Waves

Chem 3240 · Lecture 2.1

Davit Potoyan

What is a wave?

• A wave is a self-propagating disturbance
• Transfers energy, momentum, information
• Does not transport matter
• Governed by the wave equation, a PDE

Two key families: traveling waves (propagate) and standing waves (oscillate in place)

Kinds of waves

• Medium disturbance: sound, strings, water
• Quantum waves: complex wavefunctions for electrons, atoms
• Electromagnetic waves: need no medium, travel in vacuum
• Gravitational waves: ripples in spacetime

Transverse vs longitudinal

• Transverse: disturbance is perpendicular to propagation
• Longitudinal: disturbance is along propagation

Defining a wave mathematically

• A disturbance \(u\) depends on space and time: \(u = f(x, t)\)
• A surfer riding the wave sees it frozen at \(x'\)
• A shore observer sees the front move: \(x = x' + vt\)

A right-moving wave of fixed shape:

\[u(x,t) = f(x-vt)\]

Set the front constant: \(x - vt = const \Rightarrow x = vt + const\) (moves right). The form \(f(x+vt)\) moves left.

Periodic traveling waves

A sine wave traveling along \(x\):

\[y(x,t)= A \sin(kx-\omega t+\phi)\]

• Amplitude \(A\): max disturbance
• Wave number \(k\): periodicity in space
• Angular frequency \(\omega = kv\): periodicity in time
• Phase \(\phi\): starting point

Complex representation

• Easier to compute with complex exponentials, take real or imaginary part at the end

\[u(x,t) = Ae^{i(kx-\omega t)}\]

• Real and imaginary parts are cosine and sine
• The phasor rotates as \(t\) advances at fixed \(x\)

Periodicity in space and time

• Repeats every wavelength \(\lambda\) in space: \(k\lambda = 2\pi\)

\[k=\frac{2\pi}{\lambda}\]

• Repeats every period \(T\) in time, frequency \(\nu = 1/T\): \(\omega T = 2\pi\)

\[\omega=\frac{2\pi}{T} = 2\pi\nu\]

Wavelength, frequency, and speed are linked by \(\omega = kv\), i.e. \(\lambda \nu = v\)

The classical wave equation

\[u(x,t) = Ae^{i(kx-\omega t)}\]

The classical wave equation

\[u(x,t) = Ae^{i(kx-\omega t)}\]

• Two \(x\)-derivatives: \(u_{xx} = -k^2\, u\)
• Two \(t\)-derivatives: \(u_{tt} = -\omega^2\, u\)
• Their ratio eliminates \(u\): \(\dfrac{k^2}{\omega^2} = \dfrac{1}{v^2}\)

The classical wave equation

\[u(x,t) = Ae^{i(kx-\omega t)}\]

• Two \(x\)-derivatives: \(u_{xx} = -k^2\, u\)
• Two \(t\)-derivatives: \(u_{tt} = -\omega^2\, u\)
• Their ratio eliminates \(u\): \(\dfrac{k^2}{\omega^2} = \dfrac{1}{v^2}\)

\[\frac{\partial^2 u(x,t)}{\partial x^2 } = \frac{1}{v^2}\frac{\partial^2 u(x,t)}{\partial t^2}\]

Solutions are wave functions of space and time

Combining waves: interference

• Superposition: if \(u_A\) and \(u_B\) solve the wave equation, so does \(u_C = u_A + u_B\)
• Interference: combining waves yields greater, lower, or equal amplitude

Interference of two phases

Summing two waves differing only in phase:

\[\Psi_{\text{total}} = 2 \cos\left( \frac{\phi_1 - \phi_2}{2} \right) e^{i \left( kx - \omega t + \frac{\phi_1 + \phi_2}{2} \right)}\]

• In phase (\(\phi=0\)): amplitude doubles
• Out of phase (\(\phi=\pi\)): cancels

Live: dial the phase

wave 1 \(= \sin(kx)\), wave 2 \(= \sin(kx+\phi)\), sum with envelope \(2\cos(\phi/2)\)

viewof phi = Inputs.range([0, 6.283], {step: 0.01, value: 1.0, label: "phase φ (rad)"})

{
  const xs = d3.range(0, 12.57, 0.03);
  const series = (fn) => xs.map(x => ({x, y: fn(x)}));
  return Plot.plot({
    width: 1000, height: 360,
    x: {label: "kx"},
    y: {domain: [-2.2, 2.2], label: "u"},
    marks: [
      Plot.ruleY([0], {stroke: "#ccc"}),
      Plot.line(series(x => Math.sin(x)), {x: "x", y: "y", stroke: "#7fb3d5", strokeWidth: 1.5}),
      Plot.line(series(x => Math.sin(x + phi)), {x: "x", y: "y", stroke: "#e59866", strokeWidth: 1.5}),
      Plot.line(series(x => Math.sin(x) + Math.sin(x + phi)), {x: "x", y: "y", stroke: "#C8102E", strokeWidth: 2.5})
    ]
  });
}

md`Sum amplitude right now: **${(Math.abs(2*Math.cos(phi/2))).toFixed(2)}** (in phase → 2, out of phase → 0)`

Takeaway

A wave is a disturbance described by \(u=f(x \pm vt)\) that obeys the classical wave equation \(u_{xx} = \frac{1}{v^2}u_{tt}\), with \(k=2\pi/\lambda\), \(\omega=2\pi\nu\), and \(\omega=kv\); superposing waves produces interference.