Show how the solution \((1.5 .5)\) of the wave equation can be transformed into a general traveling-wave solution.

Short Answer

Expert verified
**Question:** Transform the given solution \((1.5, 0.5)\) into a general traveling-wave solution of the wave equation \(u_{tt} = c^2u_{xx}\). **Answer:** The general traveling-wave solution of the wave equation is \(u(x, t) = f(x - ct) + g(x + ct) = u(1.5 - 0.5c, 0.5) + u(1.5 + 0.5c, 0.5)\), where \(f(x-ct)\) and \(g(x+ct)\) are functions that satisfy the wave equation and depend on the wave speed \(c\).

Step by step solution

01

Rewrite the given solution as a function

Given the solution \((1.5, 0.5)\), we can define the displacement function \(u(x, t)\) as: \(u(x, t) = u(1.5 -c(0.5), 0.5) = u(1.5 - 0.5c, 0.5)\).
02

Find the general traveling-wave solution

Now, we can rewrite the displacement function as a sum of two functions \(f\) and \(g\). We are looking for functions \(f(x - ct)\) and \(g(x + ct)\) such that their sum \(u(x, t)\) satisfies the wave equation. Note that: \(u(x,t) = f(x - ct) + g(x + ct) = u(1.5 - 0.5c, 0.5)\).
03

Comparing the two forms of the solution

Since the given displacement function \(u(x, t)\) and the general traveling-wave solution must be equal, we can set the \(f\) and \(g\) terms equal to the corresponding terms in the given solution: \(f(x - ct) = u(1.5 - 0.5c, 0.5)\) \(g(x + ct) = u(1.5 + 0.5c, 0.5)\)
04

Rewrite the functions f and g

We can rewrite \(f\) and \(g\) to better match the traveling-wave solution. For the function \(f\), we can write: \(f(x - ct) = u(1.5 - 0.5c, 0.5)\) \(f(z) = u(1.5 - 0.5c, 0.5)\), where \(z = x - ct\) Similarly, for the function \(g\), we can write: \(g(x + ct) = u(1.5 + 0.5c, 0.5)\) \(g(w) = u(1.5 + 0.5c, 0.5)\), where \(w = x + ct\)
05

Finalize the general traveling-wave solution

In the end, we find the general traveling-wave solution to be: \(u(x, t) = f(x - ct) + g(x + ct) = u(1.5 - 0.5c, 0.5) + u(1.5 + 0.5c, 0.5)\). The given solution \((1.5, 0.5)\) has been transformed into a general traveling-wave solution in the form of a sum of two functions, \(f\) and \(g\), which depend on the wave speed \(c\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Two 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.

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free