Wave equation with mesh, "The ranges ... cannot be combined to a region"

Question

I am attempting to solve the wave equation with the boundary displacement oscillating in time, using mesh.

Here is a working program for the heat equation with the same boundary conditions:

<< NDSolve`FEM`;
mesh = ToElementMesh[Disk[], "MaxCellMeasure" -> {"Area" -> 0.1, "Length" -> 1}];
heatsol = NDSolveValue[
  {Derivative[0, 2, 0][u][t, x, y] + Derivative[0, 0, 2][u][t, x, y] -
   Derivative[1, 0, 0][u][t, x, y] == 0,
   DirichletCondition[u[t, x, y] == Sin[2 Pi t], True],
   u[0, x, y] == 0},
  u, {t, 0, 1}, {x, y} \[Element] mesh];

heatframes = Table[Plot3D[heatsol[t, x, y], {x, y} \[Element] mesh,
  PlotRange -> {-1, 1}],  {t, 0, 1, 0.1}];

Export[NotebookDirectory[] <> "heat.GIF", heatframes, "DisplayDurations" -> 0.1];

This works fine and produces the expected behaviour:

However, if I change the first derivative in time to a second derivative in time to get the wave equation, I get an error:

NDSolveValue[
  {Derivative[0, 2, 0][u][t, x, y] + Derivative[0, 0, 2][u][t, x, y] -
   Derivative[2 (* changed from 1 *) , 0, 0][u][t, x, y] == 0,
   DirichletCondition[u[t, x, y] == Sin[2 Pi t], True],
   u[0, x, y] == 0},
  u, {t, 0, 1}, {x, y} \[Element] mesh]

"NDSolveValue::fememrc: The ranges {{0.,1.},<<1>>} cannot be combined to a region. Please specify a combined region. >>"

Why does changing the derivative from 1st order (heat) to 2nd order (wave) result in this error?

Answer 1

When you change from the heat equation to the wave equation you must add one more initial condition : not only the surface u[0,x,y] must be known but also one need a condition that specify the movement of the surface, for example the surface is still :

(Derivative[1, 0, 0][u][tt, x, y] /. tt-> 0) == 0

  heatsol=NDSolveValue[
      {Derivative[0, 2, 0][u][t, x, y] + Derivative[0, 0, 2][u][t, x, y] -
       Derivative[2 (* changed from 1 *) , 0, 0][u][t, x, y] == 0,
       DirichletCondition[u[t, x, y] == Sin[2 Pi t], True],
       u[0, x, y] == 0,
       (Derivative[1, 0, 0][u][tt, x, y] /. tt-> 0) == 0},
  u, {t, 0, 1},{x, y} \[Element] mesh]

heatframes = Table[Plot3D[heatsol[t, x, y], {x, y} \[Element] mesh,
  PlotRange -> {-3, 3}],  {t, 0, 1, 0.1}];

Export[ "heat.GIF", heatframes, "DisplayDurations" -> 0.1];