Resonance in a 2D box with one side driven

• Feb 21st 2010, 08:08 AM
jessica282828
Resonance in a 2D box with one side driven
The problem:

Small amplitude waves disturb a gas contained in a 2D box 0< x < L, 0< y < b. You may assume that the flow is described by a velocity potential (phi) which satisfies the 2D wave equation.

One side of the box is driven, so that partial d(phi)/dx = aexp(-iwt) at x=0 (real part assumed).

Obtain the solution; phi = [accos(w(L-x)/c)] / [wsin(wL/c)], where c is the constant in the wave equation d^2(phi)/dt^2 = c^2 (grad)^2(phi),

For what values of w is this solution inadmissible? What happens if w takes one of these values?

[Sorry about the poor layout, struggling without a maths keyboard... :S]

My solution:

I am happy getting to the solution for phi above.

I can also see that the solution is inadmissible for sin(wL/c) = 0.
i.e. wL/c = n(pi), n an integer.
i.e. w = n(pi)c/L.

So solution is inadmissible for w = n(pi)c/L.

At this frequency you get resonance and, instead of a time periodic solution, you get growth in time.

My questions:

But how can i show this mathematically? I want to find the solution for phi when w = n(pi)c/L and show that you get growth with time. I tried substituting this value for w into the governing equation and then working through to a solution for phi, but i'm not getting the correct solution / getting a solution at all?! I can't work out where i am going wrong.

Any help would be much appreciated.

Thanks, Jess