Hello,
Although I submitted the same question a couple of days ago here, but I am still trying to solve the 1-dim acoustic wave equation with two different media. Actually, though this equation, I want to know how much the acoustic pulse is reflected and transmitted at the interface. Could you please help me or can give some comments ?
In the following code, I used the two continuity conditions of the acoustic wave at the interface of media, which are pressure and displacement continuity conditions.
Thank you very much in advance.
Remove["Global`"];
L = 10.0; T = 3.0; ( L : boundary, T : time )
[Rho]1 = 5; [Rho]2 = 4; ( mass density of each layer )
v1 = 5; v2 = 4; ( speed of sound of each layer )
( For continuity conditions at interface, Pressure : [Rho]1 v1^2
(D[u1[x,t],x]/.x->0)== [Rho]2 v2^2 (D[u2[x,t],x]/.x->0),
Displacement : u1[0,t]==u2[0,t] *)
Sol = NDSolve[{D[u1[x, t], t, t] == v1^2 D[u1[x, t], x, x],
D[u2[x, t], t, t] ==
v2^2 D[u2[x, t], x, x], (D[u1[x, t], x] /. x → L) ==
0, (D[u2[x, t], x] /. x → -L) ==
0, [Rho]1 v1^2 (D[u1[x, t], x] /.
x → 0) == [Rho]2 v2^2 (D[u2[x, t], x] /. x → 0), (D[
u1[x, t], t] /. t → -1) == 0, (D[u2[x, t], t] /. t → -1) ==
0, u1[x, -1] == Exp[-20 (x - L/2)^2], u2[x, -1] == 0}, {u1,
u2}, {x, -L, L}, {t, -1, T}, MaxSteps → Infinity,
Method → {“MethodOfLines”,
“SpatialDiscretization” → {“TensorProductGrid”}}][[1]]
Plot3D[u1[x, t] /. Sol, {x, -L, L}, {t, -1, T}, PlotRange → All]