Finding a Solution to Wave Problem: Uxx+Utt=0

[Arben]

Hello everybody.

I don't know how to find solution and to draw it for this wave problem:
Uxx+Utt=0
U(x,0)=0
Ut(x,0)=1 a )for 0<=x<=1 and
Ut(x,0)=0 otherwise.
I have to draw the grafic for t=0, 1, 2 for x element from -5 to 5.

 

[tiny-tim]

Welcome to PF!

Uxx+Utt=0
U(x,0)=0
…

Hi Arben! Welcome to PF!

Hint: 0 = (D_xD_x - D_tD_t)U

= (D_x + D_t)(D_x - D_t)U.

= D_x+tD_x-tU.

Does that help?

 

[HallsofIvy]

In other words, since this is a wave equation with c= 1, any solution can be written in the form U(x,t)= f(x+ t)+ g(x- t) so U(x, 0)= f(x)+ g(x)= 0=> f(x)= -g(x).

U_t(x,t)= f'(x+t)- g'(x-t). Further, since f(x)= -g(x), f'(x)= -g'(x) so U_t(x,0)= f'(x)-g'(x)= 2f'(x). ;Since you are given U_t(x,0) you can integrate to get f.