wave vector k. Moreover, note that b2 is the same for two values, rffiffiffi x g ; (13) V¼ 6 k k

(9)

In particular, any superposition of 2-D plane waves with wavenumber x2/g satisfies Eq. (9). J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

(11)

of the component V of v0 in the direction of k. Therefore, the solution [Eqs. (6), (11), and (12)] remains valid in a fluid, where V(z) is discontinuous and switches between the two Oleg A. Godin: Shear acoustic-gravity waves

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FIG. 3. (Color online) Variation with height of physical parameters in a shear acoustic-gravity wave in an unbounded quiescent fluid. The fluid consists of a gas half-space z > 0 and a liquid half-space z < 0. Cutoff frequencies of the shear wave are x ¼ x0 and xþ ¼ 2.5 x0. Amplitudes of the vertical displacement w3 (a), Eulerian pressure perturbation p (b), and wave energy density (c) are shown on the logarithmic scale as functions of the non-dimensional height k0z, where k0 ¼ x02/g, for five frequencies: x ¼ 0.9x0 (1), 1.2x0 (2), 1.7x0 (3), 2.0x0 (4), and 3.1x0 (5). Mass density profile (line 6) is shown in panel (a).

values given by Eq. (13) on a horizontal interface or interfaces. This is similar to the observation made by Mollo-Christensen30 regarding Gerstner waves in incompressible fluids. 1074

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Let background flow velocity profiles be given by v0 ¼ ½V ð1Þ ; v02 ðzÞ; 0 and v0 ¼ ½V ð2Þ ; v02 ðzÞ; 0 at z < z0 and z > z0, respectively. Here V ð1Þ 6¼ V ð2Þ are constants. From Eq. (13) we find that the shear wave [Eq. (11)] will exist in such a medium, for arbitrary v02(z), at frequency 2 x ¼ 2gjV ð1Þ þ V ð2Þ jðV ð2Þ  V ð1Þ Þ and with wave vector  1 ð 1Þ ð 2Þ k ¼ ½2x V þ V ; 0; 0. The shear wave exists also in a fluid, where there are multiple interfaces with the same V ð1Þ and V ð2Þ values. If V ð1Þ þ V ð2Þ ¼ 0, the shear wave becomes a stationary perturbation with x ¼ 0. Shear waves [Eqs. (6), (8), and (9) and Eqs. (6), (11), and (12)] exist despite the absence of shear rigidity in inviscid fluids. Instead, the restoring force is provided by gravity. In addition to being incompressible, fluid motion is potential when either background flow is uniform or absent. Assuming that b ¼ 1 when v0 ¼ 0, it is easy to check that the oscillatory velocity v ¼ r(ip/xbq0). The oscillatory displacement and velocity fields in the shear waves are independent of the background sound speed and density fields. All components of the oscillatory displacement increase exponentially with height. Eulerian pressure perturbation is independent of the sound-speed field, is proportional to the unperturbed density, and has a more complicated than w, generally non-monotone vertical dependence. Despite the exponential growth of the displacement amplitude with z, Eulerian pressure amplitude decreases with height when the rate of the mass density decrease is higher than the rate of the displacement increase, which occurs at lower wave frequencies [Fig. 3(b)]. Similarly, the time-averaged wave energy density28 decreases with height, when the rate of mass density decrease is higher than twice the rate of the displacement increase [Fig. 3(c)]. In the example shown in Fig. 3, the strong changes in the vertical profiles of the pressure and wave energy density, which occur around k0z ¼ 0, 0.5, and 4, are due to changes in the mass density decrease rate d ðlnqÞ=dz at these heights [Fig. 3(a)] as well as the mass density discontinuity at the gas–liquid interface z ¼ 0. The exact solution [Eqs. (6), (8), and (9)] of hydrodynamic equations for acoustic-gravity waves in quiescent fluids was previously reported in Ref. 24, while its extension [Eqs. (6), (11), and (12)] to moving media is new. Exact solutions of the linearized equations of motion [Eqs. (1) and (2)] are only approximate solutions of the exact, nonlinear hydrodynamic equations. From estimates of the nonlinear terms discarded in the process of linearization of the hydrodynamic equations, it follows31 that the amplitude of the oscillatory displacement w of fluid parcels should be much smaller than the wavelength 2p/k (or, equivalently, wave-induced perturbations in the flow velocity should be much smaller than the phase speed x/k of the wave) for the linear approximation to be valid. The linear solutions [Eqs. (8) and (11)] predict exponential growth of the displacement amplitude with height. Unless the growth is arrested by various dissipation mechanisms, which are present in the Earth’s atmosphere and become increasingly important at larger heights,32 the nonlinear effects should be taken into account at z > zn, where zn  k1 ln(2p/kjW0j). Investigation of the nonlinear effects is beyond the scope Oleg A. Godin: Shear acoustic-gravity waves

of this paper. It can be shown, however, that the exact, nonlinear hydrodynamic equations of inviscid, compressible fluids also admit shear-wave solutions.33 C. Alternative derivations

Of course, the exact solutions derived in Sec. II B can be obtained without using the particular form [Eqs.(1) and (2)] of the linearized hydrodynamic equations. Any other set of linearized equations of mechanics of compressible fluids can be used instead. For instance, in quiescent media, one can start from Gill’s34 momentum equations (6.4.4)–(6.4.5), the continuity equation (6.4.6), and Eq. (6.14.6), which follows from the equation of state. For shear waves, i.e., under conditions (6), Gill’s continuity and state equations are met, while momentum equations reduce to our Eq. (7), from which the solution Eq. (8) follows. In a fluid with uniform horizontal flow, c ¼ c(r) and q0 ¼ q0(z), the acoustic-gravity wave equation derived in Ref. 35 becomes ! d2 1 d2 p~ r~ p N2 2 (14)  r  r p~ ¼ 0  dt2 q0 c2 dt2 q0 h q0 2

gq1 0 dq0 =dz

2

2

 g =c , N is the in our notation, where N ¼ buoyancy frequency, and p~ and w3 are related by the equation35     d2 @ 1 g @ w p~: (15) þ g ¼  þ 3 dt2 @z q0 c2 @z Due to Eq. (5), the wave equation (14) is met automatically by shear waves. For the solution with p~  0 to be nontrivial, i.e., to describe an actual wave motion, w3 should be non-zero. Imposing this requirement, from Eq. (15) we find the dispersion equation (12) of shear waves and the solution Eq. (11) for w3. Consider waves with the harmonic dependence expðik  r  ix tÞ on horizontal coordinates and time in a stratified fluid with horizontal background flow, where all parameters of the medium depend only on z. Vertical dependences of the Eulerian pressure and the vertical component of the oscillatory displacement satisfy a set of firstorder, ordinary differential equations   @p g þ 2 p ¼ q0 x2 b2  N 2 w3 ; @z c ! @w3 g k2 1 p  2 w3 ¼  : @z c x2 b2 c2 q0

(16)

(17)

These or very similar equations (sometimes referred to as residual equations23) have been employed by various authors.36–40 In shear waves, p ¼ q0 gw3 according to Eq. (6). Substituting this relation into Eqs. (16) and (17), one obtains @w3 =@z ¼ g1 x2 b2 w3 and @w3 =@z ¼ k2 gx2 b2 w3 , respectively. The compatibility of the latter two equations gives the dispersion relation (12) of shear waves, and then the solution (11) for w3 readily follows from either of the Eqs. (16) and (17). J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

The algebraic simplicity of the shear wave solution allows one to check quickly the validity of differential equations for acoustic-gravity waves proposed in the literature. For instance, Boyles41 proposed the following compact wave equation for acoustic-gravity waves in three-dimensionally inhomogeneous, quiescent fluids   rp 1 @ 2 p prp0 rp0 1 :  ¼  þ r r q0 q0 q0 c2 @t2 q0 c2 q20 c4 It is easy to check that, unlike various correct descriptions of the acoustic-gravity waves considered above, Boyles’s wave equation is not satisfied by the shear wave Eqs. (6) and (9) even in the case of harmonic dependence on horizontal coordinates and time. A more detailed inspection shows that the pressure-density relation p ¼ c2q between the Eulerian pressure p and density q perturbations was used in Ref. 41 instead of the correct relation28 p þ w  rp0 ¼ c2 ðq þ w  rq0 Þ, eventually leading to the erroneous acoustic-gravity wave equation. In another book42 and related literature on tsunami generation, which is cited in Ref. 42, the wave equation     @2 p p 2 D  c ¼0 @t2 q0 q0 is utilized to describe acoustic-gravity waves in the ocean modeled as a quiescent, compressible fluid in a uniform gravity field. This wave equation is also incorrect as revealed by substitution of the exact, shear wave solution Eqs. (6) and (9). For the shear waves in quiescent fluids, we have Dð p=q0 Þ ¼ 0, and the wave equation of Ref. 42 is obviously not satisfied. The problems with the wave equation can be traced back to discarding some spatial derivatives of the background density q0 in its derivation and the a priori supposition that the fluid motion is a potential one, which is actually not the case for generic acoustic-gravity waves. D. Extension to dissipative media

Consider acoustic-gravity waves in a quiescent, viscous fluid. Viscosity does not affect continuity and state equations, but the viscous forces (stresses) need to be accounted for in the Euler equation.31 When there is no background flow, there are no viscous forces in the absence of waves, and Eq. (4) remains valid for the background state of a viscous fluid. By taking into account the gravity forces in Eq. (15.5) of Ref. 31 and linearizing this equation with respect to the wave amplitude, the linearized Euler equation in viscous fluids becomes    @vj @p @ @vm @vj ¼ qgj  þ g þ q0 @t @xj @xm @xj @xm    @ 2 þ (18) f g rv ; @xj 3 where summation over repeated indices j, m ¼ 1, 2, 3 is implied. Here, q is the Eulerian perturbation in density of the fluid, v is the wave-induced fluid velocity, g and f are shear Oleg A. Godin: Shear acoustic-gravity waves

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(or first) and bulk (or second) viscosity coefficients. The viscosity coefficients are non-negative.41 Viscosity will have no effect on fluid motion if the terms containing g and f in Eq. (18) vanish. For the shear wave solution [Eqs. (6), (8), (9)] derived for quiescent, ideal fluids in Sec. II B, Dv ¼ 0 and r  v ¼ 0. Hence, the shear wave remains an exact solution of linearized hydrodynamic equations of viscous fluids provided g ¼ const; there are no restrictions on the bulk viscosity. Note that the shear viscosity g is independent of pressure in ideal gases; in the atmosphere, g only weakly depends on air temperature.32 Therefore, g ¼ const is a reasonable assumption for acoustic-gravity waves in the atmosphere. Note that, unlike g, the kinematic viscosity g/q0 rapidly increases with height, which leads to very strong dissipation of generic acoustic-gravity waves in the middle and upper atmosphere.32 For particular types of the shear wave [Eqs. (6), (8), and (9)], even spatially variable shear viscosity will have no effect on the wave. For instance, when W(x, y) in Eq. (8) is chosen to be independent of the horizontal coordinate y, the viscous stresses, i.e., viscosity-related terms in Eq. (18), vanish for arbitrary differentiable dependence g(y). Viscosity does not lead to dissipation of the shear waves [Eqs. (6), (8), and (9)] in unbounded fluids as long as g ¼ const. In viscous fluids, however, waves should meet additional boundary conditions besides continuity of p~ and the normal component of w at fluid–fluid interfaces or the condition p~ ¼ 0 at a free surface. The shear wave alone generally does not meet the full set of boundary conditions in viscous fluids, leading to generation of viscous waves and wave energy dissipation in the vicinity of boundaries and interfaces.43,44 Viscous dissipation of the shear wave is similar to the well-studied damping of surface gravity waves in incompressible fluids with constant viscosity, which is also associated with fluid boundaries only.31 As a caveat, all our findings regarding the effects of viscosity on the shear waves refer to linear waves only and will not necessarily hold at very large z, where nonlinear effects may become nonnegligible. So far, we have neglected thermal conductivity. In the absence of viscosity and thermal conductivity, wave propagation is an adiabatic process, and entropy density remains constant in moving fluid parcels. In real fluids, the entropy density satisfies the so-called general equation of heat transfer.31 Linearization of the heat transfer equation (49.4) of Ref. 31 with respect to the wave amplitude gives   @S þ v  rS0 ¼ r  ðjrT0 þ j0 rT Þ; q0 T0 (19) @t where S0, T0, and j0 are the entropy density, temperature, and thermal conductivity in the background state, while S, T, and j are the wave-induced Eulerian perturbations of the respective physical quantities. In stationary media, the heat flux is divergence-free in the absence of waves: r  ðj0 rT0 Þ ¼ 0:

(20)

Equation (19) means that entropy density variations in the fluid parcel occur due to variations in the heat conducted to 1076

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the parcel. Generally, viscous dissipation contributes to entropy changes,31 but these effects are of the second order in the wave amplitude and do not affect linear waves. The shear waves [Eqs. (6), (8), (9)] will be unaffected by thermal conductivity and remain an exact solution of the hydrodynamic equations of real fluids, if the right-hand side of Eq. (19) is identically zero. In the shear wave, thermodynamic parameters are advected with the fluid (i.e., remain constant in moving fluid parcels). In terms of the Eulerian perturbations, it means that28 j þ w  rj0 ¼ 0;

T þ w  rT0 ¼ 0:

(21)

Taking into account that Dw ¼ 0 and r  w ¼ 0 in the shear wave, we find from Eqs. (20) and (21) that the right-hand side of the linearized heat transfer equation (19) is identically zero, when j0 rT0 ¼ const:

(22)

Thus, thermal conductivity of the fluid does not affect the shear waves [Eqs. (6), (8), and (9)], when the heat flux density in the absence of waves is constant. This is the case, in particular, when the background state is isothermal. Moreover, Eq. (20) ensures that the condition (22) is always met in layered media. Radiative exchange due to absorption and radiation of electromagnetic energy by the fluid can be a significant mechanism of acoustic-gravity wave damping in the Earth’s atmosphere and especially in stars.32 Thermal electromagnetic radiation by a fluid parcel is controlled by its temperature. Advection of density and temperature in the shear waves [Eqs. (6), (8), and (9)] means that the waves will not change the balance between absorption and radiation of electromagnetic energy in a stationary medium and will not be affected by the radiative exchange, if the intensity of the background electromagnetic field can be considered constant over spatial scales that are large compared to the waveinduced displacements w of fluid parcels. The above analysis of the effects of viscosity and thermal conductivity on the shear waves translates, without change, to uniformly moving layered fluids. This is easy to prove by considering the problem in a reference frame moving with the background flow. Furthermore, the results regarding the thermal conductivity remain valid for those non-uniform, discontinuous profiles of the background flow velocity, which are found in Sec. II B to support the shear waves, since the boundary conditions for the heat flux31 are unaffected by the horizontal flows. Conclusions regarding the effects of viscosity do not apply in the latter case because discontinuous flow velocity profiles are not possible in viscous fluids. III. POINT SOURCE IN A LAYERED MEDIUM A. Excitation of normal modes by a point source

Consider a quiescent, horizontally stratified ideal fluid. Let acoustic-gravity waves be generated by a monochromatic point source of mass with the amplitude a0 of the Oleg A. Godin: Shear acoustic-gravity waves

volume injection rate. The source is located at the point r0 ¼ (0, 0, z0). (One can think of the source of mass as a small sphere undergoing periodic radial pulsations.) Such a source is described mathematically28 by adding the term ðixÞ1 a0 qc2 dðr  r0 Þexpðix tÞ in the right-hand side of Eq. (2), where d() is the Dirac delta function. Let the functions Pðk; zÞ and W ðk; zÞ describe variations of pressure and vertical displacement with height in waves with harmonic dependence expðik  rÞ on horizontal coordinates. These functions meet Eqs. (16) and (17) for free waves. Designate Pð1Þ ðk; zÞ and Pð2Þ ðk; zÞ the solutions to Eqs. (16) and (17) that satisfy conditions at z ! 1 (or at the lower boundary, in the case of a bounded medium) and at z ! þ1 (or at the upper boundary), respectively. In terms of these solutions, the pressure field generated by the point source can be represented as the integral21   N 2 ðz0 Þ pðr; z0 Þ ¼ ixq0 ðz0 Þ 1  a0 x2 ð þ1 ð1Þ P ðk; z< ÞPð2Þ ðk; z> Þ  4pWr ðk; z0 Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ  H01 k x2 þ y2 kdk; (23) where Wr ðk; zÞ ¼ Pð1Þ ðk; zÞ@Pð2Þ ðk; zÞ=@z  Pð2Þ ðk; zÞ@Pð1Þ ðk; ð Þ zÞ=@z, z< ¼ min(z, z0), z> ¼ max(z, z0), and H01 ðÞ is a Hankel function. Although the factor 1  N2(z0)/x2 in front of the integral in Eq. (23) tends to zero when frequency tends to N(z0), the field does not vanish because the factor is compensated by a similar factor in the Wronskian Wr(k, z0) in the integrand. This becomes clear when the Wronskian is represented in the following equivalent form:  Wr ðk; zÞ ¼ qðzÞ x2  N 2 ðzÞ h i  Pð1Þ ðk; zÞW ð2Þ ðk; zÞ  Pð2Þ ðk; zÞW ð1Þ ðk; zÞ ; which follows from Eq. (16). When the functions Pð1Þ ðk; zÞ and Pð2Þ ðk; zÞ are linearly dependent, i.e., the same solution of Eqs. (16) and (17) satisfies conditions on both upper and lower boundaries (or at respective infinities), the Wronskian Wr ¼ 0, and there is a pole in the integrand. These poles correspond to normal modes, or the discrete spectrum of the problem.21,28,36 Following Refs. 28 and 36 in calculating residues in the poles, for the contribution of an individual normal mode with k ¼ ks into the field p(r, z0) due to the point source, we find

xa0 ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 H ps ðr; z0 Þ ¼ ks x þ y Pð1Þ ðks ; zÞPð1Þ ðks ; z0 Þ; 4Q 0 (24) where Q¼

ð zþ z

h i2 dz Pð1Þ ðks ; zÞ : q 0 ðzÞ

(25)

Integration in Eq. (25) is over the entire vertical extent of the waveguide. If the fluid is unbounded, z6 ¼ 61; if the fluid is bounded from above by a free surface at z ¼ h, then J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

z ¼ 1; zþ ¼ h; and if the fluid has two free surfaces at z ¼ h6 ; h < hþ , then z6 ¼ h6 . Note that the pressure ps(r, z0) is invariant with respect to interchange of the source and receiver positions, in agreement with the reciprocity principle28,45 for waves in compressible fluids. The derivation of Eqs. (24) and (25) for acoustic-gravity waves is analogous to the textbook derivation28 for the acoustic waves and will not be reproduced here. As in the acoustic case,46 in a waveguide with two horizontal boundaries, where there is no continuous spectrum, the same result can be derived directly from Eqs. (1) and (2) by using the orthogonality relation40 for normal modes of acoustic-gravity waves.

B. Excitation of the shear wave by a point source

In compressible fluids, there exists only one normal mode, in which the fluid motion is incompressible. The mode wavenumber and shape function are given by exact, explicit expressions Pð1Þ ðks ; zÞ ¼ q0 ðzÞexpðks zÞ; ks ¼ x2 =g according to Eqs. (6), (8), and (9). For this shear-wave normal mode, general Eqs. (24) and (25) become ! 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi xa0 x2 ðzþz0 Þ=g ð1Þ x q ðzÞq0 ðz0 Þe x2 þy2 ; H0 ps ðr;z0 Þ ¼ 4Q 0 g (26) Q¼

ð zþ

  q0 ðzÞexp 2x2 z=g dz:

(27)

z

Equation (26) shows that the shear wave [Eqs. (6), (8), and (9)] will be excited by a point source with a finite, non-zero amplitude when, and only when, the quantity Q [Eq. (27)] is finite. This is not surprising. The result reflects the known property of acoustic waveguides that a solution to the governing equations, which satisfies boundary conditions (and/or conditions at infinity in the case of fluid of infinite vertical extent), will be a proper normal mode (i.e., a mode of the discrete spectrum) only if a certain L2 norm of the solution is finite.47 Equation (24) shows that the absolute value of Q [Eq. (25)] serves as such a norm for acoustic-gravity waves in quiescent fluids. This is consistent with earlier results on orthonormality of normal modes of acoustic-gravity waves.40 Consider the power radiated by a monochromatic point source of acoustic gravity waves. The power carried by proper normal modes is conveniently calculated as the flux of the vector28 Reð pv Þ through a cylindrical surface x2 þ y2 ¼ const. Here the asterisk denotes complex conjugation. The power fluxes due to individual normal modes are additive40 because of the mode orthogonality. Using Eqs. (8) and (26), for the power flux in the shear-wave normal mode, we find ! x 2 2 2x2 z0 : (28) ja jq ðz0 Þexp Js ¼ g 8Q 0 0 Note that the power carried by the shear waves from the point source is inversely proportional to the norm Q [Eq. (27)]. Oleg A. Godin: Shear acoustic-gravity waves

1077

For sources with a fixed mass injection rate q0(z0)a0, the radiated shear wave power flux increases steadily with the source height. Depending on density stratification, the opposite may be true for sources with a fixed volume injection rate. When there is a gas–liquid interface in the medium, the density is typically much larger on the liquid side of the interface. If a source with a0 ¼ const is located at a distance on the order of wavelength from a gas–liquid interface, then Eq. (28) shows that much more shear wave energy is radiated by the source in the liquid than in the gas. This suggests that oceanic sources may be much more efficient in generating the shear waves in the coupled ocean-atmosphere system than tropospheric sources. C. Cutoff frequencies of the proper shear-wave normal mode

When zþ and z are finite and the maximum and minimum of the mass density q0(z) are positive and finite, Q [Eq. (27)] is also positive and finite. Hence, the wave [Eqs. (6), (8), and(9)] is a proper normal mode in any finite fluid layer with piecewise continuous density q0(z) and free boundaries. The normal mode exists at all frequencies (as long as the assumptions hold that wave dissipation and the Earth’s rotation are negligible). If the fluid is unbounded, Q will be finite only if the density tends to zero rapidly enough at z ! þ1 and remains finite or tends to infinity slowly enough at z ! 1. More specifically, the integral in Eq. (27) converges at the upper limit, if q0(z) ! 0 faster than exponentially (i.e., limz!þ1 q0 ðzÞexpðazÞ ¼ 0 for every a > 0), and the integral diverges at the upper limit, if q0(z) remains finite at z ! þ1 or tends to zero more slowly than exponentially. In the case of the exponential behavior of q0(z) at z ! þ1, i.e., when q0 ðzÞ qþ expðaþ zÞ at sufficiently large z, the integral converges at the upper limit if x < xþ and diverges if x xþ. The upper cutoff frequency xþ ¼ (gaþ/2)1/2. Quite similarly, the integral in Eq. (27) converges at the lower limit, if q0(z) remains finite at z!1 or tends to infinity more slowly than exponentially. If q0(z) ! þ1

 faster than exponentially (i.e., limz!1 q1 z az Þ ¼ 0 for 0 ð Þexpð every a > 0), the integral diverges at the lower limit. In the case of exponential behavior of q0(z) at z ! 1, i.e., when qðzÞ q expða zÞ at sufficiently large (z) and a > 0, the integral converges at the lower limit if x > x and diverges if x  x, where the lower cutoff frequency x ¼ (ga/2)1/2. In the particular case of the layered medium consisting of two fluid half-spaces, each with a constant sound speed and an exponential stratification of mass density, the same frequency range (ga/2)1/2 < x < (gaþ/2)1/2 of the shear-wave proper mode existence was obtained, from rather different considerations, as a part of a systematic analysis21 of transmission of acoustic-gravity waves through gas–liquid interfaces. In that problem, the shear-wave normal mode is one of the surface waves supported by the gas–liquid interface. It was found in Ref. 21 that the shear wave is responsible for a very large, on the order of the ratio of liquid and gas densities (e.g., three orders of magnitude in the case of air–water interfaces), 1078

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

increase in the transparency of the interface at frequencies close to the upper cutoff frequency xþ of the shear-wave normal mode. Spatial localization of the shear-wave normal mode depends on its frequency (Fig. 3). In an unbounded fluid, the energy of the normal mode is concentrated at large negative z at frequencies close to the lower cutoff frequency and at large positive z at frequencies close to the upper cutoff frequency. Depending on the density stratification, the normal mode energy can be localized in different layers at different frequencies. For the proper shear-wave normal mode to exist, wave energy density has to tend to zero at z !61; the wave energy density cannot have monotone vertical dependence. When this dependence has a strong maximum or maxima at z ¼ zmj, the normal mode is localized at corresponding heights in the sense that the bulk of the power flux carried by the normal mode is transported at z zmj. When, as in one of the examples shown in Fig. 3 (line 3), strong maxima exist simultaneously in the liquid and in the gas, excitation of the shear-wave normal mode by sources in the liquid will be strongly manifested in the gas, and vice versa. Of course, when the fluid has an upper (lower) free boundary, only conditions at z!1 (respectively, at z!þ1) determine whether Q [Eq. (27)] is finite and, hence, the shear wave [Eqs. (6), (8), and (9)] is a proper normal mode. When the integral in the right-hand side of Eq. (27) diverges, Eqs. (6), (8), and (9) still define an exact solution to the governing equations, which is then a normal mode of the continuous spectrum46,47 rather than a proper normal mode. Put differently, the shear wave [Eqs. (6), (8), and (9)] always exists in any unbounded or having a free boundary compressible fluid with a piecewise continuous layered density, but the physical meaning and significance of the wave depend on whether Q [Eq. (27)] is finite. For illustration, consider two particular cases. 1. Inhomogeneous layer between isothermal gas half-spaces

Let the fluid be an isothermal gas at jzj > z0, with the sound speed, absolute temperature, molecular weight, and the ratio of specific heats at constant pressure and constant volume being c6, T6, l6, and c6 at z > z0 and z < z0, respectively. No assumptions (other than the density being horizontally stratified) are made about fluid properties at z0 < z < z0. In a perfect gas, the sound speed is related to temperature by c6 ¼ ð c6 RT6 =l6 Þ1=2 ; where R is the universal gas constant.43 In isothermal gases, the density stratification is given by22   q0 ðzÞ ¼ q exp gc z=c2 ; z < z0 ;

q0 ðzÞ ¼ qþ exp gcþ z=c2þ ; z > z0 : (29) From the above discussion of Eq. (27) it follows that a proper shear-wave normal mode exists in a finite frequency band x < x < xþ, where pffiffiffiffiffiffiffiffiffiffi x6 ¼ gc1 c6 =2 ; (30) 6 provided the upper cutoff frequency xþ is larger than the lower cutoff frequency x. If x xþ, no such proper Oleg A. Godin: Shear acoustic-gravity waves

normal mode exists. In terms of the temperatures, the condition of the proper normal mode existence becomes Tþ T < : lþ l

(31)

In the case of gases of the same composition, temperature in the upper half-space should be lower than in the lower half-space. In Ref. 22, pp. 568–571, Lamb considered linear acousticgravity waves propagating along a horizontal interface z ¼ 0 of two isothermal gases of different temperature and analyzed surface waves, which can be supported by the interface. Lamb’s environmental model is a particular case z0 ¼ 0 of the model considered in the present section. Lamb assumed, erroneously, that the particle displacement necessarily decreases with height at z > 0 in surface waves and, therefore, was not able to find the surface wave with the dispersion equation x2 ¼ gk, which represents incompressible motion of the fluid.

Let the fluid in the background state have constant entropy density at z > z0 and at z < z0. When entropy is constant, dp0/dq0 ¼ c2, and from Eq. (4) we find ! ðz 2 ð 0 Þ 0 q0 ðzÞ ¼ q0 ðz0 Þexp g c z dz ; z < z0 ; z0

q0 ðzÞ ¼ q0 ðz0 Þexp g

c

!

2 ð 0 Þ

z dz

0

;

z > z0 :

(32)

z0

We further assume that the sound speed   has finite limits c6 at z!61. Then q0 ðzÞ  exp gz=c26 at z! 61, and for the upper and lower cutoff frequencies of the proper shear-wave normal mode we obtain [cf. Eq. (30)] pffiffiffi x6 ¼ g= 2c6 : (33) According to Eq. (33), xþ > x and, hence, the proper normal mode exists when cþ < c, which is the case for the ocean-atmosphere environment and is typically the case for a gas overlaying a liquid. It is remarkable that, while the wave field [Eqs. (6), (8), and (9)] is independent of the sound speed, compressibility of the medium re-enters the problem through conditions of the proper mode existence. In a wave physics context, treatment of fluids as incompressible is usually justified by wave speeds being small

k2  k 1 ¼ 

ð z2 k1 z1

ð z2



IV. DISPERSION EQUATIONS IN MORE GENERAL ENVIRONMENTS

Consider acoustic-gravity waves in a layered, moving fluid. In the absence of waves, the fluid occupies a layer z1 < z < z2. Let waves with harmonic dependence expðik  r  ix tÞ on horizontal coordinates and time satisfy boundary conditions p~ ¼ ixZð jÞ ðk; xÞw3 ;

2. Inhomogeneous layer between isentropic fluid half-spaces

ðz

compared to sound speed.34 It is interesting that, while representing incompressible motion, the shear wave [Eqs. (6), (8), and (9)] can propagate faster than the local sound speed in the layer z0 < z < z0 between homogeneous half-spaces. According to Eq. (33), at frequencies approaching the upper (lower) cutoff frequencies, the phase speed of the shear-wave normal mode is 21/2 times larger, while the group speed is just 21/2 times smaller, than the sound speed at z! þ1 (respectively, at z! 1).

j ¼ 1; 2:

Generally, Z(1) and Z(2) may depend on the horizontal wavenumber k and wave frequency and have the meaning of impedance of the lower and upper boundary, respectively. In particular, Z ¼ 0 on a free surface and Z ¼ 1 on a rigid surface, where w3 ¼ 0. There is no time-averaged energy flux through the boundary when its impedance is purely reactive: ReZ ¼ 0, wave energy is injected into the waveguide, if ReZ > 0, and wave energy leaves the waveguide, if ReZ < 0.28 Let P1(z) and U1(z) be the vertical dependencies of the Eulerian pressure p and the vertical oscillatory displacement w3 in a shear wave in a quiescent layer z1 < z < z2 with free boundaries, sound speed c(z) and density q(z). Designate k1 the wave vector of the shear wave; k1 ¼ x2/g according to Eq. (9). Let P2(z) and U2(z) be the vertical dependencies of p and w3 in a normal mode of the fluid layer z1 < z < z2 with background flow velocity v0(z), sound speed and density profiles c(z) þ dc(z) and q0(z) þ dq0(z), and boundary impedances Z(1) and Z(2). Designate the wavenumber of the normal mode k2. In moving fluids, it depends on the direction of the horizontal wave vector. When Z(1), Z(2), dc, dq0, and v0 are all sufficiently small, the normal mode dispersion relation and shape function are close to those in the shear wave. Then, k2 can be found using the perturbation theory for normal modes of acoustic-gravity waves.36,40 With accuracy to the terms of the second order in the environmental perturbations, from Eq. 29 in Ref. 40, we have

!1  i P21 ix3 h 2 U1 ðz2 ÞZ ð2Þ  U12 ðz1 ÞZ ð1Þ dz q0 2

  k1  v0 dz x2 ðP1  q0 gU1 Þ2 dc þ k12 P21 þ q20 x4 U12 3 c x q z1 0      4  2 dq0 x2 2 2 2 2 2 : þ  k  q gU  q x  g k U P 1Þ 0 1 ð 1 0 1 1 c2 2q0 þ

(34)

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

(35)

Oleg A. Godin: Shear acoustic-gravity waves

1079

Since P1 ¼ q0gU1 and k1 ¼ x2/g in the shear wave, coefficients in front of dc and dq0 are identically zero in the integrand in Eq. (35). Hence, as expected, variations in the sound speed and density do not affect the shear wave. Using Eq. (8), we simplify the general perturbation relation Eq. (35) to read k1 k2  k 1 ¼ x

!1

ð z2 e

2k1 z

From the boundary conditions and Eqs. (6) and (37), we find

q0 dz

k2  k1 ¼ A þ OðA2 Þ; k1 !1 ð z2 2k1 z e2k1 z1 qðz1 Þ; A ¼ k1 e qdz

z1

  i 2k1 z1 ð1Þ e  Z  e2k1 z2 Zð2Þ 2  ð z2 2k1 e2k1 z Vq0 dz ;

normal modes of acoustic gravity waves. From Eq. (14) in Ref. 40, we have an exact relation: ð k12  k22 z2 dz 2 ðP1 U2  P2 U1 Þjz¼z P1 P2 : (37) z¼z1 ¼ x2 z1 q 0

(36)

(38)

z1

z1

where V ¼ k11 k1  v0 is the projection of the flow velocity on the direction of the wave vector. Equation (36) describes small perturbations of the shear wave dispersion relation by the background flow and finite impedances of the boundaries. The normal mode in a moving fluid is generally not a pure shear wave. For particular x, Z(1), Z(2), and v0, Eq. (36) can be used to test whether the condition jk2  k1 j k1 is met, and the normal mode is close to the shear wave. For instance, flow-induced perturbations will be small for an arbitrary direction of the normal mode propagation if the average flow velocity Ðz Ðz hv0 i ¼ ð z12 e2k1 z q0 dzÞ1 z12 e2k1 z v0 q0 dz is small compared to the phase speed of the shear wave. Results for infinite fluid without boundaries or with a single boundary are obtained from Eq. (36) by extending the integration over z to the entire vertical extent of the fluid and letting Z ¼ 0 for a non-existent boundary or boundaries. It is easy to check that, when Z(1) ¼ Z(2) ¼ 0 and V ¼ const, Eq. (36) agrees with the exact result Eq. (12) for shear waves in uniformly moving unbounded fluids. Let the fluid occupy the half-space z < z2 and the flow velocity v0 is non-zero in a layer z0 < z < z2 of finite vertical extent. Let the fluid density be given by the first of Eq. (32) and the sound speed c ¼ c at sufficiently large z. Then, as shown in Sec. III C, the unperturbed shear wave normal mode has the cutoff frequency x (33). When wave frequency approaches the cutoff frequency, the wavenumber perturbation tends to zero, according to Eq. (36). This is not surprising since, when x ! x, all the wave energy and the power flux carried by the normal mode are concentrated in the unperturbed half-space z < z0. The structure of Eq. (36) suggests that a normal mode of acoustic-gravity waves remains close to the shear waves not only when impedances Z(1) and Z(2) are small, but also when the boundaries are located at such z1, 2 that the product exp(2k1z)q0(z) is small there. More generally, one should expect that there exists a normal mode, which is close to the shear wave, whenever environmental perturbations are confined to a region, the relative contribution of which into the norm Q [Eq. (27)] is much smaller than unity. For instance, consider a quiescent fluid with a free boundary at z ¼ z2 and a rigid boundary at z ¼ z1. Since Z(1) is infinite, Eq. (36) cannot be used in this problem. Instead, we start from the generalized orthogonality relation40 of 1080

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

provided that A 1. Note that k2 < k1, i.e., the presence of a rigid boundary decreases the phase speed of the normal mode. In a particular case, where relative density variations are much smaller than relative variations

in exp(2k1z) within the  layer z1 < z < z2, we get A 2= exp 2k1 ðz2  z1 Þ  1 , and the condition A 1 is equivalent to k1(z2  z1) 1. Thus, wave motion remains nearly incompressible, when the wavelength of the shear wave is small compared to the vertical extent of the fluid. It is straightforward to check that the perturbation result [Eq. (38)] for compressible fluids agrees in this particular case with the exact dispersion equation23 x2 ¼ kgtanhkðz2  z1 Þ of surface waves in an incompressible fluid.

V. DISCUSSION AND SUMMARY

Shear acoustic-gravity waves are similar to waves in incompressible fluids, but the phase and group speeds of the shear normal modes can be as large as the sound speed. Of course, all the waves in incompressible fluids are shear waves. Of all the various wave solutions of the hydrodynamic equations for incompressible fluids, only a specific class remains applicable to compressible fluids. Namely, for a solution to be valid in both incompressible and compressible fluids, pressure in every moving fluid parcel (i.e., the Lagrangian pressure) should remain constant. Our results show that it is, indeed, possible in bounded and unbounded inhomogeneous fluids in a uniform gravity field. With the pressure and density holding steady in fluid parcels, the pressure and density at a given point in space, i.e., the Eulerian pressure and density, vary periodically in time in a continuous shear wave as different fluid parcels pass through the observation point. While the sound speed has no effect on the shear wave field in compressible fluids, the sound-speed profile was found in Sec. III C to control the cutoff frequencies of the shear normal mode in open waveguides. The shear wave is described by an exact analytic solution of linearized hydrodynamic equations. In inviscid, quiescent fluids, the exact solution [Eqs. (6), (8), and(9)] applies to inhomogeneous fluids with an arbitrary piecewise continuous density profile and an arbitrary sound speed. In contrast, the other known exact solution for acoustic-gravity waves, the Lamb wave, applies only to fluids with a constant sound speed and an exponential density stratification.22,32 As has Oleg A. Godin: Shear acoustic-gravity waves

been illustrated in Sec. II C, the wide applicability and algebraic simplicity of the shear wave exact solution make it a convenient tool for verifying analytical and numerical models of acoustic-gravity wave fields. Exact solutions of the acoustic wave equation in quiescent layered media have long been the subject of active research, see reviews in Chap. 3 of Ref. 44, and Chap. 4 of Ref. 28. It is well known that exact analytic solutions for a range of frequencies are available only for specific soundspeed and density profiles, and no explicit analytic solution can be found for generic layered medium.28,44 Since sound waves can be considered as a special (limiting) case of acoustic-gravity waves, when g ! 0, it might seem impossible for an exact solution for a generic stratified fluid to exist in the general case of acoustic-gravity waves, when no such solution exists in the specific case of sound. The apparent contradiction is resolved by noting that the shear acousticgravity waves have the dispersion relation x2 ¼ gk in quiescent media, and their frequency tends to zero in the limit g ! 0. Thus, the shear wave solution, which is valid in a wide range of frequencies of acoustic-gravity waves, reduces to the stationary (x ¼ 0) solution of the acoustic wave equation; see Sec. 1.2.3 of Ref. 44 for a brief discussion of the stationary solutions of the acoustic wave equation for generic layered fluids. The shear waves are supported not only by quiescent fluids but also by moving fluids with certain flow-velocity profiles (Sec. II B). Remarkably, the shear wave remains an exact solution of the governing equations in viscous fluids as long as there are no spatial variations in the shear viscosity of the quiescent fluid; the bulk viscosity can be an arbitrary function of coordinates. Then, the shear wave propagates in viscous fluids without attenuation (Sec. II D). When the heat flux within the fluid is constant in the absence of waves, the shear wave also remains unaffected, unlike other acousticgravity waves, by the thermal conductivity of the fluid. While the shear wave solution satisfies the governing equations under surprisingly general conditions, more stringent requirements need to be met for the solution to represent a proper normal mode (Sec. III). The set of proper modes of acoustic-gravity waves always includes exactly one shear-wave normal mode, when a quiescent fluid layer has two boundaries, where a constant pressure is maintained (in particular, pressure-release boundaries). As has been discussed in Sec. IV, continuous variations of the background flow velocity with height in a stratified fluid and the presence of horizontal boundaries other than constantpressure surfaces, lead to modifications of the shear-wave normal mode. We have identified conditions when these perturbations remain small. Depending on the wave frequency, even strong environmental perturbations such as replacing a pressure-release surface with a rigid horizontal boundary, can have a weak effect on the shear-wave normal mode. The shear waves extend through the entire oceanatmosphere system, can be effectively excited by underwater sources, and experience much weaker viscous attenuation in the atmosphere than other acoustic-gravity waves of comparable wavelength. These attributes suggest that shear J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

acoustic-gravity waves may play a significant role in coupling wave processes in the ocean and the atmosphere. ACKNOWLEDGMENTS

This work was supported in part by the Office of Naval Research, grant N00014-13-1-0348. Stimulating discussions with B. Cornuelle, I. M. Fuks, L. A. Ostrovsky, and N. A. Zabotin are gratefully acknowledged. 1

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