Here is another attempt to come nearer to what you want.

Define the wave equation in polar coordinates

DGL[f_] :=  D[f, r, r] + 1/r D[f, r] + 1/r^2 D[f, phi, phi] - D[f, t, t]

A solution is given by

u[r_, phi_, t_, n_, m_] := 
BesselJ[n,BesselJZero[n, m] r] (\[Alpha] Cos[n phi] + \[Beta] Sin[n phi]) (a Cos[ BesselJZero[n, m] t] - b Sin[BesselJZero[n, m] t])

See

DGL[u[r, phi, t, n, m]] // FullSimplify`

Now we try to approximate the e-function by a series of u's. Taking into account that Exp[ - r^2 ] is not dependent on phi we chose n = 0 and calculate the coefficients in the series according to

c[n_] := \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(r\ Exp[\(-r^2\)]\ \
BesselJ[0, r\ BesselJZero[0, n]] \[DifferentialD]r\)\)/\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(r\ 
SuperscriptBox[\(BesselJ[0, 
     r\ BesselJZero[0, n]]\), \(2\)] \[DifferentialD]r\)\)

Unfortunately the integral in the numerator must be evaluated numerically. But having done this we get something that could be useful.....

Attachments:

Thanks for your answers, this has helped me a lot! Im curious how you came up with the first solution Hans? Also, Is there a way to use the product ansatz to "transform" one complex DSolve computation into several simpler ones?(Sorry if this goes beyond the scope of this discussion)

With

ww = w0 Exp[I A t]
uu = u0 Exp[I Sqrt[A^2 - B^2] x]
vv = v0 Exp[ I B y]

and

f = uu vv ww // FullSimplify

you have a solution to the wave equation (but not for the ic ).

Another solution is (you may take +t as well)

f = ff[(Cos[a] x + Sin[a] y - t)]
D[f, x, x] + D[f, y, y] - D[f, t, t] // Simplify

and then, to come somewhat "nearer" to your ic

f = Exp[(Cos[a] x + Sin[a] y - t)^2]
D[f, x, x] + D[f, y, y] - D[f, t, t] // Simplify

Thanks for the replys! NDSolve is really interesting and the animations are great, but i would also really like to take a look at the analytical solutions for these equations. I tried some ic's (plane waves,gaussians,..) in 2D, now with the 4 bc's, but everytime i just got my input as output, without an error message (as with your codes).I've found something in the DSolve documentation though, under Scope->Hyperbolic PDEs, quote:

Dirichlet problem for the wave equation in a rectangle:

weqn = D[u[x, y, t], {t, 2}] == Laplacian[u[x, y, t], {x, y}];
ic = {u[x, y, 0] == (1/10) (x - x^2) (2 y - y^2), 
   Derivative[0, 0, 1][u][x, y, 0] == 0};
bc = {u[x, 0, t] == 0,
    u[0, y, t] == 0, u[1, y, t] == 0, u[x, 2, t] == 0};

The solution is a doubly infinite trigonometric series:

(sol = FullSimplify[
    u[x, y, t] /. DSolve[{weqn, ic, bc}, u, {x, y, t}][[1]], 
    K[1] \[Element] Integers && 
     K[2] \[Element] Integers]) // TraditionalForm
\!\(\*
UnderoverscriptBox[
StyleBox["\[Sum]", "InactiveTraditional"], 
RowBox[{
TemplateArgBox[
RowBox[{"K", "[", "1", "]"}]], "=", 
TemplateArgBox["1"]}], 
TemplateArgBox["\[Infinity]"]]\(\*
UnderoverscriptBox[
StyleBox["\[Sum]", "InactiveTraditional"], 
RowBox[{
TemplateArgBox[
RowBox[{"K", "[", "2", "]"}]], "=", 
TemplateArgBox["1"]}], 
TemplateArgBox["\[Infinity]"]]\*
TemplateArgBox[
FractionBox[
RowBox[{"32", " ", 
RowBox[{"(", 
RowBox[{
SuperscriptBox[
RowBox[{"(", 
RowBox[{"-", "1"}], ")"}], 
RowBox[{"K", "[", "1", "]"}]], "-", "1"}], ")"}], " ", 
RowBox[{"(", 
RowBox[{
SuperscriptBox[
RowBox[{"(", 
RowBox[{"-", "1"}], ")"}], 
RowBox[{"K", "[", "2", "]"}]], "-", "1"}], ")"}], " ", 
RowBox[{"cos", "(", 
RowBox[{"\[Pi]", " ", "t", " ", 
SqrtBox[
RowBox[{
SuperscriptBox[
RowBox[{"K", "[", "1", "]"}], "2"], "+", 
FractionBox[
SuperscriptBox[
RowBox[{"K", "[", "2", "]"}], "2"], "4"]}]]}], ")"}], " ", 
RowBox[{"sin", "(", 
RowBox[{"\[Pi]", " ", "x", " ", 
RowBox[{"K", "[", "1", "]"}]}], ")"}], " ", 
RowBox[{"sin", "(", 
RowBox[{
FractionBox["1", "2"], " ", "\[Pi]", " ", "y", " ", 
RowBox[{"K", "[", "2", "]"}]}], ")"}]}], 
RowBox[{"5", " ", 
SuperscriptBox["\[Pi]", "6"], " ", 
SuperscriptBox[
RowBox[{"K", "[", "1", "]"}], "3"], " ", 
SuperscriptBox[
RowBox[{"K", "[", "2", "]"}], "3"]}]]]\)\)

End of quote. Why does this work and the others don't? The only difference i can see are the initial conditions?

So I played around. Yeah it seems DSolve recognized something with the ICs that it liked. So that isn't helping you much. However, this might help you with some tricks related to the infinite series. Also notice I used DSolveValue to avoid getting back a rule.

weqn = D[u[x, y, t], {t, 2}] == Laplacian[u[x, y, t], {x, y}];
ic = {u[x, y, 0] == (1/10) (x - x^2) (2 y - y^2), 
   Derivative[0, 0, 1][u][x, y, 0] == 0};
bc = {u[x, 0, t] == 0, u[0, y, t] == 0, u[1, y, t] == 0, 
   u[x, 2, t] == 0};
dsol = DSolveValue[{weqn, ic, bc}, u, {x, y, t}];
ListAnimate[
 Table[Plot3D[
   Activate[dsol[x, y, t] /. Infinity -> 5], {x, 0, 1}, {y, 0, 2}, 
   PlotRange -> {{0, 1}, {0, 2}, {-0.05, 0.05}}], {t, 0, 10, 0.25}]]

Note the Activate function and the replacement of infinity.

I realize you want a nice compact equation, but you might not get it.

Interesting.

I don't get an error message, but neither a solution.....

DSolve[{-
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "2"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, t] + 
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, t] + 
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, t] == 0, 
  u[x, y, 0] == E^(-x^2 - y^2), 
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, 0] == 0, u[x, 0, 0] == 0, 
  u[0, y, 0] == 0}, u, {x, y, t}]