So basically we have Stoke's first Problem/Rayleigh plate, meaning an impulsively starting Plate with the boundary conditions $u(0,t)=t^n$ and $u(\infty,t)=0$. In theory it is easy to solve the PDE, if $u(0,t)$ wouldn't be time-dependent! Do you have an idea how to solve it with the similarity Ansatz, taking an Ansatz like $\eta=y/g(t)$ and $u^*(\eta)=u(x,t)/A(t)$?
Stokes's first Problem/Rayleigh plate $u(0,t)=t^n$
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