Fourier Series and PDEs: Sheet 6

2. Interpretation of d'Alembert's solution

A stretched string occupies the semi-infinite interval $-\infty < x \leq 0$. Let $f$ be a twice differentiable function and let \[ y(x,t) \equiv f(x-ct) - f(-x-ct). \]

a) Show that $y(x,t)$ is a solution of the wave equation with wave speed $c$ and that $y(0,t) = 0$ for all $t$. Describe what is happening here in terms of incident and reflected waves.

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b) If, by contrast, \[ y(x,t) \equiv f(x-ct) + f(-x-ct), \]

show that $y$ is once again a solution of the wave equation. What boundary condition does $y$ now satisfy at $x = 0$? Describe what is happening here in terms of incident and reflected waves.

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3. Using d'Alembert's for a prescribed initial velocity

a) i. By introducing new independent variables $r = x - t$ and $s = x + t$, show that if $y(x,t)$ is a solution of the wave equation $y_{tt} = y_{xx}$, in which the wave speed $c = 1$, then there are functions $F$ and $G$ such that \[ y(x,t) = F(x - t) + G(x + t). \]

ii. Deduce that, if $y$ satisfies the initial conditions $y(x,0) = f(x)$, and $y_t(x,0) = g(x)$, for $-\infty < x < \infty$, then \[ y(x,t) = \frac{1}{2} \left[ f(x - t) + f(x + t)\right] + \frac{1}{2} \int_{x-t}^{x + t} g(s) \, \de{s}. \]

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b) i. Given that $f(x) = 0$ and $g(x) = |x|/(1+|x|)^2$, show that if $x > t > 0$, then \[ y(x,t) = \frac{1}{2} \ln \left( \frac{1+x+t}{1+x-t}\right) + \frac{t}{t^2 -(1+x)^2}. \]

ii. Find $y(x,t)$ if $t > x > 0$.

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4. A localized solution and d'Alembert's

a) Find the solution, at every time $t > 0$, of the initial-value problem \[ y_{tt} = c^2 y_{xx}, \quad -\infty < x < \infty, t > 0 \]

with initial conditions \[ y(x,0) = \begin{cases} \epsilon \cos\left(\frac{\pi x}{2a}\right) & |x| \leq a, \\ 0 & |x| > a, \end{cases}, \qquad y_t(x,0) = 0. \]

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b) Sketch the graph of $y$ versus $x$ at the times $t = a/(2c)$, $t = a/c$, and $t = 3a/(2c)$.

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