 Solution

Expand in the complete Fourier sine series $\tilde{y}(\omega ,x)={\sum }_{n\ge 1}{b}_n(\omega )\sin (k_{n}x)$ and apply Fourier inversion. Substitution of part (ii) givesTo check the source normalization, apply ${\partial }_t^2-{\partial }_x^2$ mode by mode. It cancels the denominator; the remaining inverse transform gives $2\pi F(t)$, and completeness givesThe endpoints vanish termwise. These identities establish the equation in the distributional sense, with classical validity wherever the series and derivatives converge suitably.

The displayed real-axis integrals in the paper are not ordinary convergent integrals at their poles. One definite choice is the Cauchy principal value at each $\omega =\pm k_n$. If a causal solution is wanted, use $[k_n^2-(\omega -i0{)}^2{]}^{-1}$ instead. Its inverse transform is $H(t)\sin (k_{n}t)/k_n$, so the causal solution iswhen the past integral exists. The principal-value choice instead uses the time-symmetric kernel $\sin (k_n|t|)/(2k_n)$. No temporal initial data are given, so homogeneous sine and cosine modes remain permissible additions. This is the causal Fourier resolvent of a point-driven fixed string.