Chapter 1: Problem 2
Show how the solution \((1.5 .5)\) of the wave equation can be transformed into a general traveling-wave solution.
Chapter 1: Problem 2
Show how the solution \((1.5 .5)\) of the wave equation can be transformed into a general traveling-wave solution.
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Get started for freeTwo transverse symmetrical sawtooth waves, each having the form at \(t=0\) shown in the figure are traveling in opposite directions on a stretched string. Investigate the resulting disturbance and plot the wave pattern at several times in the time interval \(0 \leq t \leq l / c\). Does the pattern in the range \(0 \leq x \leq l\) correspond to Fig. \(1.6 .2\), which is a plot of \(\eta(x, l)\) as given by (1.6.9)?
A uniform string of linear mass density \(\lambda_{0}\) and under a tension \(\tau_{0}\) has a small bead of mass \(m\) attached to it at \(x=0\). Find expressions for the complex amplitude and the power reflection and transmission coefficients for sinusoidal waves brought about by the mass discontinuity at the origin. Do these coefficients hold for a wave of arbitrary shape?
Three long identical strings of linear mass density \(\lambda_{0}\) are joinzd together at a common point forming a symmetrical Y. Thus they lie in a plane \(120^{\circ}\) apart. Each is given the same tension \(\tau_{0 .}\) A distant source of sinusoidal waves sends transverse waves, with motion perpendicular to the plane of the strings, down one of the strings. Find the reflection and transmission coefficients that characterize the junction.
Show that energy flow and total energy are related by the continuity equation (see Prob. 1.8.4) $$ \frac{\partial P}{\partial x}=-\frac{\partial E_{1}}{\partial t} $$ by multiplying each side of the wave equation \((1,11,6)\) by \(\partial_{\eta} / \partial l\) and carrying out a development analogous to that leading to \((1.11 .15)\). Using \((1.11 .15)\) and \((1.11 .20)\), show that \(P, g_{x}\), and \(E_{1}\) are all solutions of the usual wave equation, with the wave velocity \(c=\left(r_{0} / \lambda_{0}\right)^{1 / 2}\).
The two waves \(\eta_{1}=A_{1} \cos (\kappa x-\omega t)\) and \(\eta_{2}=A_{2} \cos (\kappa x+\omega t)\) travel together on a stretched string. Show how the wave amplitude ratio \(A_{1} / A_{2}\) can be found from a measurement of the so-called standing-wave ratio \(S\) of the resulting wave pattern, which is defined as the ratio of the maximum amplitude to the minimum amplitude of the envelope of this pattern. Note that \(S\) and the position and amplitude of the minima (or maxima) wave disturbance uniquely determine the two traveling waves that give rise to the standing-wave pattern.
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