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Chapter 5: Problem 1

Supply convincing symmetry arguments establishing that the reflected and refracted plane waves at a plane interface have vector wave numbers lying in the plane of incidence. Why must all three waves have the same frequency?

Short Answer

Expert verified
## Question: Demonstrate that the reflected and refracted plane waves have vector wave numbers in the plane of incidence using symmetry arguments. Explain why all three waves must have the same frequency. ## Answer: The vector wave numbers of reflected and refracted plane waves lie within the plane of incidence due to the symmetry properties of the medium and the boundary conditions. The normal component of the wave number vector remains unchanged during reflection and refraction, ensuring that the wave number vector stays within the plane of incidence. Additionally, the frequency of all three waves (incident, reflected, and refracted) must be conserved to maintain energy conservation throughout the reflection and refraction process.

Step by step solution

01

Understanding the Problem

To solve this problem, it is essential to understand the concepts of reflection, refraction, plane waves, and the plane of incidence. The plane of incidence is the plane containing both the incident and reflected rays, while the waves' reflection and refraction occur due to the change in the medium. Plane waves are propagating waves where the wavefront moves with a constant phase velocity.
02

Reviewing the Symmetry Arguments

The symmetry arguments rely on the idea that the medium's properties are symmetric at the plane interface. These symmetry properties imply that the waves' behavior on both sides of the interface is equivalent. Therefore, the reflected and refracted waves should have similar characteristics, including the direction of their vector wave numbers.
03

Establishing Wave Number Vectors in the Plane of Incidence

As the plane of incidence contains both the incident and reflected rays, we can argue that the normal vector to this plane (perpendicular to the plane) does not change during reflection and refraction. When a wave is reflected or refracted, it must maintain its perpendicular component, the wave number vector's normal component, to satisfy the boundary conditions. This normal component remains unchanged during reflection and refraction, ensuring that the wave number vector lies within the plane of incidence.
04

Showcasing the Symmetry Argument

When a plane wave reaches the interface, it interacts with the medium and gets reflected and refracted. Since all three waves (incident, reflected, and refracted) lie within the plane of incidence, we can apply the symmetry argument and deduce that their wave number vectors must also lie within the plane of incidence. If the wave number vectors were not in the plane of incidence, then the symmetry properties and boundary conditions would not be satisfied.
05

Explaining the Same Frequency for All Three Waves

The waves' frequency must be conserved during reflection and refraction, as the frequency is a property of the wave that depends on its source and not on the medium. The conservation of energy dictates that the total energy before and after the interaction (reflection and refraction) must remain the same; in other words, the sum of the energies of the reflected and refracted waves must equal the incident wave's energy. As energy is proportional to the square of the amplitude and frequency, the frequency must remain constant for all three waves to maintain energy conservation throughout the interaction.

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Most popular questions from this chapter

By changing variables to spherical coordinates \(r, \theta, \phi\), such that \(x=r \sin \theta \cos \phi\), \(y=r \sin \theta \sin \phi, z=r \cos \theta\), show that the laplacian of \(p\) becomes $$ \nabla^{2} p=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial p}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial p}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} p}{\partial \phi^{2}} $$

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Formula (5.7.10) does not include the standing waves for which one or two of the mode numbers \(l, m, n\) are zero. Establish the more accurate formulat $$ N=\frac{\omega_{\max }^{3}}{6 \pi^{2} c_{f}^{2}} V+\frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A+\frac{\omega_{\max }}{16 \pi c_{f}} L $$ where \(V=a b c, A=2(a b+a c+b c)\), and \(L=4(a+b+c)\) †P. M. Morse,"Vibration and Sound," \(2 \mathrm{~d}\) ed, chap. 8, McGraw-Hill Book Company, New York, \(1948 .\) ± See Morse, op. cit., p. 394 .
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