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PAGE 1

Wave Equation (d'Alembert solution)

the waves we've looked at are standing waves (not going anywhere)

A diagram showing a standing wave as a solid arc above and a dashed arc below, with a vertical double-headed arrow in the center indicating oscillation.

but it's actually made of two traveling waves that move away from each other

A diagram showing two separate wave pulses moving in opposite directions. The left pulse has an arrow pointing right, and the right pulse has arrows pointing both left and right, illustrating wave propagation.

this is also "visible" from 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 \[ 2 \sin A \cos B = \sin(A+B) + \sin(A-B) \]

\[ y(x,t) = \underbrace{\frac{1}{2} \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi}{L}(x+at)\right)}_{\substack{\uparrow \\ \text{each wave} \\ \text{is half magnitude} \\ \text{wave traveling LEFT}}} + \underbrace{\frac{1}{2} \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi}{L}(x-at)\right)}_{\substack{\uparrow \\ \text{wave traveling RIGHT}}} \]
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d'Alembert solved the wave equation in 1747 (60 years before Fourier)

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

\[ -\infty < x < \infty, \quad t > 0 \]

\( \underbrace{\quad}_{\text{no BCs}} \)

\( a \): speed of wave propagation

IC: \( y(x,0) = f(x) \) initial displacement

\( y_t(x,0) = g(x) \) " velocity

d'Alembert realized that an observer moving with one of the traveling waves would see a wave with constant magnitude

\( \rightarrow \) use as coordinate system

right-moving wave: wave speed is \( a \)

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

for left-moving wave: \( x + at = \text{constant} \)

PAGE 3

Solving the Wave Equation via Change of Variables

let

  • \( \xi = x + at \)
  • \( \eta = x - at \)

\[ y_{tt} = a^2 y_{xx} \quad y(x,t) \]

\[ y_x = y_{\xi} + y_{\eta} \]

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

\[ y_t = a(y_{\xi} - y_{\eta}) \]

\[ y_{xx} = y_{\xi\xi} + 2y_{\xi\eta} + y_{\eta\eta} \]

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

the wave becomes:

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

integrate:

\[ y_{\eta}(\xi, \eta) = \phi(\eta) \]

(with respect \( \xi \))

again:

\[ y(\xi, \eta) = \phi(\eta) + \psi(\xi) \]

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

\[ y(x,t) = \underbrace{\phi(x-at)}_{\text{wave moving RIGHT}} + \underbrace{\psi(x+at)}_{\text{wave moving LEFT}} \]

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Applying Initial Conditions

initial conditions:

  • \( y(x,0) = f(x) \)
  • \( y_t(x,0) = g(x) \)

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

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

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

at \( t = 0 \)

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

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

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

\[ -a \phi(x) + a \psi(x) = \int_{x_0}^{x} g(s) ds \]
Solve 1st and 3rd eqs simultaneously

\( \vdots \)

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

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

PAGE 5

Sub into \( y(x,t) = \phi(x-at) + \psi(x+at) \)

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

Example

\( f(x) = \sin(x) \) initial displacement

\( g(x) = 0 \) initial velocity

\( a = 1 \)

\[ y(x,t) = \frac{1}{2} \left[ \sin(x-t) + \sin(x+t) \right] \]
  • half of initial displacement to the right
  • half of initial displacement to the left
A graph of a sine wave on a coordinate system with x and y axes. The wave oscillates periodically around the x-axis.
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Example

\[ f(x) = \begin{cases} 1 & -1 < x < 1 \\ 0 & \text{else} \end{cases} \]

\( g(x) = 0 \)

\( a = 1 \)

\[ y(x,t) = \frac{1}{2} \left[ f(x-t) + f(x+t) \right] \]
A small sketch of a rectangular pulse function centered at the origin on a coordinate system, with boundaries at -1 and 1 on the x-axis.
A graph showing a single rectangular pulse of height 1 centered at the origin, spanning from x = -1 to x = 1.
A graph showing a stepped pulse function. It has a central peak of height 1 between -0.5 and 0.5, and two lower steps of height 0.5 on either side.
A graph showing two separate rectangular pulses of height 0.5 moving away from the origin. One pulse is on the left and one is on the right, each with a width of 2.
PAGE 7

Example: Hammered String

\( f(x) = 0 \)

\( g(x) = \begin{cases} V_0 & \text{if } |x| < K \\ 0 & \text{else} \end{cases} \)

\( a = 1 \n

A rectangular pulse function g(x) centered at the origin on the x-axis, with height V_0 between -K and K. This represents the initial velocity of a hammered string.

Initial velocity (hammered string)

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

\( \underbrace{\hspace{100pt}}_{\text{area under } g(x) \text{ on } x-t < x < x+t} \)

If \( t < K \)

A trapezoidal wave on a coordinate system with axes x and y. The wave has a flat top at height V_0 t and sloping sides moving outwards as indicated by arrows.

Height \( V_0 t \)

If \( t \) is large

A graph showing two flat horizontal lines moving away from the origin along the x-axis, representing the wave pulse splitting and traveling in opposite directions.

does not return to 0!