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Consider a two dimensional flexible membrane whose equilibrium position occupies a region in the horizontal x y-plane. Suppose this membrane vibrates up and down with u(x, y,t) denoting the vertical displacement of the point (x, y) of the membrane at time t. The vertical displacement function u(x, y,t) satisfies the two dimensional wave equation

∂^2u/ ∂t^2 +c^2 ∇^2 u

where c^2 =T/ρ for T and ρ the membranes tension and density respectively. Suppose that the rectangular membrane 0 ≤ x ≤ a, 0 ≤ y ≤ b is released from rest with given initial displacement u(x, y, 0) = f (x, y). Further, suppose that we have the following boundary conditions u(0, y,t) = u(a, y,t) = u(x, 0,t) = u(x,b,t) = 0, and ut(x, y, 0) = 0. Find an expression for the vertical displacement of the membrane.

user927154
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  • By $ut(x, y, 0)$, do you mean $u_t(x,y,0)$? In any event, the boundary conditions are clear, but I can't see how enough information is given about the initial conditions to define any solution. – bob.sacamento May 13 '21 at 14:17
  • Sir, can I get your e-mail address to send you the question entirely? – user927154 May 13 '21 at 14:21
  • Yes, that is what I meant. The later. – user927154 May 13 '21 at 14:23
  • I would rather you edit the question and supply any further information there. If you do that, everyone, not just me, can see the full question and you are more likely to get the help you need. As it is, there is no info about the initial displacement of the membrane, so there is no way to define any solution. – bob.sacamento May 13 '21 at 14:30
  • @bob.sacamento The initial displacement is given as $f(x,y)$. I assume the solution is meant to be expressed as some sort of sum of integrals over that. – eyeballfrog May 13 '21 at 14:34

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