A uniform elastic medium is characterized by a wave impedance \(Z\) and a wave velocity c. Waves can travel in the \(x\) direction according to the ordinary one-dimensional wave equation. Now suppose that the properties of the medium vary so that both \(Z\) and \(c\) are slowly varying functions of \(x\). (a) Show that the wave equation for the displacement becomes $$ \frac{\partial^{2} \psi}{\partial x^{2}}+\frac{d \ln [Z(x) c(x)]}{d x} \frac{\partial \psi}{\partial x}=\frac{1}{c^{2}(x)} \frac{\partial^{2} \psi}{\partial x^{2}} $$ (b) Assume according to the WKB method a traveling-wave solution of the form $$ \psi=A(x) e^{i[S(x)-\omega t]} $$ and show that $$ A^{2}(x) Z(x) c(x) \frac{d S(x)}{d x}=\text { const } $$ and that approximately $$ \left(\frac{d S}{d x}\right)^{2}=\frac{\omega^{2}}{c^{2}(x)}-\frac{1}{4}\left(\frac{d \ln Z}{d x}\right)^{2}-\frac{1}{2}\left(\frac{d \ln Z}{d x}\right)\left(\frac{d \ln c}{d x}\right) $$ (c) Apply this result to the exponential horn discussed in Sec. \(4.3\) and verif \(y\) that it gives the horn wave velocity (4.3.9).

Wave impedance is a fundamental concept in the study of waves, particularly in elastic media such as air or water. It is defined as the ratio of the pressure amplitude to the velocity amplitude of a wave and is represented mathematically as the product of the medium's density (\rho) and the wave velocity (c). In the context of the exercise, wave impedance, denoted as Z, is crucial because it characterizes how waves transmit through different media. The wave equation, tailored for variable impedance, takes the form:
\[\begin{equation}\frac{\partial^{2} \psi}{\partial x^{2}} + \frac{d \ln [Z(x) c(x)]}{d x} \frac{\partial \psi}{\partial x} = \frac{1}{c^{2}(x)} \frac{\partial^{2} \psi}{\partial t^{2}}\end{equation}\]
In this situation, the variability of Z means that the medium's properties are changing with respect to position. Therefore, waves will experience different impedances as they travel, affecting their propagation. The wave impedance is also involved in the WKB method, revealing how its variation influences the amplitude and phase of a wave solution, showcasing the liaison between wave mechanics and medium characteristics.

Wave velocity, often symbolized by c, is the speed at which a wavefront propagates through a medium. It significantly impacts how wave phenomena such as reflection, refraction, and diffraction occur in different environments. Sawtooth equation involving wave velocity enables us to predict the spread of waves in non-uniform media:
\[\begin{equation}\frac{\partial^{2} \psi}{\partial x^{2}} + \frac{d \ln [Z(x) c(x)]}{d x} \frac{\partial \psi}{\partial x} = \frac{1}{c^{2}(x)} \frac{\partial^{2} \psi}{\partial t^{2}}\end{equation}\]
When c varies with position, it indicates that the speed of wave propagation changes, which can lead to phenomena such as phase shifts or the bending of wave paths. In the exercise, variations in wave velocity are vital for understanding how waves advancian through changing media conditions and is integral to analyzing wave behavior with the WKB method, as seen in the relationship:
\[\begin{equation}\left(\frac{d S}{d x}\right)^2 = \frac{\omega^2}{c^2(x)} - \frac{1}{4}\left(\frac{d \ln Z}{d x}\right)^2 - \frac{1}{2}\left(\frac{d \ln Z}{d x}\right)\left(\frac{d \ln c}{d x}\right)\end{equation}\]
This resulting equation reflects how wave velocity's spatial variation influences the phase function S(x) within a traveling wave solution.

The Wentzel-Kramers-Brillouin (WKB) method provides a semoanalyticn solution to differential equations with a slowly varying parameter, widely used in quantum mechanics and wave theory. It approximates the behavior of a wave in a medium where the properties, such as impedance and wave velocity, change with location. By assuming a traveling-wave form:
\[\begin{equation}\psi=A(x) e^{i[S(x)-\omega t]}\end{equation}\]
The WKB method aids in finding expressions for the amplitude A(x) and the phase function S(x). The resulting equations from the exercise:
\[\begin{equation}A^2(x) Z(x) c(x) \frac{d S(x)}{d x} = \text{const}\end{equation}\]
\[\begin{equation}\left(\frac{d S}{d\ camera}\right)^2 = \frac{\omega^2}{c^2(x)} - \frac{1}{4}\left(\frac{d \ln Z}{d x}\right)^2 - \frac{1}{2}\left(\frac{d \ln Z}{d x}\right)\left(\frac{d \ln c}{d x}\right)\end{equation}\]
show that both the amplitude and the phase of the solution depend on the medium's characteristics. The constant term in the first equation implies that even as the wave's amplitude and the media's properties change, their combined effect on the wave's transmission remains consistent. The second equation suggests that the phase function, which determines where the peaks and troughs of the wave occur, is also affected by changes in Z and c. Understanding these relationships can help us predict wave behavior in complex situations like varying media or potential barriers and is an integral concept for advanced wave analysis.