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Mobile App Chapter 14: Problem 7
If \(v_{p}\) is the phase velocity of a guided wave and \(v_{s}\) its group velocity, show that \(v_{p} v_{k}=c^{2}\)
Short Answer
Expert verified
Question: Demonstrate the relationship between phase velocity (v_p), group velocity (v_s), and the speed of light (c) for a guided wave.
Answer: For a guided wave, the relationship between phase velocity (v_p), group velocity (v_s), and the speed of light (c) is given by the equation \(v_p v_s = c^2\).
Step by step solution
01
Define phase velocity and group velocity
Phase velocity \((v_p)\) is defined as the ratio of the angular frequency \((ω)\) to the wavenumber \((k_p)\): \(v_p = \dfrac{ω}{k_p}\). Group velocity \((v_s)\) is defined as the derivative of the angular frequency with respect to the wavenumber: \(v_s = \dfrac{dω}{dk_s}\).
02
Express angular frequency in terms of wavenumber
We can relate the angular frequency and wavenumber using the dispersion relation of a guided wave. For simplicity, let's consider the dispersion relation of a waveguide to be: \(ω^2 = c^2 k^2\). Here, \(ω\) is the angular frequency and \(k\) is the total wavenumber (which includes both the \(k_p\) and \(k_s\) components).
03
Solve for phase velocity
To find an expression for \(v_p\), we need to express the angular frequency in terms of \(k_p\). From the dispersion relation, we have \(ω = c k\). Using the expression for phase velocity, \(v_p = \dfrac{ω}{k_p} = \dfrac{c k}{k_p}\).
04
Solve for group velocity
To find an expression for \(v_s\), we need to differentiate the angular frequency with respect to \(k_s\). From the dispersion relation, \(ω = c k\), we have \(\dfrac{dω}{dk_s} = c \dfrac{dk}{dk_s}\). We know that \(k = k_p + k_s\), so \(\dfrac{dk}{dk_s} = 1\). Thus, \(v_s = \dfrac{dω}{dk_s} = c\).
05
Multiply phase velocity and group velocity
Now we multiply the expressions for phase and group velocities: \(v_p v_s = \left(\dfrac{c k}{k_p}\right) (c) = \dfrac{c^2 k}{k_p}\).
06
Express wavenumber in terms of phase and group velocities
From the dispersion relation and the expression for phase velocity, we know that \(ω = c k\) and \(v_p = \dfrac{c k}{k_p}\). Rearranging the phase velocity expression gives us \(k = v_p k_p\).
07
Substitute wavenumber expression into the product of phase and group velocities
In step 5, we found that \(v_p v_s = \dfrac{c^2 k}{k_p}\). Now substitute the expression we found for \(k\) in step 6: \(v_p v_s = \dfrac{c^2 (v_p k_p)}{k_p} = c^2\).
Therefore, we have successfully shown that for a guided wave, \(v_p v_s = c^2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Phase Velocity
Phase velocity is a measure of how fast a single wave phase, such as a crest, travels through a medium. It's defined by the equation:
\( v_p = \frac{\omega}{k_p} \)
where \( \omega \) is the angular frequency and \( k_p \) is the phase wavenumber. Importantly, the phase velocity can provide insights into the nature of the wave, and under certain conditions, it may exceed the speed of light without contradicting the principles of relativity; an occurrence not possible for the transmission of information or energy.
\( v_p = \frac{\omega}{k_p} \)
where \( \omega \) is the angular frequency and \( k_p \) is the phase wavenumber. Importantly, the phase velocity can provide insights into the nature of the wave, and under certain conditions, it may exceed the speed of light without contradicting the principles of relativity; an occurrence not possible for the transmission of information or energy.
Group Velocity
Group velocity relates to the propagation speed of the envelope of a group of waves and is denoted by:
\( v_s = \frac{d\omega}{dk_s} \)
It represents the speed at which energy or information travels through a medium. Group velocity is especially significant in dispersive media where different frequencies travel at different speeds, leading to the spread of wave packets over time.
\( v_s = \frac{d\omega}{dk_s} \)
It represents the speed at which energy or information travels through a medium. Group velocity is especially significant in dispersive media where different frequencies travel at different speeds, leading to the spread of wave packets over time.
Dispersion Relation
The dispersion relation is a fundamental concept in wave physics, linking the angular frequency \( \omega \) of a wave to its wavenumber \( k \) through a functional relationship. For certain media, this relation can be expressed as
\( \omega^2 = c^2 k^2 \)
The form of the dispersion relation indicates whether a medium is dispersive or non-dispersive. It affects how the phase and group velocities relate to one another, as well as how wave packets evolve over time.
\( \omega^2 = c^2 k^2 \)
The form of the dispersion relation indicates whether a medium is dispersive or non-dispersive. It affects how the phase and group velocities relate to one another, as well as how wave packets evolve over time.
Waveguide
A waveguide is a structure that confines and directs the propagation of waves, such as electromagnetic or sound waves. Characterized by its ability to support guided waves, the waveguide enforces boundary conditions that affect the dispersion relation of the waves it carries. These conditions determine the possible modes of propagation and, consequently, the wave's phase and group velocities within the guide.
Angular Frequency
Angular frequency, denoted by \( \omega \), is a measure of how rapidly a wave oscillates in time, typically measured in radians per second. It's linked to the more common concept of frequency, \( f \), by the relationship
\( \omega = 2\pi f \).
In wave equations, angular frequency is crucial in describing both phase and group velocity and in defining the behavior of waves through their dispersion relations.
\( \omega = 2\pi f \).
In wave equations, angular frequency is crucial in describing both phase and group velocity and in defining the behavior of waves through their dispersion relations.
Wavenumber
Wavenumber, represented by \( k \), quantifies the spatial frequency of a wave, indicating the number of wave cycles per unit of distance. In mathematical terms,
\( k = \frac{2\pi}{\lambda} \),
with \( \lambda \) being the wavelength. Wavenumber plays an integral role in dispersion relations and is key to expressing both phase and group velocities.
\( k = \frac{2\pi}{\lambda} \),
with \( \lambda \) being the wavelength. Wavenumber plays an integral role in dispersion relations and is key to expressing both phase and group velocities.
Guided Waves
Guided waves are waves that are confined within a boundary, such as those within a waveguide. These waves' behavior, including their velocity and mode patterns, are determined by the geometry of the guide and the material properties. Guided waves exhibit dispersion, meaning their phase and group velocities are typically functions of frequency, echoing the intimate relationship between dispersion relations, phase velocity, and group velocity.
