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Chapter 6: Problem 1

Show that the sum of two traveling harmonic waves \(A_{1} \cos \left(\omega t-k z+\varphi_{1}\right)\) and \(A_{2} \cos \left(\omega t-k z+\varphi_{2}\right)\) traveling in the \(+z\) direction and having the same frequency \(\omega\) is itself a harmonic traveling wave of the same kind. That is, the sum can be written in the form \(A \cos (\omega t-k z+\varphi)\). Find out how \(A\) and \(\varphi\) are related to \(A_{1}, A_{2}\), \(\varphi_{1}\), and \(\varphi_{2} .\) (Hint: The use of complex numbers or a rotating vector diagram helps immensely.)

Short Answer

Expert verified
The resultant wave is another traveling wave with amplitude \(A = \sqrt{(A_{1}\cos \varphi_{1} + A_{2}\cos \varphi_{2})^2 + (A_{1}\sin \varphi_{1} + A_{2}\sin \varphi_{2})^2}\) and phase \(\Phi = \tan^{-1}((A_{1}\sin \varphi_{1} + A_{2}\sin \varphi_{2})/(A_{1}\cos \varphi_{1} + A_{2}\cos \varphi_{2}))\).

Step by step solution

01

Express Waves Using Complex Exponentials

The wave can be represented using Euler's formula: \[ A_{1} \cos(\omega t - kz + \varphi_{1}) = \text{Re}\{ A_{1} e^{i(\omega t - kz + \varphi_{1})} \} \] \[ A_{2} \cos(\omega t - kz + \varphi_{2}) = \text{Re}\{ A_{2} e^{i(\omega t - kz + \varphi_{2})} \} \] By expressing the cosine terms as complex exponentials, it becomes easier to find their sum.
02

Add the Complex Exponentials

Add the two complex exponentials obtained from the waves:\[A_{1} e^{i(\omega t - kz + \varphi_{1})} + A_{2} e^{i(\omega t - kz + \varphi_{2})}\] Using algebraic properties, this can be expressed as:\[ e^{i(\omega t - kz)} \left(A_{1} e^{i\varphi_{1}} + A_{2} e^{i\varphi_{2}}\right) \]
03

Simplify the Sum of Complex Phasors

The expression now consists of two parts: a traveling wave factor \(e^{i(\omega t - kz)}\) and a phasor sum \(A_{1} e^{i\varphi_{1}} + A_{2} e^{i\varphi_{2}}\). The phasor sum can be simplified using vector addition in the complex plane. Let:\[ r = A_{1} e^{i\varphi_{1}} + A_{2} e^{i\varphi_{2}} \]This phasor \(r\) can be expressed in magnitude and phase form as \( R e^{i\Phi} \).
04

Find Magnitude and Phase of the Resultant Wave

The magnitude \(R\) of the resultant phasor \(r\) is given by the square root of the sum of squares of the real and imaginary parts:\[ R = \sqrt{(A_{1} \cos \varphi_{1} + A_{2} \cos \varphi_{2})^2 + (A_{1} \sin \varphi_{1} + A_{2} \sin \varphi_{2})^2} \]The phase \(\Phi\) is given by:\[ \Phi = \tan^{-1}\left(\frac{A_{1} \sin \varphi_{1} + A_{2} \sin \varphi_{2}}{A_{1} \cos \varphi_{1} + A_{2} \cos \varphi_{2}}\right) \]
05

Write the Sum as a Single Harmonic Wave

The sum of the two original waves can now be written using the magnitude and phase obtained:\[ A \cos(\omega t - kz + \Phi) \]where \(A = R\) and \(\Phi\) is the new phase angle derived in Step 4. This confirms that the sum of the two traveling harmonic waves is another wave of the same form.

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Key Concepts

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

Harmonic Waves
Harmonic waves are fundamental types of waves that repeatedly occur at fixed intervals. These types of waves are typically represented in the form of a cosine or sine function, which illustrates their periodic nature. In mathematical terms, a harmonic wave moving in the positive z-direction can be denoted as \(A \cos(\omega t - kz + \varphi)\). Here, \(A\) represents the amplitude or peak value of the wave, \(\omega\) is the angular frequency, \(k\) is the wave number, and \(\varphi\) is the phase angle.
Harmonic waves are prevalent in various physical phenomena, such as sound waves, electromagnetic waves, and water waves. Understanding the representation of these waves is crucial as it enables the analysis of more complex wave patterns that often occur in nature.
In this exercise, we look at the addition of two harmonic waves having the same frequency and wave number but different amplitudes and phase angles. The goal is to show that their sum results in another harmonic wave of the same form.
Complex Exponentials
Complex exponentials are a powerful mathematical tool used to simplify the analysis of oscillatory phenomena, such as wave superposition. By expressing a cosine function as a complex exponential using Euler's formula, \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), we can handle oscillatory components algebraically, rather than geometrically.
For example, the wave \( A_{1} \cos(\omega t - kz + \varphi_{1}) \) can be expressed as the real part of \( A_{1} e^{i(\omega t - kz + \varphi_{1})} \).
This transformation makes it easier to add waves since the properties of exponents鈥攕uch as addition in the exponent when multiplying鈥攕implify calculations.
  • This conversion seems esoteric but allows for elegant solutions, particularly with superposition.
  • It offers a unified framework that aligns well with various fields, such as engineering and physics, for deeper insights into wave behavior.
Phasor Addition
Phasor addition involves the process of summing angles and vector representations of harmonic oscillators. When dealing with multiple waves, it can be cumbersome to sum all the trigonometric components directly. Instead, we use phasors, which are vectors in the complex plane, to represent the amplitudes and phases.
In a phasor diagram, each wave's phasor is drawn as an arrow whose length corresponds to the wave's amplitude (\(A\)) and direction (angle) indicates the phase (\(\varphi\)).
Adding waves then becomes a task of vector addition in the complex plane:
  • The phasors \(A_{1} e^{i\varphi_{1}}\) and \(A_{2} e^{i\varphi_{2}}\) are added together to yield a resultant phasor \(r = A_{1} e^{i\varphi_{1}} + A_{2} e^{i\varphi_{2}}\).
  • This resultant phasor can easily be converted back into term of amplitude and phase using its magnitude \(R\) and angle \(\Phi\).
Phasor addition not only makes the problem more manageable but also brings clarity to understanding how various wave components interact.
Resultant Amplitude and Phase
After expressing waves with complex exponentials and performing phasor addition, we can determine the resultant wave's amplitude and phase. The resultant amplitude \(A = R\) and phase \(\Phi\) are derived by considering both real and imaginary components of the resultant phasor.
For the resultant amplitude \(R\), use the formula: \[ R = \sqrt{(A_{1} \cos \varphi_{1} + A_{2} \cos \varphi_{2})^2 + (A_{1} \sin \varphi_{1} + A_{2} \sin \varphi_{2})^2} \]
The phase \(\Phi\) is found using the inverse tangent function: \[ \Phi = \tan^{-1}\left(\frac{A_{1} \sin \varphi_{1} + A_{2} \sin \varphi_{2}}{A_{1} \cos \varphi_{1} + A_{2} \cos \varphi_{2}}\right) \]
These expressions tell us precisely how two overlapping waves combine to form a new wave. This approach streamlines finding new wave parameters, critical for acoustics and optics.
  • Knowing \(A\) and \(\Phi\) allows for an accurate depiction of the combined wave's characteristics.
  • These resultant parameters highlight the collective effects of interference鈥攁 central theme in wave study.
Understanding the resultant amplitude and phase is essential for analyzing complex wave interactions in various scientific and engineering contexts.

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Most popular questions from this chapter

Show that any differentiable function \(f\left(t^{\prime}\right)\) of \(t^{\prime}=\) \(t-(z / v)\) satisfies the classical wave equation, i.e., show $$ \frac{\partial^{2} f\left(t^{\prime}\right)}{\partial t^{2}}=v^{2} \frac{\partial^{2} f\left(t^{\prime}\right)}{\partial z^{2}} $$ Show also that any differentiable function \(g\left(t^{\prime}\right)\) of \(t^{\prime \prime}=t+(z / v)\) satisfies the classical wave equation. Make up an example of a function \(f\left(t^{\prime}\right)\) and show explicitly that it satisfies the classical wave equation.

A frequency-modulated voltage can be written in the form (for example) $$ V=V_{0} \cos \left[\omega_{0}\left(1+a_{m} \cos \omega_{\text {nood }} t\right) t\right]=V_{0} \cos \omega t $$ with $$ \omega=\omega_{0}+\omega_{0} a_{m} \cos \omega_{\text {rnod }} t $$ One way of producing a frequency-modulated carrier wave to transmit music is by use of a "capacitative micropbone." The sound waves move a diaphragm which moves one plate of a capacitor. The capacitor then has capacitance (for example) $$ C=C_{0}\left(1+c_{m} \cos \omega_{\text {roo } 0} t\right) $$ Suppose this capacitance is part of an \(L C\) circuit with natural oscillation frequency \(\omega=\sqrt{1 / L C}\). The voltage across the capacitor is, for example, \(V=V_{0}\) cos wt. Show that for \(c_{m}\) small in magnitude compared with unity, one obtains a frequency-modulated voltage with amplitude \(a_{m}\) proportional to \(c_{m}\). Find the proportionality constant between \(c_{m}\) and \(a_{m}\).

Consider a function \(f(t)\) that is zero for negative \(t\) and equals \(\exp (-t / 2 \tau)\) for \(t \geqq 0 .\) Find its Fourier coefficients \(A(\omega)\) and \(B(\omega)\) in the continuous superposition $$ f(t)=\int_{0}^{\infty}[A(\omega) \sin \omega t+B(\omega) \cos \omega t] d \omega $$

Suppose \(f(t)\) is zero except in the interval from \(t=t_{1}\) to \(t=t_{2}\) of duration \(\Delta t=t_{2}-t_{1}\) and centered at \(t_{0}=\frac{1}{2}\left(t_{1}+t_{2}\right)\). Suppose that in this interval \(f(t)\) makes exactly one sinusoidal oscillation at angular frequency \(\omega_{0}\), starting and ending with value zero at \(t_{1}\) and \(t_{2}\left(\right.\) i.e., \(\left.\Delta t=T_{0}=2 \pi / \omega_{0}\right)\). Find the Fourier coefficients \(A(\omega)\) and \(B(\omega)\) in the continuous superposition $$ f(t)=\int_{0}^{\infty}\left[A(\omega) \sin \omega\left(t-t_{0}\right)+B(\omega) \cos \omega\left(t-t_{0}\right)\right] d \omega $$ Make a rough plot of the Fourier coefficients versus \(\omega\) and a sketch of \(f(t)\).

Show that for light of index \(n(\lambda)\), $$ \frac{1}{v_{g}}=\frac{1}{v_{\varphi}}-\frac{1}{c} \lambda \frac{d n(\lambda)}{d \bar{\lambda}} $$ where \(\lambda\) is the vacuum wavelength of the light.
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