Exercise 13.2.5
From Maxwell’s equations we can find a way to convert the wave equation to a first-order form that uses only first-order derivatives in space:
\[ u_t = c^2(v_y - w_x),\\ v_t = u_y\\ w_t = -u_x, \] subject to \(u=0\) on the boundary
- Show that a solution satisfies \(u_t = c^2(u_{xx} + u_{yy})\)
\[ v_{ty} = u_{yy}\\ w_{tx} = -u_{xx} \]
Now what?
- Solve with \(c=2\) in the rectange \([-3,3] \times [-1,1]\), \(u(x,y,0) = \exp(x-x^2)(9-x^2)(1-y^2)\), and \(v=w=0\) at \(t=0\). Use \(m=50\) for x and \(n=25\) for y, solve for \(0 \leq t \leq 6\), and make an animation.