How to solve Wave Equation numerically?

Questions
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1

I’m trying a wave equation with NDSolve, but I’m having problems with the boundary conditions. This is the equation I want to solve:

D[y[x, t],t, t] == D[y[x, t], x, x]

How do I enter the boundary conditions in form of a function? E.g a sine wave or a Gaussian wave package?

2

Just need to make sure the B.C. and the I.C.'s are consistent. Here is an example with boundary condition at x=0 which is sin function.

eq = D[y[x, t], t, t] == D[y[x, t], x, x];
bc = {y[0, t] == t Sin[t], y[2 Pi, t] == 0};
ic = {y[x, 0] == Sin[x], (D[y[x, t], t] /. t -> 0) == 0};
sol = NDSolve[Flatten@{eq, ic, bc}, y, {x, 0, 2 Pi }, {t, 0, 10}]

which gives

{{y->InterpolatingFunction[{{0.,6.28318530717959},{0.,10.}},<>]}}

Now you can plot the solution

Plot3D[Evaluate[y[x, t] /. sol], {t, 0, 10}, {x, 0, 2 Pi }, 
 PlotRange -> All, AxesLabel -> {"t", "x", "y(x,t)"}]

Untitled.png
Untitled.png411×296 33.4 KB

Or you can animate the solution in time:

Animate[Grid[{{"t=", i, " seconds"},
   {Plot[Evaluate[y[x, t] /. sol /. t -> i], {x, 0, 2 Pi }, 
     PlotRange -> {{0, 2 Pi}, {-8, 8}}, AxesLabel -> {"x", "y(x,t)"}, 
     ImageSize -> 300, PlotStyle -> Red], SpanFromLeft}}], {i, 0,10, .001}]

wave.gif

3

Excellent reply Nasser! We just wanted to bring to everyone’s attention that there is a very detailed tutorial on this topic in a related discussion: ** Solve a simple wave equation with System Modeler: Equation-based Approach **