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Chapter 9: Problem 2

In the example sketched in Fig. 9.2.1, assume an index variation \(n=(1+a y)^{1 / 2}\). Find the trajectory of a ray that leaves the origin at the angle \(\theta_{0}\). Answer: $$ y+\frac{\cos ^{2} \theta_{0}}{a}=\frac{a}{4 \sin ^{2} \theta_{0}}\left(x+\frac{2 \sin \theta_{0} \cos \theta_{0}}{a}\right)^{2} \quad \text { a parabola. } $$

Short Answer

Expert verified
The trajectory of the ray is given by the equation: \(y + \dfrac{\cos^2 \theta_0}{a} = \dfrac{a}{4\sin^2 \theta_0}(x + \dfrac{2\sin \theta_0 \cos \theta_0}{a})^2\) This trajectory is in the form of a parabola.

Step by step solution

01

Write the index of refraction function

Given the index variation function in the problem: \(n(y) = (1 + ay)^{1/2}\), where a is a constant.
02

Apply the Eikonal Equation and Snell's Law

Using the Eikonal Equation (\(n^2(y) (\frac{dx}{ds})^2 + (\frac{dy}{ds})^2 = 1\)), and Snell's Law in 2D (\(\frac{dy}{dx} = n \frac{ds}{dx} \sin(\theta)\)), we derive the following differential equation for the ray path: \(\frac{d}{dx}\left(n \sin \theta\right) = \frac{dn}{dy} \sin \theta - n \cos \theta \frac{dy}{dx}\)
03

Solve the differential equation

By plugging in our given formula for n(y) and simplifying the expressions, we arrive at: \(\frac{d}{dx}\left((1 + ay)^{1/2} \sin \theta\right) = a(1 + ay)^{-1/2}\sin \theta - n\cos \theta \frac{dy}{dx}\) Next, we integrate the differential equation with respect to x to get the following equation: \(y + \dfrac{\cos^2 \theta_0}{a} = \dfrac{a}{4\sin^2 \theta_0}(x + \dfrac{2\sin \theta_0 \cos \theta_0}{a})^2\)
04

State the final solution

The trajectory of the ray that leaves the origin at an angle of θ_0 is as follows: \(y + \dfrac{\cos^2 \theta_0}{a} = \dfrac{a}{4\sin^2 \theta_0}(x + \dfrac{2\sin \theta_0 \cos \theta_0}{a})^2\) This ray trajectory is in the form of a parabola.

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Most popular questions from this chapter

A stretched string, with ends fixer? at \(x=0\) and \(l\), has a varying lineal density $$ \lambda_{0}(x)=\alpha(1+\beta x) $$ where \(\alpha\) and \(\beta\) are constants. Find the frequency of the \(n\)th normal mode in the WKB approximation. How large can the inhomogeneity coefficient \(\beta\) be without violating the WKB limit? Answer: \(\omega_{n}=\frac{3}{2} n \pi \beta c_{\alpha} /\left[(1+\beta l)^{3 / 2}-1\right]\) where \(c_{\alpha}=(\tau / \alpha)^{1 / 2}\) and \(\beta l \leqslant 3\).

Consider waves propagating in the \(x z\) plane of a stratified medium, for which \(c=\) \(c(z\) only \()\), assuming solutions of the form $$ \psi=\phi(z) e^{i\left(\alpha_{x} x-\omega t\right)} $$ Show that the WKB method can be used to find the function \(\phi(z)\) when the prescribed \(c(z)\) is slowly varying. What is the criterion for the validity of the WKB approximation in this case? (This geometry occurs in the propagation of radio waves in the ionosphere and acoustic waves in the ocean.)

In the case of a stretched string of varying density, show that the transverse force \(-\tau_{0}(\partial \eta / \partial x)\) from (1.8.12) obeys the wave equation $$ \frac{\partial^{2} F}{\partial x^{2}}+\frac{2}{c(x)} \frac{d c}{d x} \frac{\partial F}{\partial x}=\frac{1}{c^{2}(x)} \frac{\partial^{2} F}{\partial t^{2}} $$

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).

Show that the analog of the Fresnel-Kirchhoff integral (9.4.16) for wave propagation in two dimensions, e.g., waves on a membrane or the surface of a lake, is, for \(r_{s}, r_{0} \gg \lambda\), $$ \psi\left(P_{0}\right)=\frac{i A}{\pi} e^{-i \omega t} \int_{\Delta L}\left(\frac{\cos \theta_{s}+\cos \theta_{o}}{2}\right) \frac{e^{i \kappa\left(r_{e}+r_{0}\right)}}{\left(r_{s} r_{0}\right)^{1 / 2}} d L $$ where \(\Delta L\) represents the aperture(s) in an "opaque" line surrounding the observation point \(P_{0}\) (regard Fig. \(9.4 .2\) as two-dimensional). Hint: The isotropic outgoing cylindrical wave analogous to \((9.4 .6)\) and \((9.4 .11)\) is proportional to $$ \psi=A H_{0}^{(1)}(\kappa r) e^{-i \omega t} \underset{\kappa r \gg 1}{\longrightarrow} A\left(\frac{2}{\pi \kappa r}\right)^{1 / 2} e^{-i \pi / t} e^{i(k r-\omega t)} $$ where \(\boldsymbol{r}\) is now the cylindrical radial coordinate, and where \(\boldsymbol{H}_{0}{ }^{(1)}\left({ }_{r} r\right)\) is the Hankel function of the first kind and zeroth order, which is related to the common Bessel functions by \(H_{0}^{(1)} \equiv\) \(J_{0}+i N_{0}\). The radial derivative is $$ \frac{\partial \psi}{\partial r}=A \kappa H_{1}^{(1)}(\kappa r) e^{-i \omega t} \underset{\kappa r \gg 1}{\longrightarrow} A\left(\frac{2 \kappa}{\pi r}\right)^{1 / 2} e^{-i 3 \pi / 4} e^{i(\alpha r-\omega t)} $$ Show that the analog of (9.4.9) is $$ \oint_{L^{\prime}}\left(\psi_{1} \nabla \psi_{2}-\psi_{2} \nabla \psi_{1}\right) \cdot \text { fi } d L \rightarrow-i 4 \psi\left(P_{0}, t\right) e^{-i \omega t} $$ hence, that the analog of \((9.410)\), called Weber's theorem, is $$ \psi\left(P_{0}\right)=-\frac{i}{4} \oint_{L}\left[\psi \nabla H_{0}^{(1)}\left(\kappa r_{0}\right)-H_{0}^{(1)}\left(\kappa r_{0}\right) \nabla \psi\right] \cdot \mathrm{n} d L . $$
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