Subfield: Optics

Seismic waves in a stratified medium 

Suppose that a short earthquake happens at the surface of some planet. The seismic waves can be assumed to originate from a line source situated at $z=x=0$, where $x$ is the horizontal coordinate and $z$ is the depth below the surface (Fig. 3). The seismic wave source can be assumed to be much longer than any other length considered in this question.

As a result of the earthquake, a uniform flux of the so-called longitudinal $P$ waves is emitted along all the directions in the $x-z$ plane that have positive component along the $z$-axis. Since the wave theory in a solid is generally complicated, in this problem we neglect all the other waves emitted by the earthquake. The crust of the planet is stratified so that the P-wave speed $v$ depends on depth $z$ according to $v=v_{0}\left(1+z / z_{0}\right)$, where $v_{0}$ is the speed at the surface and $z_{0}$ is a known positive constant.

<img_4472>

Figure 3. Coordinate system used in part B.

The effects of the surface curvature of the planets can be neglected. You might need to use the formula

$$
(1+x)^{\varepsilon} \approx 1+\varepsilon x \text {, when }|x| \ll 1 \text {. }
$$
Context question:
B.1 Consider a single ray emitted by the earthquake that makes an initial angle $0<\theta_{0}<\pi / 2$ with the $z$-axis and travels in the $x-z$ plane. What is the horizontal coordinate $x_{1}\left(\theta_{0}\right) \neq 0$ at which this ray can be detected at the surface of the planet? It is known that the ray path is an arc of a circle. Write your answer in the form $x_{1}\left(\theta_{0}\right)=A \cot \left(b \theta_{0}\right)$, where $A$ and $b$ are constants to be found.
Context answer:
\boxed{$x_{1}(\theta_{0})=2 z_{0} \cot \theta_{0}$}


Extra Supplementary Reading Materials:

If you were unable to find $A$ and $b$, in the following questions you can use the result $x_{1}\left(\theta_{0}\right)=A \cot \left(b \theta_{0}\right)$ as given. Suppose that total energy per unit length of the source released as $\mathrm{P}$ waves into the crust during the earthquake is $E$. Assume that waves are completely absorbed when they reach the surface of the planet from below.

Question: B.2 Find how the energy density per unit area $\varepsilon(x)$ absorbed by the surface depends on the distance along the surface $x$.

Solution:
B.2
In two dimensions, $\frac{E}{\pi} \mathrm{d} \theta_{0}$ is the energy carried by rays that are emitted within interval $\left[\theta_{0}, \theta_{0}+\mathrm{d} \theta_{0}\right)$. On the other hand, the energy carried by rays that arrive at $[x, x+\mathrm{d} x)$ is $\varepsilon \mathrm{d} x$. Therefore,

$$
\varepsilon=\frac{E}{\pi}\left|\frac{\mathrm{d} \theta_{0}}{\mathrm{~d} x}\right|
$$

Using the result of question B.1,

$$
\frac{\mathrm{d} x}{\mathrm{~d} \theta_{0}}=-\frac{A b}{\sin ^{2}\left(b \theta_{0}\right)}=-A b\left(1+\cot ^{2}\left(b \theta_{0}\right)\right)=-\frac{b\left(A^{2}+x^{2}\right)}{A} .
$$

Hence,

$$
\varepsilon(x)=\frac{E A}{\pi b\left(A^{2}+x^{2}\right)}=\frac{2 E z_{0}}{\pi\left(4 z_{0}^{2}+x^{2}\right)}
$$


$\varepsilon(x)=\frac{E A}{\pi b\left(A^{2}+x^{2}\right)}=\frac{2 E z_{0}}{\pi\left(4 z_{0}^{2}+x^{2}\right)}$.

Final answer: $\varepsilon(x)=\frac{2 E z_{0}}{\pi(4 z_{0}^{2}+x^{2})}$