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Advanced Engineering Mathematics, International Student Edition

Peter V. O'Neil

Chapter 17

The Wave Equation - all with Video Answers

Educators

Section 1

The Wave Equation and Initial and Boundary Conditions

03:53

Problem 1

Let $y(x, t)=\sin (n \pi x / L) \cos (n \pi c t / L)$. Show that $y$ satisfies the one-dimensional wave equation for any positive integer $n$.

Foster Wisusik
Foster Wisusik
Numerade Educator
02:12

Problem 2

Show that $z(x, y, t)=\sin (n x) \cos (m y) \cos \left(\sqrt{n^2+m^2} c t\right)$ satisfies the two-dimensional wave equation for any positive integers $n$ and $m$.

James Kiss
James Kiss
Numerade Educator
05:11

Problem 3

Let $f$ be any twice-differentiable function of one variable. Show that
$$
y(x, t)=\frac{1}{2}[f(x+c t)+f(x-c t)]
$$
satisfies the one-dimensional wave equation.

WM
William Mead
Numerade Educator
01:04

Problem 4

Show that $y(x, t)=\sin (x) \cos (c t)+\frac{1}{c} \cos (x) \sin (c t)$ satisfies the one-dimensional wave equation, together with the boundary conditions
$$
y(0, t)=y(2 \pi, t)=\frac{1}{c} \sin (c t) \quad \text { for } t>0
$$
and the initial conditions
$$
y(x, 0)=\sin (x), \frac{\partial y}{\partial t}(x, 0)=\cos (x) \text { for } 0<x<\pi
$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:31

Problem 5

Formulate a boundary value problem (partial differential equation, boundary and initial conditions) for vibrations of a rectangular membrane occupying a region $0 \leq x \leq a, 0 \leq y \leq b$ if the initial position is the graph of $z=f(x, y)$ and the initial velocity (at time zero) is $g(x, y)$. The membrane is fastened to a stiff frame along the rectangular boundary of the region.

James Kiss
James Kiss
Numerade Educator
01:14

Problem 6

Formulate a boundary value problem for the motion of an elastic string of length $L$, fastened at both ends and released from rest with an initial position given by $f(x)$. The string vibrates in the $x, y$ plane. Its motion is opposed by air resistance, which has a force at each point of magnitude proportional to the square of the velocity at that point.

Hast Aggarwal
Hast Aggarwal
Numerade Educator