13E (Further Complex Methods)

The string is initially at rest, so

y (x, 0) = y_{t} (x, 0) = 0

t

converts the wave equation into

p^{2} y = c^{2} \frac{\partial ^{2} y}{\partial x ^{2}} .

For

Re p > 0

, the solution that decays as

x \to \infty

y = A (p) e^{- p x / c}

y (0, p) = f (p)

fixes

A = f

, and therefore

y (x, p) = f (p) e^{- p x / c} .

ii

For

f (t) = sin ω t

f (p) = \frac{ω}{p ^{2} + ω ^{2}}, y (x, p) = \frac{ω e ^{- p x / c}}{p ^{2} + ω ^{2}} .

y (x, t) = \frac{1}{2 πi} \int_{γ - i \infty}^{γ + i \infty} \frac{ω e ^{p (t - x / c)}}{p ^{2} + ω ^{2}} d p .

When

t > x / c

, close the contour to the left. The residues at

p = \pm iω

sum to

sin (ω (t - x / c))

. When

t < x / c

, close it to the right, where there are no poles, and the integral vanishes. Hence

y (x, t) = {0, sin (ω (t - x / c)), t < x / c, t \geq x / c .

Equivalently,

y (x, t) = Θ (t - x / c) sin (ω (t - x / c))

c

without reflection or dispersion; a point at

x

begins moving after the propagation delay

x / c

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