Hooke's Law for Plane Waves

Figure 3: An infinitesimal section of an enclosed fluid with volumetric dilation.

• Consider the infinitesimal section of fluid MNOP with cross-sectional area $A$ diagrammed in Fig. 3. Under the influence of a pressure change $p$, the section is displaced to points M$^{\prime}$N$^{\prime}$O$^{\prime}$P$^{\prime}$, undergoing a volume dilation in the process.

• Assuming section MNOP has a volume $Q$, the volume of M$^{\prime}$N$^{\prime}$O$^{\prime}$P$^{\prime}$ is given by
    $$Q + dQ = A dx \left(1 + \ensuremath{\frac{\partial {\xi}}{\partial {x}}}\right).$$ (3)

• For small dilation, Hooke's law for fluids provides an accurate approximation to the relationship between the change in applied pressure, the resulting strain, and the bulk modulus $K$,
    $$p = -K \, \frac{dQ}{Q} = - K \ensuremath{\frac{\partial {\xi}}{\partial {x}}}.$$ (4)

• The temperature within sound waves tends to rise where the air is compressed and fall where it is expanded. Because the wavelengths of sound are generally large, the pressure maxima and minima are thus spread far apart and the temperature gradients are too small to produce significant heat conduction within the medium. This behaviour is referred to as an adiabatic process.

• The adiabatic bulk modulus for sound waves can be expressed in terms of the ratio of specific heats of air at constant pressure and constant volume $\gamma = C_p/C_v = 1.4$ and the average atmospheric pressure $P_0$, or the density of air $\rho$ and the speed of sound $c$ as $K_{ad} = \gamma P_{0} = \rho c^{2}$.

• The speed of sound does not vary with atmospheric pressure but it does vary with temperature. For air at temperature $\Delta T$ in degrees Celsius,
    $$c \approx 332 (1 + 0.00166 \Delta T)\, \textrm{m s}^{-1}.$$ (5)