Thank you so much @Rouben Rostamian  for such a wonderful solution! as @ecterrab  said it's a beauriful/skilled presentation

@ecterrab Thank you for your help. I used the following Maple code, but when it comes to casesplit command I don't get a solution, it just copy the command line + the equations. Is there something wrong in the calling sequence?

> with(PDEtools)

> pdesys := [rho*(diff(diff(Q(x, z, t), t, t), x))+rho*(diff(diff(W(x, z, t), t, t), z))-(lambda+2*mu)*(diff(diff(Q(x, z, t), x, x)+diff(Q(x, z, t), z, z), x))-mu*(diff(diff(W(x, z, t), x, x)+diff(W(x, z, t), z, z), z)) = 0, rho*(diff(diff(Q(x, z, t), t, t), z))-rho*(diff(diff(W(x, z, t), t, t), x))-(lambda+2*mu)*(diff(diff(Q(x, z, t), x, x)+diff(Q(x, z, t), z, z), z))+mu*(diff(diff(W(x, z, t), x, x)+diff(W(x, z, t), z, z), x)) = 0]

> casesplit(pdesys, [diff(Q(x, z, t), t, t), diff(W(x, z, t), t, t)])