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