 Causal Fourier resolvent of a point-driven fixed string

ID: causal-fourier-resolvent-of-a-point-driven-fixed-string

For

y_{tt} - y_{xx} = F (t) δ (x - x_{0})

q_{n}^{''} + k_{n}^{2} q_{n} = 2 F (t) sin (k_{n} x_{0}) / L

. With the

e^{- iω t}

Fourier transform convention, the causal inverse of this oscillator uses the displayed denominator. Its inverse transform is

H (t) sin (k_{n} t) / k_{n}

q_{n} (t) = 2 sin (k_{n} x_{0}) / (L k_{n}) \int_{- \infty}^{t} F (s) sin [k_{n} (t - s)] d s

when the past integral exists. The plain denominator

k_{n}^{2} - ω^{2}

requires a pole prescription: its Cauchy principal value gives the time-symmetric kernel

sin (k_{n} ∣ t ∣) / (2 k_{n})

instead. Homogeneous sine and cosine modes are additional freedom unless temporal data or causality are specified.

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