A rectangular plate of sides a and b has its edges fixed in the xy plane and set into transverse vibration. If the initial displacement is f(x,y) and the initial velocity is g(x,y), find the displacement u(x,y,t).
Does anybody know how to solve this partial differential equation?
Assume the usual separation of variables . Substitute and separate into 3 ODE, i.e.
, , and
Now, bring in the BC's for the first two ODE's giving (details omitted)
which leads to
which you can solve (again leading to sin's and cos's). Then match your initial condition.