18.3.4.1 Solution pph-a

Question:

Show that the given solution

$$u(x,y) = \sin(nx)\sin(nt)$$

with $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m_2$ does indeed satisfisfy the wave equation

$$u_{tt} = u_{xx}$$

and the boundary conditions

$$u(x,0)=0 \quad u(0,t)=0 \quad u(\pi ,t)=0 \quad u(x,\frac{m_1}{m_2}\pi)=0$$

How about twice that solution? Ten times? How about if $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2m_2$? How about if $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $10m_2$? So how many solutions are there really to this single problem?

Answer:

Plug it in.

There are infinitely many solutions.