Thermo-viscous damping of acoustic waves in narrow channels: A comparison of effects in air and water

Perforations in plates and structures have traditionally been used as a sound manipulation tool (Estrada et al., 2008; Leppington and Levine, 1973; Putra and Thompson, 2010) with a significant breadth of work having been carried out over the years, involving analytical, computational, and experimental approaches. This research has been performed mainly in order to study the influence of microstructure on transmission and reflection of incident acoustic energy. In recent years the term metasurface has been coined by the metamaterials community, as a microstructured surface that is capable of almost complete absorption of sound, usually in a very narrowband manner with some of the microstructure proposed being quite exotic (Jiménez et al., 2017; Jiménez et al., 2016). A classical related problem, still of great importance and studied with significant interest, is the problem of acoustic propagation through a screen with periodic slits or channels (Christensen et al., 2008). The canonical problem of transmission through a single channel, slit, or duct with rigid walls has been studied widely over the years (Allard and Atalla, 2009; Gomperts and Kihlman, 1967; Gompeters, 1964; Oldham and Zhao, 1993; Stinson and Champoux, 1992; Wilson and Soroka, 1965). It is well-known that resonances are set up within the channel, associated with relations between the wavelength and the length of the slit, save for what are known as end-corrections. Although the term has not classically been employed, more recently these resonances have been termed the Fabry-Perot resonances, terminology that appears to have crossed over from the electromagnetics community. Christensen et al. (2008) provided a model for the emergence of such resonances associated with an inviscid fluid. As pointed out by Ward et al. (2015), however, neglecting thermo-viscous effects is a significant assumption, particularly at small lengthscales and the latter authors showed that including attenuation in-air results in the phase speed along a narrow duct or channel being substantially reduced, even when the channel width is an order of magnitude greater than the boundary layer parameter $δ¯ν=ν¯/ω¯$⁠. Here $ν¯$ denotes kinematic viscosity and $ω¯$ is the radian frequency. In the present analysis, we show that Stokes's boundary layer thickness $δ¯s=2π2ν¯/ω¯≈9δ¯ν$ is a better indicator than $δ¯ν$ of the extent to which the channel-wall boundary layers disturb the motion of the fluid; $δ¯ν$ underestimates, somewhat, the true extent of the boundary layer.

More generally then, over the years it has been observed that thermo-viscous boundary layer effects can have a significant impact upon the propagation of sound along narrow, rigid-walled channels giving substantial acoustic absorption within the audio frequency range in air. This phenomenon is used routinely in the acoustics industry as a means to soften boundaries and attenuate unwanted sound. Indeed, the propagation of acoustic waves along channels, ducts, tubes, or slits has been studied extensively over the years, with work dating back to the late nineteenth century. Helmholtz (1863) studied the impact of viscosity and estimated the absorption due to viscous effects. Kundt (1868) tested the theory experimentally and noticed that absorption was higher than expected, presumably due to thermal effects. Kirchhoff (1868) introduced the general theory of thermo-visco acoustics based on the Navier-Stokes equations and the Fourier-law of heat conduction for a perfect gas in a circular tube. He also gave approximate expressions in the case of a wide tube, i.e., when the radius is much larger than the viscous/thermal boundary layer thickness. Rayleigh (1896) gave approximate solutions for the narrow tube limit, i.e., when the radius is much smaller than the viscous/thermal boundary layer thickness and for the two-dimensional narrow channel case. Zwikker and Kosten (1949) introduced an approximate model for wide tubes that allowed for the derivation of a simpler solution in which viscous and thermal effects decoupled. Weston (1953), based on Kirchhoff's model, developed approximations for tubes of various diameters (narrow, wide, and very wide). Tijdeman (1975) rewrote Kirchhoff's model in terms of non-dimensionalized parameters, which allowed him to estimate the relative importance of each term and to show that the two main factors involved are the reduced viscous wavenumber and the reduced acoustic wavenumber. Bruneau et al. (1989) presented a general framework for a perfect gas in a bounded domain; a general formula for the dispersion relation with a no-slip boundary condition for the velocity and an isothermal boundary condition for the temperature is obtained and applied to simple geometries. Stinson (1991) analyzed Kirchhoff's original model and showed that Zwikker and Kosten's solution is recovered in the appropriate limit. He also extended Kirchhoff's theory to tubes of arbitrary cross-section. Beltman (1999a) provided an extensive literature review on the various models and approximations developed in thermo-viscous acoustics for a perfect gas. All those models were compared and their domain of validity was assessed. Solutions for canonical problems for each approximate/exact model were also provided. In subsequent work, this theory was then applied to a variety of engineering problems, such as the spherical resonator, the classic circular tube, and a miniaturized acoustic transducer (Beltman, 1999b). The last model is of particular interest as it involved a fluid-structure interaction problem.

It is clear that most, if not all, the literature above is focused on a perfect gas. The study of thermo-viscous acoustic propagation in narrow water filled channels or slits appears to be lacking although work has been done on the viscous liquid-filled elastic tube, usually having relatively thin walls. This area of research was initiated by Del Grosso (1971) who considered multimode propagation in an inviscid fluid-filled elastic tube. It has been followed up recently with work that includes viscous effects, carried out in a number of papers (Baik et al., 2010; Dokumaci, 2014; Elvira-Segura, 2000; Liangh and Scarton, 2002). We note, however, that the frequency range considered in these works is generally high since the studies are mainly associated with non-destructive evaluation. The low frequency regime appears to be unexplored territory.

The purpose of the analysis presented in this paper is therefore two-fold, first to provide a general framework with which to study thermo-viscous acoustic attenuation in narrow channels that is applicable to an arbitrary Newtonian fluid. In particular, we wish to study the impact of thermo-viscous effects upon dense fluids of low compressibility, for which the perfect gas assumption is clearly inappropriate, and to determine the extent to which boundary layer influences are felt throughout the channel both in air and in water. Second, we wish to determine the conditions under which viscous and thermal boundary effects may influence acoustic propagation in water, when account is taken of the coupling between water and a real (elastic) channel wall material. (A pre-requisite to this study is the development of a theoretical framework for a general Newtonian fluid.) In the present study, we restrict our attention to the influence of thermo-viscous effects upon the lowest order symmetric duct mode, as only this mode would propagate along a narrow, rigid-walled tube filled with an inviscid fluid; future studies will consider higher-order modes.

In Sec. II we summarize the equations that arise when Beltman's analysis is extended to an arbitrary Newtonian fluid by using the linearized, quasi-equilibrium equations of state for a viscous fluid presented in Chapter 3 of Dunn et al. (2015), thereby enabling the study of the impact of thermo-viscous effects upon sound propagation in water. These equations are then applied to study the propagation of sound in narrow rigid-walled two-dimensional channels in Sec. III, where analytic expressions are derived for the dispersion equations of natural modes incorporating thermo-viscous effects. From this analysis, we are able to determine the conditions under which thermo-viscosity has a significant impact upon the propagation of guided acoustic waves by examining the phase-speed and attenuation of such waves along the channel for two different thermal boundary conditions: insulating and conducting. In particular, we show that the thermal expansion coefficient of the fluid has a significant impact upon the relative influence of viscous and thermal effects.

In Sec. IV we present results associated with propagation in air and in water along rigid-walled channels, and consider the different behavior of these two fluids. We show that their behaviors are captured by simple approximate analytic expressions for the complex phase-speed that are valid up to the MHz frequency range. We also compare our theoretical predictions to the in-air measurements of Ward et al. (2015) and find that our results are broadly consistent with their data, thereby confirming our analysis, which should be valid for any simple (Newtonian) fluid. Finally, in Sec. V, we consider acoustic coupling with compliant boundaries. As is known this is particularly important in the water-filled channel, unlike the air-filled case where all boundaries can, in general, be considered rigid. In particular, we show that for narrow, water-filled channels that have widths of the same order of magnitude as Stokes's boundary layer thickness, fluid-structure interaction has a dramatic impact upon the characteristics of the fundamental acoustic mode of propagation. Attenuation is increased by about 40 dB m^–1 compared to the rigid-walled assumption and the phase speed is slowed by almost 70%. We offer conclusions in Sec. VI.

Whilst variables and parameters employed herein are defined in the text, where they are first encountered, a notation summary is provided in the  Appendix for the convenience of the reader.

In this section, the equations governing linear thermo-viscous acoustics are presented in non-dimensional form. By combining these equations in the manner described by Beltman (1999a), for time-harmonic propagation, the vorticity field is shown to satisfy the Helmholtz equation and fractional temperature fluctuations of the fluid can be described by a pair of Helmholtz equations. From these solutions the velocity and pressure fields can be constructed. The study is restricted to two dimensions, which is appropriate for the models of interest.

Below we state the linearized governing equations for thermo-viscous fluids with dimensional variables and parameters having an over-bar and their non-dimensional counterparts being free of bars. In non-dimensional form then, these equations are (see, e.g., Chapter 3 of Dunn et al., 2015)

where boldface symbols are used to denote vector quantities. h denotes non-dimensionalized fluctuations of entropy per unit mass, whilst p is the non-dimensional acoustic pressure. They are defined by

s and θ are, respectively, the fractional fluctuations of density and temperature, i.e.,

In the above, $p¯, ρ¯, h¯$⁠, and $T¯$ denote, respectively, the fluctuating (dimensional) pressure, density, entropy, and absolute temperature of the fluid, with $p¯0, ρ¯0, h¯0$⁠, and $T¯0$ being their equilibrium values (taken to be spatially invariant); $c¯0$ is the barotropic sound speed of the fluid. Other relations between dimensional and non-dimensional variables are those for the fluid particle velocity $u¯=c¯0u=c¯0 (ux,uy,0)$⁠, Cartesian variables $(x¯,y¯)=L¯ (x,y)$ where $L¯$ is a problem specific length scale and time $t¯=(L¯/c¯0) t$⁠. Non-dimensional parameters are also introduced for convenience, in particular the first, or shear, coefficient of viscosity: $η¯=ρ¯0c¯0L¯ η$ and second coefficient of viscosity: $η¯′=ρ¯0c¯0L¯ η′$⁠, noting that $η¯$ and $η¯′$ are related to the commonly used bulk viscosity, $η¯B$⁠, through $η¯B=η¯′+(2/3)η¯$⁠. Furthermore, the thermodynamic coefficients of conductivity $K¯$⁠, specific heat at constant pressure $c¯p$⁠, and thermal expansion at constant pressure $β¯$ are non-dimensionalized as follows:

Equation (2) is associated with a non-dimensional viscous stress tensor, which, in index form, has an ijth component (see, e.g., Chapter 41 of Feynman et al., 1965)

where $u1≡ux, u2≡uy, x1≡x, x2≡y$⁠, δ_ij is the ijth component of the Kronecker-delta tensor and we employ the convention that repeated indices are summed over. The dimensional stress is $σ¯ij=ρ¯0c¯0 2σij$⁠.

Beltman (1999a) shows that the governing equations for an ideal gas decouple into a vortex field, $Ω$⁠, and two thermo-compression fields, Θ₁ and Θ₂. We find this field description to be valid in the more general case of an arbitrary Newtonian fluid whose governing equations were described in the previous section. The decomposition applied to an arbitrary Newtonian fluid within the frequency domain is given next as well as the relationships between $Ω, Θ1, Θ2$⁠, and other variables.

Since the problem here is considered two-dimensional, the vortex field can be written as $Ω=∇×u=k̂Ω$⁠, where $k̂$ indicates a unit vector pointing in the z-direction, and Ω is a scalar function. Ω satisfies the diffusion equation

and is decoupled from the thermal and compression field-variables. For harmonic excitation, $Ω(x,y,t)=Ω̃(x,y,ω)e−iωt$⁠, where $ω=ω¯L¯/c¯0$ is the non-dimensionalized frequency (or equivalently non-dimensionalized acoustic wavenumber), and Eq. (9) becomes the complex Helmholtz equation for $Ω̃$⁠, i.e.,

In terms of dimensional parameters,

where $δ¯ν$ is the boundary layer parameter of Ward et al. (2015) that was referred to in the Introduction.

Similarly, harmonic temperature fluctuations are written as $θ(x,y,t)=θ̃(x,y,ω)e−iωt$⁠, where the following decomposition is used for $θ̃$⁠:

$Θ̃1(x,y,ω)$ and $Θ̃2(x,y,ω)$ each satisfy a Helmholtz equation, viz.,

where

with

and $ζ=2η+η′=ηB+(4/3)η$⁠. $C$ represents the non-dimensionalized thermal diffusion coefficient. If in Eq. (14) and the first of Eq. (15), $A$ is replaced by the ratio of specific heats at constant pressure and volume, the solution for an ideal gas is recovered; see, for example, Stinson (1991) and Beltman (1999a).

It is readily shown for harmonic excitation, defining the velocity $u(x,y,t)=ũ(x,y,ω)e−iωt$⁠, condensation $s(x,y,t)=s̃(x,y,ω)e−iωt$ and pressure $p(x,y,t)=p̃(x,y,ω)e−iωt$⁠, that these harmonic variables can be defined in terms of $Ω̃,Θ̃1$ and $Θ̃2$ in the forms

Having summarized the governing equations and relations between dependent field variables, we now move on to defining the problem of interest.

In this section and with reference to Fig. 1, we examine the behavior of the natural modes of propagation along an infinitely-long parallel-sided channel, considering the impact of frequency, and channel width in particular, upon phase-speed and attenuation. The channel is aligned with the x-direction and lies in $|y¯|≤L¯$⁠. Its width in the z-direction is considered to be of infinite extent, and the motion of the fluid is considered to be independent of $z¯$⁠. Hence, the problem is two dimensional, governed by the equations summarized in Sec. II.

We start by setting the characteristic length scale, $L¯=L¯$⁠, i.e., equal to the half-width of the channel. Thus, in non-dimensional coordinates, the channel lies in $|y|≤1$ and is aligned along the x direction. Its walls are considered rigid so that

Solutions are sought with the dependence e^ikx, where k may be complex [with $ℑ(k)≥0$] and its permissible values are determined by the boundary conditions on the channel wall. Here we concentrate upon natural modes possessing a symmetric pressure distribution about the centre-line of the channel. This case is of particular interest because, for an inviscid fluid, only the lowest order symmetric mode is cut on at channel widths and frequencies of interest; all other modes are evanescent. For symmetric modes, we require $p̃(x,y,ω)$ to be an even function of y. This requires $s̃, θ̃$⁠, and $ũx$ to be even functions of y, whilst $Ω̃$ and $ũy$ are odd. Hence our basic k-space solutions for temperature and vorticity, which we denote by $θ̂(k,y,ω)=Θ̂1(k,y,ω)+Θ̂2(k,y,ω)$⁠, and $Ω̂(k,y,ω)$⁠, respectively, using the hat symbol to denote a k-space function, take the forms

where $γ1=(k2−κ12)1/2,γ2=(k2−κ22)1/2,α=(k2−iω/η)1/2$⁠. The functional forms of B, D, and E will clearly depend upon the thermal properties of the channel walls. We now consider two cases: thermally insulating walls and isothermal (perfectly conducting) walls.

In addition to Eq. (19), a thermally insulating channel wall must also satisfy

or its equivalent form in k-space. Imposing the boundary conditions in Eqs. (19) and (22) upon the basic solutions from Eqs. (20) and (21), noting Eq. (16), requires that for natural modes of propagation along the channel, k must satisfy the following dispersion equation:

Temperature fluctuations vanish on a perfectly conducting channel wall whose temperature is maintained at the equilibrium temperature of the fluid. Thus, the thermal boundary condition for this case is

with the corresponding dispersion equation for symmetric modes being

In Sec. IV below, we examine the numerical behavior of the dispersion equation for typical values of the thermo-viscous coefficients applicable to air and water. First, however, it is instructive to consider the limit of vanishingly small thermal expansion, which is of particular relevance to water.

It is clear from Eq. (4) that for a vanishingly small thermal expansion coefficient, pressure and density variations are decoupled from temperature fluctuations. That is as $β→0$⁠,

and the acoustic (compressional) channel modes become independent of thermal effects but may still be influenced by the viscous boundary layer. In this limit, we see from Eq. (15) that clearly $A=1+O(β2)$⁠, and $S→1−iω(ζ−C)+O(β2)$⁠, whence Eqs. (23) and (25) acquire the following common form:

Equation (27) is independent of γ₁ and hence κ₁, where in the limit $β→0$⁠, κ₁ depends only upon thermal parameters (and fluid density), indeed

whilst

which depends only upon viscous and acoustic parameters. Furthermore, as $β→0$⁠, Eqs. (16) to (18) acquire the following forms:

Clearly as $β→0, Θ̃2/β$ must remain bounded and $Θ̃2$ must vanish, if the physical variables $ũ, s̃$ and $p̃$ are to remain finite, whence from Eq. (12), $θ̃→Θ̃1$⁠. Finally, using Eq. (31) to eliminate $Θ̃2$ from Eq. (30), we see that

which, if β is set to zero at the outset, follows directly from Eqs. (1), (2), the first of Eq. (4), and the definition of Ω [given just prior to Eq. (9)]. And we note that $p̃$ now satisfies

In this limit, the visco-acoustic and thermal problems appear to decouple completely, which of course cannot be the case because viscous losses in the former problem cause thermal heating of the fluid. This anomaly arises because non-linear terms associated with viscous stresses were neglected in the energy/heat equation [Eq. (3)]. In most circumstances, this is perfectly acceptable, but in the limit $β→0$⁠, these non-linear terms become the dominant source term of the heat equation. Thus, in the limit $β→0$⁠, the visco-acoustic problem can be solved without reference to thermal effects and the viscous stresses from its solution lead to known, non-linear, source terms in the thermal problem. We note, however, that if non-linear effects are significant in the energy equation, it may be necessary to revisit the form of the equations of state; see, for example, Pierce (1978).

The theory is now implemented in the case of air and water filled channels. Relevant parameters are listed in Table I, taken from Dunn et al. (2015), and assumed independent of frequency.

In Sec. IV A, we evaluate the phase speed and attenuation of acoustic waves propagating along air-filled and water-filled channels, by finding the lowest order roots of the dispersion equations derived in Sec. III. The results are presented as a function of the channel width, $W¯=2L¯$⁠, relative to Stokes's boundary layer thickness, $δ¯s$⁠, where the latter is defined by

and $ν¯$⁠, the kinematic viscosity, is related to the dynamic viscosity, $η¯$⁠, through $ν¯=η¯/ρ¯0$⁠. The phase-speed values obtained for an air-filled channel are compared with the experimental data of Ward et al. (2015), noting that these data are parameterized against $δ¯ν/W¯$⁠, rather than $δ¯s/W¯$⁠, where

In Sec. IV B, we examine the fluid-particle-velocity profile across the channel for various channel widths in order to determine the extent to which the channel-wall boundary-layer influences the motion of the fluid and hence the propagation of acoustic waves.

Given the root, k, for any one of the dispersion Eqs. (23), (25), and (27), the complex non-dimensionalized phase-speed, v, along the channel is given by

Roots for the dispersion equations were calculated using the matlab [version 9.2.0.556344 (R2017a)] command fsolve, which finds the local zero of a user-specified function close to a given starting point.

Before presenting the results thereby obtained, we first note that in air and water both $ωζ$ and $ωC$ are very much less than unity at frequencies up to the order of 1 GHz. Under these circumstances, and provided $ω≪1$⁠, the following approximate expressions for the (dimensionless) complex phase-speed can be obtained from Eqs. (23) and (25):

and

where $δν=δ¯ν/L¯$ is Ward's (Ward et al., 2015) viscous boundary layer parameter, non-dimensionalized on the half-width $L¯$ of the duct; $δθ=δ¯θ/L¯$ is an equivalent thermal boundary layer parameter with $δ¯θ$ defined as

and Pr is the Prandtl number, which is given by

The expression for v_insul is independent of all thermal parameters and varies only with the dimensionless parameter $L¯/δ¯ν$⁠. However, v_cond is additionally dependent upon $Pr$ and the dimensionless parameter $A−1=(β¯2T¯0c¯0 2)/c¯p$⁠, noting that Eq. (38) recovers Stinson's solution for an ideal gas (Stinson, 1991) if $A$ is replaced by the ratio of specific heats at constant pressure and volume.

If the fundamental thermo-viscous parameters are frequency independent, as assumed here, we see that in the cases of thermally insulating and perfectly conducting channel-wall boundary conditions, the functional frequency dependence of their associated complex phase-speeds is contained entirely within the expressions for the boundary layer parameters $δ¯ν$ and $δ¯θ$⁠. Indeed, given the definition in Eq. (40) of Pr, it is sufficient to specify just one of the boundary layer parameters and the Prandtl number, which is frequency independent.

Although not illustrated herein, we find that for the range of channel-widths under consideration, Eqs. (37) and (38) are in almost exact agreement with values obtained using the full dispersion equations at all frequencies up to at least 1 MHz in air and 50 MHz in water. Furthermore, their validity is not confined to air and water. Indeed, these expressions are valid for any Newtonian fluid, provided the conditions specified in their derivation are satisfied, namely, $ωζ≪1, ωC≪1,$ and $ω≪1$⁠.

Returning to the solutions of the exact dispersion equations, we note from Eqs. (37) and (38) that as $L¯,W¯→∞$⁠, both v_insul and v_cond tend to unity, which is equivalent to $k→ω$⁠. Thus, an appropriate method for finding their roots, using fsolve, is to begin with a wide channel (we chose $W¯=100δ¯s$⁠), and to start the root search from k = ω. We then gradually reduce $W¯$ down to the desired value, using the root found for the previous (larger) value of $W¯$ as a new starting point for fsolve.

By way of example, Fig. 2 shows the resulting real phase-speed and attenuation along an air-filled channel as a function of $δ¯s/W¯$⁠. Here, real phase-speed is defined as $ℜ(v)$⁠, and the attenuation in dB/wavelength is $40π(ki/kr) log10(e)$⁠, where $kr=ℜ(k)$ and $ki=ℑ(k)$⁠. The curves for viscous only and thermally insulating channel-wall boundary conditions are identical, indicating that for a thermally insulating boundary condition, the lowest order symmetric duct mode is decoupled from thermal fluctuations as expected. However, a different behavior is observed for the conducting curve shown in Fig. 2, again as expected. We observe that the results of Ward et al. (2015), indicted by the black crosses of Fig. 2, lie in-between the two extreme thermal boundary conditions of conducting and insulating channel walls. Although not presented here, we find that the behavior in air, illustrated in Fig. 2, is valid at all frequencies up to the order of 1 MHz.

Figure 3 demonstrates the behavior of a water-filled channel, which is found to be valid at all frequencies below a maximum value approaching 50 MHz. The behavior of water is very similar to that of air, except that for water, thermal effects are seen to be negligible for both the thermally conducting (isothermal) and insulating channel-wall boundary conditions, primarily because of water's low coefficient of thermal expansion (as discussed previously). For water this leads to the parameter $A$ being very close to unity, $A≈1.001$ in water, whilst in air $A≈1.39$⁠. In addition, Pr in water is about ten times greater than in air. Taken together, these two factors lead to the second term in the denominator of Eq. (38) being negligible for water, thereby recovering Eq. (37).

In summary, we see that in water, thermal effects have a negligible impact upon acoustic propagation along narrow channels, and this is also true in air if the channel wall is thermally insulating, which is perhaps not surprising as this boundary condition is close to the adiabatic thermal condition expected in freely propagating acoustic waves. However, in air, thermal effects are significant when the channel wall is conducting. For the thermally insulating channel wall boundary condition, the behavior of air and water with respect to their dimensionless phase speed and attenuation along the channel in dB/wavelength is essentially identical when parameterized against the dimensionless parameter $δ¯s/W¯$⁠. Note, however, that the attenuation along a fixed distance, say 1 m, is much less in water than in air due to the higher dimensional phase speed of water, leading to a wavelength in water that is about five times greater than in air. As an example in this case of rigid boundaries where the channel width is of the order of the boundary layer thickness, the attenuation in air at 10 kHz can be over 200 dB m⁻¹, whereas in water it is less than 37 dB m⁻¹.

Similar behavior to that discussed above is observed for the fluid particle velocity profiles across the channel. Below, we plot the x- and y-components of fluid particle velocity, normalized so that $ux(y=0)=1$⁠. For fluids such as air and water, which are characterized by low viscous and diffusion coefficients, we find that for both the insulating and isothermal channel-wall boundary conditions, $ux(y)/ux(0)$ depends only upon the non-dimensionalized viscous boundary layer parameter $δν=δ¯ν/L¯$⁠; there is no dependence upon the thermal properties of the fluid. For air, this is illustrated in the upper plot of Fig. 4 for a channel-width of $10δ¯s$⁠. Both the magnitude and the phase (the latter is not shown) of $ux(y)/ux(0)$ have the same values for all three channel wall boundary conditions. The same behavior (not shown) is observed in water.

For the y-component of fluid-particle velocity, we find that in the case of a thermally insulating channel-wall, the scaled parameter $uy(y)/(η ux(0))$ depends upon only $δν$⁠. However, when the channel wall is perfectly conducting, $uy(y)/(η ux(0))$ depends, in general, also upon the additional parameters $δθ=δ¯θ/L¯$ and $A−1=β2/cp$⁠, or alternatively its behavior may be characterized in terms of $δν$⁠, the Prandtl number Pr, and $β2/cp$⁠. For air, this behavior is illustrated in the lower plot of Fig. 4 for a channel-width of $10δ¯s$⁠. We see that whilst the viscous only and thermally insulating velocity profiles are identical, the profile for a conducting wall is different, demonstrating the influence of thermal effects in this case. Although not shown here, in water, the three thermal boundary conditions have identical velocity profiles as a result of water's low coefficient of expansion.

In the absence of boundary layer effects, the y-component of fluid particle velocity would be zero everywhere, and the x-component would be constant across the channel; that is u_x would be independent of y. The red and blue, vertical, dashed lines marked on the upper plot of Fig. 4 indicate, respectively, the positions of the non-dimensionalized boundary layer thickness parameters, $δ¯ν/L¯$ and $δ¯s/L¯$⁠, relative to the channel walls on $y=±1$⁠. The black crosses show Stokes's solution for an oscillating pressure gradient within a fluid-filled half-space lying above a rigid surface, under the assumption that the boundary layers on the two walls of the channel are well separated and hence non-interacting (Batchelor, 2000). For this relatively wide channel, the fluid-particle-velocity profile for u_x is very similar to that of Stokes's half-space solution, and we observe that $δ¯s$ is a significantly better indicator than $δ¯ν$⁠, of the extent to which the channel-wall boundary layers disturb the motion of the fluid. $δ¯ν$ underestimates the true extent of the boundary layer but as we have seen above it is a very useful parameter with which to characterize the fluid's behavior when it is influenced by boundary layer effects.

Figure 5 shows the velocity profile for a much narrower channel, $W¯=δ¯s/2$⁠, in which the boundary layers on the two walls are strongly interacting. Stokes's solution for a fluid half-space is no longer relevant; rather the velocity profile resembles that of Poiseuille flow, as indicated by the red crosses. Using $δ¯ν/L¯$ as boundary layer thickness can be seen in Fig. 5 to be severely underestimating this length.

The large difference in the magnitude of the y-component of fluid particle velocity, which is observed in Figs. 4 and 5, arises mainly because the expression for $uy(y)/(η ux(0))$ contains a scaling factor of $1/δν$⁠.

Until now, all boundaries have been considered rigid, which is a sensible approximation in the air-filled channel but not necessarily so in the case of a water-filled channel. In order to investigate the effect of elastic boundaries, and in particular their impact upon phase-speed and attenuation in this case, let us consider a water-filled channel in an undamped elastic medium of infinite extent; the only damping mechanism considered is the viscosity of the fluid. The presence of semi-infinite elastic walls requires the consideration of body waves, both compressional and shear, propagating within the elastic material. Generally, this necessitates the mathematical formulation of a physical problem with some specified forcing, the solution of which will be expressed in terms of integrals around branch-cuts that are then associated with outgoing and incoming body waves and additionally, a sum of the natural modes of propagation in the waveguide. However, such an analysis is beyond the scope of the present study; we are primarily concerned with demonstrating the impact of an elastic boundary on the natural modes of propagation within the water-filled channel, and comparing this behavior to the idealized case of a rigid boundary. In order to proceed here then we must merely ensure that the choice of branch cuts is consistent with the requirements of causality. This is discussed further below.

We shall neglect all thermal effects in this analysis since as shown earlier these are negligible in the case of acoustic propagation in water (due to the low value of the thermal expansion coefficient). Recalling that as $β→0, s→p$⁠, the (time-harmonic) fluid velocity here is therefore written compactly in terms of the condensation and vorticity, i.e.,

where $s̃$ is governed by

The components of the total (time-harmonic) fluid stress tensor are

Next, referring back to Fig. 1, instead of imposing the rigid no-slip conditions [Eq. (19)] on the boundary, we now suppose that the medium in $|y|≥1$ is a linear isotropic elastic medium of infinite extent. Using a consistent non-dimensionalization scheme to that defined for the fluid with time-harmonic dependence $e−iωt$⁠, Navier's equations of linear elasticity may be written as

where λ is the Lamé modulus and μ the shear modulus, both being non-dimensionalized on $ρ¯0c¯0 2$⁠. Furthermore, $ρs=ρ¯s/ρ¯0$ denotes the non-dimensionalized solid density and $w=(wx,wy,0)=w¯/L¯$ is the non-dimensionalized elastic displacement, with a tilde denoting its time-harmonic counterpart. Furthermore, the following potentials have been introduced, $ϕ̃=∇·w̃$⁠, and $ψ̃=∇×w̃$⁠. As with the fluid vorticity, the rotation vector $ψ̃$ points in the z-direction and may be written $ψ̃=k̂ψ̃$⁠. The scalar potentials $ϕ̃$ and $ψ̃$ satisfy the Helmholtz equations

where κ_p and κ_s are the non-dimensionalized compressional and shear wavenumbers

The elastic displacement is written simply using Eq. (44) in the form

which is the analogous form, in terms of potentials, to the fluid velocity expression in Eq. (41). Finally, the (time-harmonic) elastic stress tensor is given by

As with the case of a rigid-walled channel, we seek solutions in k-space, which again we denote by the ^̂ symbol. For the fluid-structure interaction problem, it is convenient to relate the x- and k-space potentials of the elastic solid as follows:

with $ϕ̂$ and $ψ̂$ satisfying

where

The choice of cuts for γ_p and γ_s are discussed shortly. For the fluid, we write

with $ŝ$ and $Ω̂$ satisfying

where $γ2=( k2−ω2/(1−iωζ) )1/2$ and we recall that $α=(k2−iω/η)1/2$⁠.

Boundary conditions require continuity of displacement/velocity at the fluid-solid interface and continuity of the traction components across the channel wall. In k-space, these conditions require, for the velocity

and for the traction

noting that $w̃(x,y,ω)=ŵ(k,y,ω)eikx, ũ(x,y,ω)=û(k,y,ω)eikx$ and similarly for the stress components.

By inserting Eqs. (50) and (53) into Eqs. (41), (43), (48), and (49), as appropriate, and imposing the boundary conditions from Eqs. (55)–(57), we obtain the following constraints upon our solutions at y = ±1:

where

Note that $ω2/μ(=κs2/ρs=ρ¯0ω¯2L¯2/μ¯)$ and $P$ are measures of the fluid loading acting on the elastic wall, i.e., they are measures of the fluid's inertial and viscous forces relative to the elastic stresses at the channel-wall. Clearly they are important non-dimensional parameters that will influence significantly the characteristics of our solution, although we note below that in the air and water cases, as well as a wide range of other scenarios $P≪1$ and can be set to zero in order to simplify the dispersion relation. Other important non-dimensional parameters (most of which have been defined previously) include

With these definitions, the other non-dimensional parameters that arise can be written as

where $υ$ denotes Poisson's ratio.

As for the rigid-walled channel, only modes where s is symmetric in y are considered here. In this problem (that is independent of thermal effects), the k-space vorticity and condensation then take the form

For the solid, appropriate symmetric solutions to Eqs. (51) and (52) are similarly given by

in which the square root functions of Eq. (52) are chosen such that $γp→k$ and $γs→k$ as $k→∞$⁠, with the branch cuts from κ_p and κ_s taken in the upper-half plane and those from −κ_p and −κ_s taken in the lower-half plane. To be definitive in the calculations below, the branch cuts are chosen to run parallel to the imaginary axis from their respective branch points.

The dispersion equation in k for the natural modes of propagation is obtained by substituting the solution forms from Eqs. (64)–(66) into the boundary conditions from Eqs. (58)–(60), evaluated on y = ±1. Requiring the determinant of the resulting set of simultaneous equations, in the unknown coefficients E, F, X, and Y, to be zero then leads to the following dispersion equation:

where $D1=(2k2−κs2)2−4k2γsγp=0$ is Rayleigh's dispersion equation for natural modes on the surface of a stress-free elastic half-space, and $D2=k2−γsγp$ is the dispersion equation for the natural modes on the surface of a clamped elastic half-space. Furthermore, $Dvr$ is the dispersion equation [Eq. (27)] for the natural modes of propagation along a rigid-walled channel filled with a viscous fluid in which thermal effects are negligible. We note that in the limit of $μ,ρs→∞$ with μ/ρ_s = constant, Eq. (67) reduces to $Dvr=0$⁠.

Let us now consider the implementation of the above in the specific case of a fluid-filled channel in steel with density $ρ¯s=7871 kg m−3$⁠, and compressional and shear wave-speed $v¯p=6000 m s−1$ and $v¯s=3000 m s−1$⁠, respectively. These parameter values imply a shear modulus $μ¯=70.839 GPa$ and a Lamé modulus $λ¯=141.68 GPa$⁠.