How to make a initial condition of a wave equation as a gaussian wave packet travelling to the right?

Question

I am trying to animate the solution of a wave equation with initial condition being a gaussian wave packet travelling the the right. I know that the mathematical equation is something like $e^{\frac{-(x-a-vt)^2}{b^2}}$, where $a$ and $b$ are constants and $v$ is the velocity.

So I try to make the initial condition as $\psi(x,0)=e^{\frac{-(x-a)^2}{b^2}}$ and $\frac{\partial \psi(x,t)}{\partial t}\Bigr|_{t=0}=2v(x-a)/b^2e^{\frac{-(x-a)^2}{b^2}}$

Here is my code (where I have set $a=0.3$, $b=0.1, v=0.2)$:

ydum = x/L[t];
expr1 = 1/c^2 D[\[Psi][ydum, t], {t, 2}] - 
     D[\[Psi][ydum, t], {x, 2}] + (m^2 c^2)/
      h^2 \[Psi][ydum, t] /. \[Psi][ydum, t] -> \[Psi][y, t] /. 
   x -> y L[t] // Expand
m = 1;
c = 1;
h = 1;
\[Omega] = 1;
L[t_] := 2 + Sin[\[Omega] t];
sol = NDSolve[{expr1 == 0, 
   DirichletCondition[\[Psi][y, t] == 0, y > 1], \[Psi][1, t] == 
    0, \[Psi][0, t] == 0, \[Psi][y, 0] == Exp[-(y - 0.3)^2/0.01], 
   D[\[Psi][y, t], t] == 0.4 (y - 0.3)/0.01 Exp[-(y - 0.3)^2/0.01] /. 
    t -> 0}, \[Psi], {y, 0, 1}, {t, 0, 20}]
Manipulate[
 Plot[{Abs@Evaluate[\[Psi][y, t] /. y -> y/L[t]] /. sol, 
   100 Sign[y - L[t]]}, {y, 0, 10}, ExclusionsStyle -> Red, 
  PlotRange -> {{0, 4}, {0, 2}}], {t, 0, 20}]

(This is an animated gif, but not looping)

As can be seen, the wave splits into 2 parts, one going to the right and the other to the left. What I want is for the gaussian wave packet to travel to the right entirely. Can anyone give me a solution to this?

EDIT

Thanks to user21, I have figured out my mistake during the set up of the initial conditions. I should have let $v$ (the speed of the gaussian) to be equal to $c$ (the speed of the wave in the wave equation). Implementing this on Klein-Gordon Equation gives me :

kge = 1/c^2 D[\[Psi][x, t], {t, 2}] - 
   D[\[Psi][x, t], {x, 2}] + \[Psi][x, t];
c = 1;
gaussian = Exp[-(x - 0.3 - c t)^2/0.01];
ic = {\[Psi][x, 0] == gaussian /. t -> 0, 
  D[\[Psi][x, t], t] == D[gaussian, t] /. t -> 0}
sol = NDSolve[{kge == 0, 
   DirichletCondition[\[Psi][x, t] == 0, x >= 1], \[Psi][0, t] == 0, 
   ic}, \[Psi], {x, 0, 1}, {t, 0, 10}]
Manipulate[
 Plot[Evaluate[\[Psi][x, t] /. ss], {x, 0, 2}, 
  PlotRange -> {{0, 3}, {-1, 1}}], {t, 0, 10}]

which is exactly what I want minus the backside of the gaussian descending a little as it moves, probably due to the extra $+ \psi[x,t]$ term in the wave equation, but I am fine with this.

Going back to the original problem, in which I have changed my coordinate system $$(x,t) \to (y,t)$$ where $y=\frac{x}{L(t)}$ and $L(t)=2+sin(\omega t)$, and setting the right boundary to $\psi (1,t)=0$ (in the $(y,t)$ coordinate)

ydum = x/L[t];
expr1 = 1/c^2 D[\[Psi][ydum, t], {t, 2}] - 
         D[\[Psi][ydum, t], {x, 2}] + (m^2 c^2)/
          h^2 \[Psi][ydum, t] /. \[Psi][ydum, t] -> \[Psi][y, t] /. 
       x -> y L[t] // Expand
m = 1;
c = 1;
h = 1;
 \[Omega] = 1;
L[t_] := 2 + Sin[\[Omega] t];
gaussian = Exp[-(x - 0.3 - c t)^2/0.01];
ic = {\[Psi][y, 0] == gaussian /. x -> y L[t] /. t -> 0, 
  D[\[Psi][y, t], t] == D[gaussian /. x -> y L[t], t] /. t -> 0}
sol = NDSolve[{expr1 == 0, 
   DirichletCondition[\[Psi][y, t] == 0, y >= 1], \[Psi][0, t] == 0, 
   ic}, \[Psi], {y, 0, 1}, {t, 0, 10}]
Manipulate[
 Plot[{Abs@Evaluate[\[Psi][y, t]] /. sol}, {y , 0, 3}, 
  AxesLabel -> {y, \[Psi]}, PlotRange -> {{0, 4}, {0, 2}}], {t, 0, 
  10}]

The gaussian makes some wiggly lines at its base. I am very confused as to why it does that. The problem that I am solving is the same as before (at least before the wave hits the right boundary), only that the coordinate has been scaled $x \to \frac{x}{2+sin(\omega t)}$. Is there any error in my code?

Answer 1

Here is an example from the 13.0 ref page of AcousticPDEComponent. I have included the InputForm of the PDE that is generated, which you can use in your version.

vars = {p[t, x], t, {x}};
pars = <|"SoundSpeed" -> 343, "MassDensity" -> 1.2|>;
p0 = D[0.125 Erf[(x - 0.5)/0.15], x];
ics = {p[0, x] == p0, Derivative[1, 0][p][0, x] == -343*D[p0, x]};
eqn = AcousticPDEComponent[vars, pars] == 
   AcousticSoundHardValue[x == 1, vars, pars];
InputForm[eqn]

You'd use this:

(*eqn=Inactive[Div][{{-0.8333333333333334}} . 
    Inactive[Grad][p[t, x], {x}], {x}] + 
  7.083216460261739*^-6*Derivative[2, 0][p][t, 
    x] == NeumannValue[0, x == 1]*)

pfun = NDSolveValue[{eqn, ics}, p, {t, 0, 0.003}, 
   x \[Element] Line[{{0}, {1}}]];
Manipulate[
 Plot[pfun[t, x], {x, 0, 1}, Sequence[
  PlotRange -> {0, 2}, AxesLabel -> {"x", "P"}]], {{t, 0}, 0, 0.003, 
  10^-4}, SaveDefinitions -> True]