Sound Waves in Fluids

Consider a uniform fluid (at rest) whose equilibrium density and pressure are $\rho$ and $p$, respectively. Suppose that a sound wave propagates though the fluid, and that the density and pressure perturbations produced by the wave are $\tilde{\rho}$ and $\tilde{p}$, respectively. Incidentally, the quantity $\tilde{p}$ is often referred to as the acoustic pressure. In two dimensions (i.e., neglecting any variation of perturbed quantities in the $y$-direction), the equations governing the propagation of a sound wave though a uniform fluid are (Landau and Lifshitz 1959)

$$\rho\,\frac{\partial v_x}{\partial t} = -\frac{\partial\tilde{p}}{\partial x},$$ (7.191)

$$\rho\,\frac{\partial v_z}{\partial t} = -\frac{\partial\tilde{p}}{\partial z},$$ (7.192)

$$\frac{\partial\tilde{p}}{\partial t} =\frac{K}{\rho}\,\frac{\partial\tilde{\rho}}{\partial t} = -K\left(\frac{\partial v_x}{\partial x}+ \frac{\partial v_z}{\partial z}\right).$$ (7.193)

(See Section 9.12 for a more complete discussion of fluid equations.) Here, ${\bf v} = (v_x,\,0,\,v_z)$ is the perturbation to the fluid velocity produced by the wave, and

$$K = \rho\,\frac{dp}{d\rho}$$ (7.194)

is the bulk modulus. The bulk modulus is a quantity with the units of pressure that measures a given substance's resistance to uniform compression. The bulk modulus of an ideal gas is $\gamma\,p$, where $\gamma$ is the ratio of specific heats. On the other hand, the bulk modulus of water at $20^\circ\,{\rm C}$ is $K=2.2\times 10^9\,{\rm N\,m}^{-2}$ (Wikipedia contributors 2018).

It is helpful to write

$$v_x = \frac{\partial\phi}{\partial x},$$ (7.195)

$$v_z =\frac{\partial\phi}{\partial z},$$ (7.196)

where $\phi(x,z,t)$ is conventionally referred to as a velocity potential. It follows from Equations (7.192) and (7.193) that

$$\tilde{p} = -\rho\,\frac{\partial \phi}{\partial t}.$$ (7.197)

Moreover, substitution into Equation (7.194) yields the wave equation

$$\frac{\partial^{\,2}\phi}{\partial t^{\,2}} = v^{\,2}\left(\frac{... ...\,2}\phi}{\partial x^{\,2}}+\frac{\partial^{\,2}\phi}{\partial z^{\,2}}\right),$$ (7.198)

where the characteristic wave speed is

$$v = \sqrt{\frac{K}{\rho}}.$$ (7.199)

For the case of an ideal gas, for which $K=\gamma\,p$, we obtain $v=\sqrt{\gamma\,p/\rho}$. (See Section 5.4.) On the other hand, for the case of water at $20^\circ\,{\rm C}$, for which $K=2.2\times 10^9\,{\rm N\,m}^{-2}$ and $\rho =1.0\times 10^3\,{\rm kg\,m}^{-3}$, we get $v = 1483\,{\rm m\,s}^{-1}$. This prediction is in good agreement with the measured sound speed in water at $20^\circ\,{\rm C}$, which is $1481\,{\rm m\,s}^{-1}$ (Wikipedia contributors 2018).

Forming the sum of $v_x$ times Equation (7.192), $v_z$ times Equation (7.193), and $K^{\,-1}\,\tilde{p}$ times Equation (7.194), we obtain

$$\frac{\partial {\cal E}}{\partial t} + \frac{\partial{\cal I}_x}{\partial x}+ \frac{\partial {\cal I}_z}{\partial z} = 0,$$ (7.200)

where

$${\cal E} = \frac{1}{2}\,\rho\,v_x^{\,2} + \frac{1}{2}\,\rho\,v_z^{\,2}+ \frac{1}{2}\,\frac{\tilde{p}^{\,2}}{K},$$ (7.201)

$${\cal I}_x = \tilde{p}\,v_x,$$ (7.202)

$${\cal I}_z = \tilde{p}\,v_z.$$ (7.203)

Equation (7.201) can be recognized as a two-dimensional energy conservation equation. (See Section 6.5.) Here, ${\cal E}$ is the acoustic energy density, and ${\cal I}_x$ and ${\cal I}_z$ are the acoustic energy fluxes in the $x$- and $z$-directions, respectively.

Consider a situation (analogous to that illustrated in Figure 7.7) in which a sound wave is incident at an interface between two uniform immiscible fluids. Let the region $z<0$ be occupied by a fluid of equilibrium density $\rho_1$ and sound speed $v_1$, and let the region $z>0$ be occupied by a fluid of equilibrium density $\rho_2$ and sound speed $v_2$. We can write the wavevectors of the incident, reflected, and refracted waves as

$${\bf k}_i = \frac{\omega}{v_1}\,(\sin\theta_i,\,0,\,\cos\theta_i),$$ (7.204)

$${\bf k}_r = \frac{\omega}{v_1}\,(\sin\theta_r,\,0,\,-\cos\theta_r),$$ (7.205)

$${\bf k}_t =\frac{\omega}{v_2}\,(\sin\theta_t,\,0,\,\cos\theta_t),$$ (7.206)

respectively. Here, for the sake of simplicity, we have assumed that all three wavevectors lie in the same plane (as is readily demonstrated; see Section 7.8.) Moreover, in order to be valid solutions of the wave equation, (7.199), all three waves must satisfy the dispersion relation $\omega=k\,v$, where $\omega$ is the common wave frequency. Finally, $\theta_i$, $\theta_r$, and $\theta_t$ are the angles of incidence, reflection, and refraction, respectively. (See Section 7.8.)

The velocity potential in the region $z<0$ is written

$$\phi(x,z,t) = \phi_i\,\cos[\omega\,(t-\sin\theta_i\,x/v_1-\cos\theta_i\,z/v_1)]$$

$$~~~~+\phi_r\,\cos[\omega\,(t-\sin\theta_r\,x/v_1+\cos\theta_r\,z/v_1)],$$ (7.207)

where the first and second terms on the right-hand side specify the incident and reflected waves, respectively. The velocity potential in the region $z>0$ takes the form

$$\phi(x,z,t) = \phi_t\,\cos[\omega\,(t-\sin\theta_t\,x/v_2-\cos\theta_t\,z/v_2)],$$ (7.208)

where the term on the right-hand side specifies the refracted wave. The first physical matching constraint that must be satisfied at the interface is continuity of the acoustic pressure; that is,

$$[\tilde{p}]_{z=0_-}^{z=0_+}=-\left[\rho\,\frac{\partial\phi}{\partial t}\right]_{z=0_-}^{z=0_+}=0.$$ (7.209)

This contraint yields

$$\begin{multline} \rho_1\,\omega\,\phi_i\,\sin[\omega\,(t-\sin\theta_i\,x/v_1)]+\... ...] =\rho_2\,\omega\,\phi_t\,\sin[\omega\,(t-\sin\theta_t\,x/v_2)]. \end{multline}$$

The previous equation holds at all values of $x$. This is only possible if

$$\sin\theta_i = \sin\theta_r$$ (7.210)

$$\frac{\sin\theta_i}{v_1} =\frac{\sin\theta_t}{v_2}.$$ (7.211)

These two expressions are analogous to the laws of reflection and refraction, respectively, of geometric optics. (See Section 7.8.) This suggests that these laws are of universal validity, rather than being restricted to light waves. Equation (7.211) reduces to

$$\rho_1\,(\phi_i+\phi_r) = \rho_2\,\phi_t.$$ (7.212)

The second physical matching constraint that must be satisfied at the interface is continuity of the normal velocity; that is,

$$[v_z]_{z=0_-}^{z=0_+} = \left[\frac{\partial\phi}{\partial z}\right]_{z=0_-}^{z=0_+} = 0.$$ (7.213)

This constraint yields

$$\tan\theta_t\,(\phi_i-\phi_r) = \tan\theta_i\,\phi_t,$$ (7.214)

where use has been made of Equation (7.213). Equations (7.214) and (7.216) can be combined to give

$$\phi_r = \left(\frac{\rho_2\,\tan\theta_t-\rho_1\,\tan\theta_i}{\rho_2\,\tan\theta_t+\rho_1\,\tan\theta_i}\right)\phi_i,$$ (7.215)

$$\phi_t = \left(\frac{2\,\rho_1\,\tan\theta_t}{\rho_2\,\tan\theta_t+\rho_1\,\tan\theta_i}\right)\phi_i.$$ (7.216)

Equations (7.197), (7.198), and (7.204) reveal that the mean acoustic energy fluxes, normal to the interface, associated with the incident, reflected, and refracted waves are

$$\langle {\cal I}_z\rangle_i = \frac{\rho_1\,\omega\,\cos\theta_i}{2\,v_1}\,\phi_i^{\,2},$$ (7.217)

$$\langle {\cal I}_z\rangle_r = -\frac{\rho_1\,\omega\,\cos\theta_i}{2\,v_1}\,\phi_r^{\,2},$$ (7.218)

$$\langle {\cal I}_z\rangle_t = \frac{\rho_2\,\omega\,\cos\theta_t}{2\,v_2}\,\phi_t^{\,2},$$ (7.219)

respectively. Thus, it follows that the coefficients of reflection and transmission at the interface are

$$R = \frac{-\langle{\cal I}_z\rangle_r}{\langle{\cal I}_z\rangle_i} ... ...ta_t-\rho_1\,\tan\theta_i}{\rho_2\,\tan\theta_t+\rho_1\,\tan\theta_i}\right)^2,$$ (7.220)

$$T = \frac{\langle{\cal I}_z\rangle_t}{\langle{\cal I}_z\rangle_i} =... ..._2\,\tan\theta_i\,\tan\theta_t}{(\rho_2\,\tan\theta_t+\rho_1\,\tan\theta_i)^2},$$ (7.221)

respectively. It is actually possible for there to be no reflection at the interface (i.e., $R=0$), provided that $\rho_2\,\tan\theta_t=\rho_1\,\tan\theta_i$. This criterion yields

$$\tan^2\theta_i = \frac{\rho_2^{\,2}\,v_2^{\,2}-\rho_1^{\,2}\,v_1^{\,2}}{\rho_1^{\,2}\,(v_1^{\,2}-v_2^{\,2})},$$ (7.222)

which can only be satisfied if $\rho_2\,v_2> \rho_1\,v_1$ and $v_1> v_2$, or if $\rho_2\,v_2< \rho_1\,v_1$ and $v_1< v_2$. The critical angle of incidence at which there is no reflection is sometimes called the angle of intromission. However, not every pair of immiscible fluids possesses such an angle.