# Progressive wave with initial conditions f(x,0)=0, f'(x,0)=g(x)

1. Dec 4, 2008

### wakko101

Consider a long straight string that is given an initial impulse. The transverse displacement of the string y(x,t) satisfies the initial condition:

y(x,0) = 0 and y'(x,0) = G(x)

Show that the solution to the wave eq'n satisfying the intitial condition is

y(x,t) = 1/2v[H(x+vt)-H(x-vt)]

where H'(u) = G(u)

Solve for an impulse about x=0 with a step profile of the form:

0 x less than/equal to -a
-b -a < x less than/equal to 0
b 0 < x less than/equal to a
0 x > a

where a and b are positive constants.

I'm not entirely sure where to start with this. In class, we derived a solution for initial conditions of f(x,0) = f(x) and f'(x,0) = 0 by using the standing wave eq'n. Physically, that represented a long string with an initial displacement, released from rest. It's easier to picture that physical situation. This is the opposite, whereby the string has no initial displacement, but an initial velocity. Should I start with the standing wave eq'n again? Or should I go back to the wave equation (d^2f/dx^2=v^2(d^2f/dt^2)?

Any suggestions or clarifications would be appreciated.

Cheers,
W. =)