We consider an example of initial boundary value problem for the wave equation on the seni-line with Dirichlet boundary condition at x = 0:

utt(x,t) = uxx(x,t),  0 < x < ∞,  t > 0 (1)
u(0,t) = 0,  t > 0,
u(x, 0) = f(x),  0 < x < ∞,
ut(x,t) = 0,  0 < x < ∞.

Let f(x) be given by

( { (1 - cos(πx )) f(x) = --------------, 2 < x < 4 ( 2 0, otherwise

Assuming that f(x) = 0 for x < 0, we can write the odd extension f^ (x) of f(x) is

^ f(x) = f (x) - f (- x ).

The solution ^u (x,t) of

^u tt(x,t) = ^u xx(x,t),   -∞ < x < ∞,  t > 0 (2)
^u (x, 0) = f^ (x),   -∞ < x < ∞,
^u t(x,t) = 0,   -∞ < x < ∞,.
of
^ ^ ^u(x, t) = f(x-+-t) +-f-(x---t). 2


When we restrict this solution to x > 0 we obtain (recall f(x) = 0 for x < 0)

f (x + t) + f(x - t) - f(t - x ) u(x,t) = ------------------------------. 2
 (3)

solution of extended problem

Solution of the extended problem on whole line




solution of problem on half line

Solution of the  problem on half line