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Chapter 4: Problem 1

Consider a medium in which \(c_{\text {phase }}=A \omega^{n}\), where \(A\) and \(n\) are constants. Show that \(c_{\text {eroup }}=c_{\text {pasae }} /(1-n)\). For what values of \(n\) is the dispersion normal? Anomalous?

Short Answer

Expert verified
Question: Determine the relationship between the group velocity and phase velocity of a medium, and find the values of the constant 'n' for which the dispersion is normal and anomalous, given that the phase velocity is \(c_{\text {phase }}=A \omega^{n}\). Answer: The relationship between the group velocity and phase velocity is given by \(c_{\text {group }} = \frac{c_{\text {phase}}}{1 - n}\). For normal dispersion, \(n < 1\), and for anomalous dispersion, \(n > 1\).

Step by step solution

01

Derive the phase velocity with respect to angular frequency

We are given the phase velocity \(c_{\text {phase }}=A \omega^{n}\). To find the group velocity, we need to derive this expression with respect to angular frequency \(\omega\). So, we have: \(\frac{dc_{\text {phase }}}{d\omega} = \frac{d(A \omega^{n})}{d\omega} = A n \omega^{n-1}\)
02

Find the group velocity

The group velocity, denoted by \(c_{\text {group }}\), is given by: \(c_{\text {group }} = \frac{d \omega}{d k}\) We know that the phase velocity is given by \(c_{\text {phase }} = \frac{\omega}{k}\), so we can write: \(\omega = c_{\text {phase }} k = A \omega^{n} k\) Now, derive this equation with respect to \(k\): \(\frac{d\omega}{dk} = A n \omega^{n-1} \frac{d\omega}{d k} + A \omega^{n}\) Now, we can solve for the group velocity \(c_{\text {group }}\): \(c_{\text {group }} = \frac{d \omega}{d k} = \frac{A \omega^{n}}{1 - n}\)
03

Find the relationship between group and phase velocities

Now, we want to express the group velocity \(c_{\text {group }}\) in terms of the phase velocity \(c_{\text {phase }}\). We know: \(c_{\text {phase }} = A \omega^{n}\) So, we get: \(c_{\text {group }} = \frac{c_{\text {pasae}}}{1 - n}\)
04

Determine the values of \(n\) for normal and anomalous dispersion

Now that we have the relationship between the group and phase velocities, we can analyze the values of \(n\) for normal and anomalous dispersion. Normal dispersion occurs when the group velocity and phase velocity have the same signs, i.e., \(c_{\text {group }}c_{\text {phase }} > 0\) Anomalous dispersion occurs when the group velocity and phase velocity have opposite signs, i.e., \(c_{\text {group }}c_{\text {phase }} < 0\) From our relationship, we have: \(c_{\text {group }} = \frac{c_{\text {pasae}}}{1 - n}\) Since \(c_{\text {phase }} > 0\), \(n\) must satisfy: \(1 - n > 0 \Leftrightarrow n < 1\) for normal dispersion, and \(1 - n < 0 \Leftrightarrow n > 1\) for anomalous dispersion.

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