Wave Equation (d'Alembert solution)

we have been looking at standing waves

A diagram showing a standing wave oscillating between a solid upper curve and a dashed lower curve, with a vertical double-headed arrow in the center indicating the oscillation. — Figure: Standing wave oscillation

but that can also be viewed as a combination of two traveling waves moving in opposite directions

A sequence of two diagrams. The first shows a single wave pulse on a string. The second shows two smaller wave pulses moving away from each other, indicated by arrows pointing left and right. — Figure: Traveling wave components

we can see this in the Fourier series solution:

Problem A:

\[ y(x,t) = \sum_{n=1}^{\infty} A_n \cos\left(\frac{n\pi at}{L}\right) \sin\left(\frac{n\pi x}{L}\right) \]

we know \[ 2 \sin A \cos B = \sin(A+B) + \sin(A-B) \]

rewrite the solution

\[ y(x,t) = \underbrace{\frac{1}{2} \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi}{L}(x+at)\right)}_{\substack{\text{LEFT moving wave} \\ \text{magnitude } \frac{1}{2} A_n}} + \underbrace{\frac{1}{2} \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi}{L}(x-at)\right)}_{\text{same but moving RIGHT}} \]

PAGE 2

d'Alembert solution was from 1747 (60 years before Fourier)

\[ y_{tt} = a^2 y_{xx} \]

\[ \underbrace{-\infty < x < \infty}_{\text{no BCs}} \]

\[ t > 0 \]

ICs:: \[ y(x,0) = f(x) \] initial displacement; \[ y_t(x,0) = g(x) \] initial velocity

d'Alembert: observer moving w/ a wave would see a wave that never changes

→ use that as the coordinate system

follow wave moving right: \[ \frac{dx}{dt} = a \quad \rightarrow \quad x - at = \text{constant} \]

left: \[ \frac{dx}{dt} = -a \quad \rightarrow \quad x + at = \text{constant} \]

let \[ \begin{cases} \xi = x + at \\ \eta = x - at \end{cases} \]

rewrite \[ y_{tt} = a^2 y_{xx} \]

PAGE 3

\[ y_x = \frac{\partial y}{\partial x} = \frac{\partial y}{\partial \xi} \frac{\partial \xi}{\partial x} + \frac{\partial y}{\partial \eta} \frac{\partial \eta}{\partial x} \]

\( \rightarrow y_x = y_{\xi} + y_{\eta} \)

\[ y_t = a(y_{\xi} - y_{\eta}) \] \[ y_{xx} = y_{\xi\xi} + 2y_{\xi\eta} + y_{\eta\eta} \] \[ y_{tt} = a^2(y_{\xi\xi} - 2y_{\xi\eta} + y_{\eta\eta}) \]

Sub into \( y_{tt} = a^2 y_{xx} \)

we get

\[ y_{\xi\eta} = 0 \]

Integrate with respect to \( \xi \) : \( y_{\eta}(\xi, \eta) = \phi(\eta) \)

again : \( y(\xi, \eta) = \Phi(\eta) + \Psi(\xi) \)

back to \( x \) and \( t \)

\[ y(x, t) = \phi(x - at) + \psi(x + at) \]

PAGE 4

ICs:

\( y(x, 0) = f(x) \) displacement
\( y_t(x, 0) = g(x) \) velocity

\[ y(x, t) = \phi(x - at) + \psi(x + at) \]

\[ f(x) = \phi(x) + \psi(x) \]

\[ y_t(x, t) = \frac{\partial \phi}{\partial (x - at)} \frac{\partial (x - at)}{\partial t} + \frac{\partial \psi}{\partial (x + at)} \frac{\partial (x + at)}{\partial t} \]

at \( t = 0 \)

\[ g(x) = \phi'(x) \cdot -a + \psi'(x) \cdot a \]

or

\[ g(x) = -a \phi'(x) + a \psi'(x) \]

integrate from \( x_0 \) to \( x \)

\[ \int_{x_0}^{x} g(s) ds = -a \phi(x) + a \psi(x) \]

Solve 1st and 3rd eqs simultaneously

PAGE 5

\[ \phi(x) = \frac{1}{2} f(x) - \frac{1}{2a} \int_{x_0}^{x} g(s) ds \]

\[ \psi(x) = \frac{1}{2} f(x) + \frac{1}{2a} \int_{x_0}^{x} g(s) ds \]

then we get

\[ y(x,t) = \frac{1}{2} [f(x-at) + f(x+at)] + \frac{1}{2a} \int_{x-at}^{x+at} g(s) ds \]