Consider the wave equation in a two-dimensional quadrant
\[ u_{tt}-\Delta u=0\quad \text {in } \{x>0,\ y>0\}\times (0,\infty ) \]
with boundary conditions \(\partial _x u(0,y,t)=0\) and \(u(x,0,t)=0\) and initial condition \(u_0\), supported in the ball
\[ B_1((2,2))=\{(x,y):(x-2)^2+(y-2)^2\le 1\}. \]
Describe, precisely and with careful justification, the support of \(u(x, y, t)\) at all positive times \(t > 0\). Note: The support of a function \(f(x,y)\) is the closure of the set \(\{(x,y): f(x,y)\ne 0\}\).