MATH3271

FW02

Answers to Assignment 2

1. (a) Solve the one-dimensional wave equation for each of the following sets of initial conditions using the method of separation of variables. Take L = 1 and c = 1.

The solution to the one-dimensional wave equation is

where

and

(i) f(x) = 0.1sin(2px), g(x) = 0;

Since the Fourier sine series for f(x) is just 0.1sin(2px) a_n = 0 unless n = 2 and a₂ = 0.1. Also

b_n = 0 since g(x) = 0. Thus the solution is u(x,t) = 0.1sin(2px)cos(2pt).

(ii) f(x) = 0, g(x) = x(1 - x);

In this case, a_n = 0 since f(x) = 0 and if n ¹ 1

Thus and b_2k+1 = 0 for k > 0.

Now

The solution is

(iii) f(x) = 1 - cos(2px), g(x) = x/4 if 0 < x £ 0.4, f(x) = (1 - x)/6 if 0.4 < x £1.

Note that if n = 2 the last term above is zero. Thus a_2k = 0 and

Also

Thus the solution is