= Solution
Separate the <Klein-Gordon equation> as $\phi(t,x)=e^{\pm i\omega t}X(x)$. The spatial equation is $X''+k^2X=0$, with $\omega^2=k^2+m^2$. The first <Dirichlet boundary condition> removes the cosine solution, and the second requires $\sin(kL)=0$. Thus
$$
\boxed{k_n=\frac{n\pi}{L},\quad n=1,2,\ldots,\qquad
\omega_n=\sqrt{k_n^2+m^2}.}
$$
There is no nonzero $n=0$ Dirichlet mode. The real field is built from both complex-conjugate time dependences, with mode functions
$$
\boxed{u_n(t,x)=\frac{e^{-i\omega_nt}\sin(k_nx)}{\sqrt{\omega_nL}},\qquad
u_n^*(t,x)=\phi_n^{(+)}(t,x).}
$$
The sine functions satisfy $\int_0^L\sin(k_nx)\sin(k_mx)dx=(L/2)\delta_{nm}$. Consequently the factor $1/\sqrt{\omega_nL}$ gives unit Klein-Gordon inner product for each positive-frequency mode and the canonical equal-time field <commutator> after quantization. A general solution is a superposition of these modes, rather than a claim that a single mode exhausts all solutions.
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