Sound Waves in Fluids

The Nature of Longitudinal Disturbances

Imagine a row of balls connected by springs. If you push one ball, it compresses the spring to its neighbour, which then pushes the next ball. The disturbance travels down the line, but the individual balls only move back and forth along the line of propagation. This is a longitudinal wave. In a fluid, there are no shear forces to support transverse motion, so sound must be longitudinal. The "spring" in this analogy is the compressibility of the fluid, and the "mass" is the inertia of the fluid elements. When a region of fluid is compressed, its pressure increases. This high-pressure region exerts a force on the adjacent lower-pressure region, accelerating it. As the fluid moves, it expands, lowering its pressure. This interplay between pressure gradients and fluid inertia creates a travelling wave of compression and rarefaction.

Deriving the Wave Equation for Fluids

To derive the wave equation, we consider a small fluid element of length \(\Delta x\) and cross-sectional area \(A\). Let \(s(x,t)\) be the displacement of the fluid element from its equilibrium position. The change in length of the element is \(\Delta s = \partial s / \partial x \cdot \Delta x\). The fractional change in volume, which is the strain, is therefore \(\partial s / \partial x\). For small displacements, the change in density \(\Delta \rho\) is related to the strain by \(\Delta \rho = -\rho_0 \frac{\partial s}{\partial x}\), where \(\rho_0\) is the equilibrium density. The negative sign indicates that compression (negative strain, \(\partial s/\partial x < 0\)) leads to an increase in density.

Newton's second law applied to the fluid element gives the equation of motion. The net force on the element is the difference in pressure forces at its two ends: \(F_{\text{net}} = -A \frac{\partial P}{\partial x} \Delta x\). The mass of the element is \(m = \rho_0 A \Delta x\). Thus, \(m \frac{\partial^2 s}{\partial t^2} = -A \frac{\partial P}{\partial x} \Delta x\), which simplifies to \(\rho_0 \frac{\partial^2 s}{\partial t^2} = -\frac{\partial P}{\partial x}\). To close this system, we need a relationship between pressure and density. For sound waves, the process is rapid enough that there is no time for heat exchange, making it adiabatic. The adiabatic equation of state is \(P = K \rho^\gamma\), where \(\gamma = C_P/C_V\) is the ratio of specific heats. Linearising this around equilibrium values \(P_0\) and \(\rho_0\), we find that the pressure variation \(\Delta P\) is proportional to the density variation \(\Delta \rho\): \(\Delta P = \frac{\gamma P_0}{\rho_0} \Delta \rho\). Substituting the expressions for \(\Delta \rho\) and \(\Delta P\) into the equation of motion yields the standard wave equation for the displacement \(s(x,t)\).

where the wave speed \(c\) is given by \(c = \sqrt{\frac{\gamma P_0}{\rho_0}}\). Using the ideal gas law \(P_0 = \frac{\rho_0 R T}{M}\), where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass, we can express the speed of sound in terms of temperature and molecular weight.

Pressure and Density Waves

While the displacement \(s(x,t)\) satisfies the wave equation, it is often more convenient to work with pressure \(P(x,t)\) or density \(\rho(x,t)\). Since \(\Delta P\) is proportional to \(\Delta \rho\), and \(\Delta \rho\) is proportional to \(\partial s / \partial x\), the pressure and density variations also satisfy the wave equation. For a harmonic wave \(s(x,t) = S_0 \cos(kx - \omega t)\), the pressure variation is \(P'(x,t) = \rho_0 c \omega S_0 \sin(kx - \omega t)\). Note that the pressure wave is \(90^\circ\) out of phase with the displacement wave. At points of maximum compression or rarefaction (where \(|\partial s / \partial x|\) is largest), the displacement is zero. This is a crucial distinction: where the displacement is zero the pressure variation is greatest, and where the displacement is greatest the pressure variation is zero. The amplitude of the pressure wave is related to the displacement amplitude by \(P_0' = \rho_0 c \omega S_0\). This relationship shows that for a given displacement amplitude, the pressure amplitude increases with frequency and density.

Speed of Sound in Different Media

The formula \(c = \sqrt{\gamma P_0 / \rho_0}\) provides a powerful tool for estimating the speed of sound in various fluids. In gases, \(P_0/\rho_0\) is proportional to temperature, so the speed of sound depends primarily on temperature and the molecular composition (via \(\gamma\) and \(M\)). In air at \(20^\circ\mathrm{C}\) (\(293\ \mathrm{K}\)), with \(\gamma \approx 1.4\) and \(M \approx 0.029\ \mathrm{kg/mol}\), the speed is approximately \(343\ \mathrm{m/s}\). In liquids, the bulk modulus \(K\) replaces \(\gamma P_0\), and since liquids are much less compressible than gases, \(K\) is much larger. Although the density of water is about \(800\) times that of air, its bulk modulus (about \(2.2\ \mathrm{GPa}\)) is roughly \(1.6 \times 10^{4}\) times larger than \(\gamma P_0 \approx 0.14\ \mathrm{MPa}\) for air, resulting in a speed of sound in water of about \(1480\ \mathrm{m/s}\). In solids, the speed of sound is even higher due to the high elastic moduli, although the derivation for solids is more complex due to the presence of shear waves.