= Solution
The fixed edge at $y=0$ is automatically satisfied by $\sin(qy)$. At $y=a$, the displacement must vanish at every $x,t$, so $\sin(qa)=0$. Thus the allowed transverse <normal modes> have
$$
\boxed{q_n=\frac{n\pi}{a},\qquad n=1,2,3,\ldots.}
$$
The case $n=0$ gives zero displacement for this sine solution and is not a physical propagating mode. Negative $n$ only reverses the amplitude sign and does not create a distinct transverse mode. This discrete transverse spectrum is a consequence of the <Dirichlet boundary conditions>, rather than quantization of the longitudinal <wavenumber> in an indefinitely long guide.
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