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
(i) f(x) = 0.1sin(2px), g(x) = 0;
Since the Fourier sine series for f(x) is just 0.1sin(2px) an = 0 unless n = 2 and a2 = 0.1. Also
bn = 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, an = 0 since f(x) = 0 and if n ¹ 1
Thus and b2k+1 = 0 for k > 0.
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 a2k = 0 and
Thus the solution is