PDE Wave equation with phi(x) as initial boundaries

[Robconway]

Homework problem:


For the wave equation:

Utt-Uxx=0, t>0, xER

u(x,0)=

1, |x|<1

0, |x|>1

sketch the solution u as a function of x at t= 1/2, 1, 2, and 3


I am able to use d'Alemberts and solve for u however the boundaries and the odd/even reflections are throwing me off and I don't know how to do this. Any help would be great thank you.

 

[LCKurtz]

Homework problem:For the wave equation:

Utt-Uxx=0, t>0, xER

u(x,0)=

1, |x|<1

0, |x|>1

sketch the solution u as a function of x at t= 1/2, 1, 2, and 3I am able to use d'Alemberts and solve for u however the boundaries and the odd/even reflections are throwing me off and I don't know how to do this. Any help would be great thank you.

What boundaries? This is an infinite string with $x$ domain $(-\infty,\infty)$. You have given an initial displacement but the d'Alembert solution also requires an initial velocity $u_t(x,0)$. Did you leave that out? Is it $0$? If so, just plot the traveling waves.