What values of a and b satisfy the following wave equation?

[s_gunn]

Homework Statement

For which values of the constants a and b does u(x,t) = sin(ax)sin(bt) satisfy the wave equation U_tt = C²u_xx?

Homework Equations

The Attempt at a Solution

I've taken the partial differentials:
U_x = acos(ax)sin(bt)
U_xx = -a²sin(ax)sin(bt)
U_t = bcos(bt)sin(ax)
U_tt = -b²sin(bt)sin(ax)

and subbed them into the equation to get:
-b²sin(bt)sin(ax)+c²a²sin(ax)sin(bt)=0
so:
(-b²+c²a²)(sin(bt)sin(ax)) = 0

But then I'm lost!

I have a feeling that I may have gone in completely the wrong direction so any help would be greatly appreciated!

 

[Gokul43201]

You have not gone in the wrong direction.

If p*q=0, what can you say about p and q?

 

[s_gunn]

I know that each of them must equal zero making
b = ca and a = b/c
but is rearranging them to make a the subject really all their asking for?! Am i missing something!?

 

[Gokul43201]

I know that each of them must equal zero making
b = ca

This is correct, but you skipped a step, and in the process lost one of the solutions.

But once have the other solution, that's basically all you are asked to do.

 

[s_gunn]

Thanks Gokul43201, sometimes you just need a nudge in the right direction!