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Chapter 5: Problem 20

General sinusoidal wave. Write the traveling wave \(\psi(\pi, t)=A \cos (\omega t-k z)\) as a superposition of two standing waves. Write the standing wave \(\psi(z, t)=A \cos \omega t\) \(\cos k z\) as a superposition of two traveling waves traveling in opposite directions. Consider the following superposition of traveling waves: $$ \psi\langle z, t)=A \cos (\omega t-k z)+R A \cos (\omega t+k z) $$ Show that this sinusoidal wave can be written as a superposition of standing waves given by $$ \psi(z, t)=A(1+R) \cos \omega t \cos k z+\Lambda(1-R) \sin \omega t \sin k z $$ Thus the same wave can be thought of as a superposition either of standing waves or of traveling waves.

Short Answer

Expert verified
The wave can be expressed as both a superposition of standing and traveling waves.

Step by step solution

01

Examine the traveling wave

The given traveling wave is \( \psi(\pi, t) = A \cos(\omega t - k z) \). This represents a wave moving in the positive \( z \)-direction.
02

Expressing Traveling Wave as a Superposition of Standing Waves

Using the identity \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), we rewrite the traveling wave as:\[ \psi(z, t) = A (\cos \omega t \cdot \cos kz + \sin \omega t \cdot \sin kz) \]This expression represents a superposition of two standing waves: one involving the product of two cosine terms and the other involving the product of two sine terms.
03

Examine the Standing Wave

The given standing wave is \( \psi(z, t) = A \cos \omega t \cos k z \). This is a part of the expression derived in the previous step, and represents a wave that oscillates in time but remains stationary in space.
04

Expressing Standing Wave as a Superposition of Traveling Waves

Using the identity \( \cos a \cos b = \frac{1}{2}(\cos(a+b) + \cos(a-b)) \), we rewrite the standing wave as:\[ \psi(z, t) = \frac{A}{2} [ \cos(\omega t + kz) + \cos(\omega t - kz) ] \]This expression represents a superposition of two traveling waves moving in opposite directions.
05

Analyze Superposition of Traveling Waves

Consider the superposition \( \psi(z, t) = A \cos(\omega t - kz) + R A \cos(\omega t + kz) \). Using the identity for cosine,\( \cos a + R \cos b = (1+R) \cos(\frac{a+b}{2}) \cos(\frac{a-b}{2}) + (1-R) \sin(\frac{a+b}{2}) \sin(\frac{a-b}{2}) \),we split it into components:\[ \psi(z, t) = A(1+R) \cos \omega t \cos kz + A(1-R) \sin \omega t \sin kz \]
06

Establish Equivalence of Superpositions

The expressions obtained show that any sinusoidal wave can be expressed either as a superposition of traveling waves or as a superposition of standing waves:\[ \psi(z, t) = A(1+R) \cos \omega t \cos kz + A(1-R) \sin \omega t \sin kz \]It confirms that both viewpoints provide equivalent descriptions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Traveling Waves
Traveling waves are fundamental in understanding wave motion. They represent waves that carry energy through space or a medium. A standard representation of a traveling wave is given by the function \[ \psi(z, t) = A \cos(\omega t - kz) \] where:
  • \( A \) is the amplitude, representing the wave's maximum value.
  • \( \omega \) is the angular frequency, related to how quickly the wave oscillates in time.
  • \( k \) is the wave number, indicating the number of wave cycles per unit distance.
The expression \( \omega t - kz \) represents the phase of the wave. As time passes, the wave moves in the positive \( z \)-direction. This is the characteristic of traveling waves: they transport energy from one location to another.
To express a traveling wave as a combination (or superposition) of standing waves, one uses trigonometric identities. This approach reveals the underlying properties of wave interactions, key for more complex wave systems like quantum mechanics.
Standing Waves
Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. They appear stationary and do not transport energy in space, as each point on the wave oscillates in place.
A typical expression for a standing wave is:\[ \psi(z, t) = A \cos \omega t \cos kz \]In this representation:
  • \( \cos \omega t \) represents the time-dependent oscillation.
  • \( \cos kz \) captures the spatial variation, with "nodes" at points where the wave has zero amplitude. These nodes and "antinodes" (points of maximum amplitude) are characteristic of standing waves.
To understand standing waves as combinations of traveling waves, the trigonometric identity:\[ \cos a \cos b = \frac{1}{2}(\cos(a+b) + \cos(a-b)) \] shows that stationary waveforms result from the interaction of waves moving in opposite directions. This perspective helps in fields like acoustics and optics, where understanding wave superposition is crucial.
Trigonometric Identities
Trigonometric identities are mathematical tools that simplify complex wave equations. They are essential for transforming expressions of traveling waves into standing waves and vice versa.
A key identity used in wave problems is:
  • \( \cos(a - b) = \cos a \cos b + \sin a \sin b \) - This identity helps in expressing a single traveling wave as the sum of standing waves.
  • \( \cos a \cos b = \frac{1}{2}(\cos(a+b) + \cos(a-b)) \) - It shows how standing waves can be viewed as superpositions of two traveling waves moving in opposite directions.
These identities are not just theoretical; they have practical applications in physics and engineering. They aid in analyzing wave behavior in various settings, from musical instruments to signal processing.

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Most popular questions from this chapter

Multiple internal refiection in a microscope slide. Make a sketch showing a ray coming in from the left and hitting a slab of glass tilted at some angle. Show the first transmitted ray, the second (i.e., that transmitted after two internal reflections), the third, .... Now look at a point or line souree through a microseope slide. Hold the slide close to your eye. Starting at normal incidence, gradually tilt the slide. Look for the "virtual sources" due to multiple reflections. (The effect is greater near grazing incidence.) Look also for the light that emerges, not by transmission out of the surface of the slide, bat from the end. This is the "internally trapped" light, which finally escapes when it reaches the end surface at near-normal incidence rather than at the near-grazing incidence at which it encounters the sides of the slide.

Nonreflecting coating. A glass lens has been coated with a nonreflecting coating that is one quarter-wavelength in thickness tn the coating for light of cactum wavelength \(\lambda_{0}\). The index of refraction of the coating is \(\sqrt{n}\); that of the glass is \(n\). Take the index of refraction to be constant, independent of frequency, over the visible frequency spectrum. Let \(I_{\text {ref }}\) denote the time-averaged reflected intensity and \(I_{0}\) the incident intensity, for light at normal incidence. Show that the fractional reflected intensity has the following dependence on the wavelength of the incident light: $$ \frac{I_{\text {rot }}}{I_{0}}=4\left[\frac{1-\sqrt{n}}{1+\sqrt{n}}\right]^{2} \sin ^{2} \frac{1}{2} \pi\left(\frac{\lambda_{0}}{\lambda}-1\right) $$ where \(\lambda\) is the vacuum wavelength of the incident light. Take \(n=1.5\) for glass. Suppose \(\lambda_{0}=5500 \mathrm{~A}\) (green light), Then \(I_{\mathrm{ret}}\) is zero for green. What is \(I_{\mathrm{ret}} / I_{0}\) for blue light of vacuum wavelength \(4500 \mathrm{~A}\) ? What is it for red light of vacuum wavelength \(6500 \AA ?\)

Compare the amplitude and intensity reflection coefficients for light normally incident on a smooth water surface (index \(n=1.33\) ) for the two cases of incidence from air to water and from water to air.

Continuity of a wave at a boundary. For light (or other electromagnetic radiation) incident from medium 1 to medium 2, we found that, provided the magnetic permeability of the medium is unity (or does not change at the discontinuity) and provided the "geometry" is constant (parallel-plate transmission line of constant cross-sectional shape or slab of material in free space), then the reflection and transmission coefficients for the electric field \(E_{x}\) and magnetic field \(B_{v}\) are given by $$ \begin{array}{ll} R_{E}=\frac{k_{1}-k_{2}}{k_{1}+k_{2}}, & T_{E}=1+R_{E}=\frac{2 k_{1}}{k_{1}+k_{2}} \\ R_{B}=\frac{k_{2}-k_{1}}{k_{2}+k_{1}}, \quad T_{B}=1+R_{B}=\frac{2 k_{2}}{k_{2}+k_{1}} \end{array} $$ where \(k=n \omega / c\) and \(n\) is the index of refraction. Show that the reflection and transmission coefficients for \(E_{z}\) imply that \(E_{x}\) and \(\partial E_{z} / \partial z\) are both continuous at the discontinuity, i.e., that they have the same instantaneous values on either side of the discontinuity. (By the field on the left side (medium 1) we mean, of course, the superpasition of the incident and reflected waves.) Similarly, show that the reflection and transmission coefficients for the magnetic fleld \(B_{y}\) imply that \(B_{v}\) is continuous at the boundary but that \(\partial B_{y} / \partial z\) is not continuous. Show that \(\partial B_{v} / \partial z\) increases by a factor \(\left(k_{2} / k_{1}\right)^{2}=\left(n_{2} / n_{1}\right)^{2}\) in crossing from medium 1 to medium 2. It is important to notice that we mean the total field, not just the part traveling in a particular direction.

Termination of waves on a string. (a) Suppose you have a massless dashpot having two moving parts 1 and 2 that can move relative to one another along the \(x\) direction, which is transverse to the string direction \(z\). Friction is provided by a fluid that retards the relative motion of the two moving parts. The friction is such that the force needed to maintain relative velocity \(\dot{x}_{1}-\dot{x}_{2}\) between the two moving parts is \(Z_{d}\left(\dot{x}_{1}-\dot{x}_{2}\right)\), where \(Z_{i} d\) is the impedance of the dashpot. The input (part 1 ) is connected to the end of a string of impedance \(Z_{1}\) stretching from \(z=-\infty\) to \(z=0 .\) The output (part 2 ) is connected to a string of impedance \(Z_{2}\) that extends to \(z=+\infty\). Show that a wave incident from the left experiences an impedance at \(z=0\) which is the same as that it would experience if connected to a "load" consisting of a string stretching from \(z=0\) to \(+\infty\) and having impedance \(Z_{L}\) given by $$ \mathrm{Z}_{L}=\frac{\mathrm{Z}_{\mathrm{d}} \mathrm{Z}_{2}}{\mathrm{Z}_{d}+\mathrm{Z}_{2}}, \quad \text { that is, } \quad \frac{1}{\mathrm{Z}_{L}}=\frac{1}{\mathrm{Z}_{d}}+\frac{1}{\mathrm{Z}_{2}} \text { . } $$ Thus it is as if the dashpot and string 2 were impedances connected "in parallel" and driven by the incident wave. (b) Show that if string \(Z_{2}\) extends only to \(z=\frac{1}{4} \lambda_{2}\), where \(\lambda_{2}\) is the wavelength in medium 2 (assuming we have a harmonic wave with a single frequency), and there is terminated by a dashpot of zero impedance (frictionless), the wave incident at \(z=0\) is perfectly terminated. Show that the output connection of the dashpot at \(z=0\) cannot tell whether it is connected to a string of infinite impedance or is instead connected to a quarter-wavelength string that is "short-circuited" by a frictionless dashpot at \(z=\frac{1}{4} \lambda_{2 \cdot}\) In either ease the output connection remains at rest.
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