Waves

Chem 3240 · Lecture 2.1

Davit Potoyan

Why waves, in a chemistry course?

• Chapter 1: electrons diffract, and Bohr’s orbits are standing waves that have to fit
• Quantum mechanics is written in the language of waves
• Today: the vocabulary (\lambda, \nu, phase), the equation of motion of a wave, and how waves add
• Next lecture: waves with fixed ends produce integers

What is a wave?

• A self-propagating disturbance: carries energy and information, not matter
• Transverse: medium moves perpendicular to travel (string, light)
• Longitudinal: medium moves along travel (sound)
• The red bead never travels; only the pattern does

A shape that moves: f(x - vt)

• Surfer rides the wave: shape frozen at x'
• Shore sees it slide: x = x' + vt

u(x,t) = f(x') = f(x - vt)

• Hold the peak fixed: x - vt = \text{const} moves right at speed v
• f(x + vt) moves left

Periodic traveling waves

u(x,t) = A\sin(kx - \omega t + \phi)

• Amplitude A, wave number k, angular frequency \omega = kv, phase \phi

Wavelength, period, speed

Repeats every \lambda in space and every T in time: k\lambda = 2\pi and \omega T = 2\pi

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

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

v = \frac{\omega}{k} = \lambda\nu

• One wavelength per period. The medium sets v, so fixing \lambda fixes \nu

Complex representation

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

• Real part cosine, imaginary part sine
• At fixed x: a phasor turning at rate \omega
• Derivatives become multiplications by ik and -i\omega
• Take the real part at the very end

The classical wave equation

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

Two x-derivatives, two t-derivatives

\frac{\partial^2 u}{\partial x^2} = (ik)^2 u = -k^2 u, \qquad \frac{\partial^2 u}{\partial t^2} = (-i\omega)^2 u = -\omega^2 u

Divide and use \omega = kv: the ratio is k^2/\omega^2 = 1/v^2

\frac{\partial^2 u}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}

• Holds for every shape f(x \mp vt) by the chain rule: the equation of motion of waves

What the equation says: curvature pulls

• Newton’s law for a piece of string: acceleration follows curvature. Neighbors pull a trough up and a crest down; a straight piece feels nothing
• v^2 = \mathcal{T}/\mu, tension over mass per length: tighter or lighter strings carry faster waves

Example: read a wave off its formula

u(x,t) = \sin(2x - 10t + \pi/2) in SI units. Find A, \lambda, \nu, T, and v.

A = 1, \qquad k = 2\ \text{m}^{-1} \Rightarrow \lambda = \pi\ \text{m}, \qquad \omega = 10\ \text{s}^{-1} \Rightarrow \nu = \tfrac{5}{\pi}\ \text{Hz},\; T = \tfrac{\pi}{5}\ \text{s}

v = \frac{\omega}{k} = 5\ \text{m/s to the right}, \qquad \sin(\theta + \pi/2) = \cos\theta

• Match to A\sin(kx - \omega t + \phi) and read off; only v = \omega/k needs a calculation

Superposition and interference

• Linear equation: u_A + u_B is again a solution
• Equal waves, phase difference \phi:

\text{sum} = 2\cos\tfrac{\phi}{2}\; e^{i(\theta + \phi/2)}

• \phi = 0: amplitude doubles
• \phi = \pi: waves cancel

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: "#107895", strokeWidth: 1.5}),
      Plot.line(series(x => Math.sin(x + phi)), {x: "x", y: "y", stroke: "#e07b00", 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)`

Standing waves: two waves, opposite directions

• x and t separate: the shape \sin kx stays put, its height \cos\omega t oscillates
• Nodes at kx = n\pi, spaced \lambda/2, never move
• Clamped ends must be nodes: only L = n\lambda/2 fits

\sin(kx-\omega t) + \sin(kx+\omega t) = 2\sin(kx)\cos(\omega t)

Can you surf a light wave?

• Our first move was to ride along with the wave, x' = x - vt. It works because water, sound, and strings have a medium
• Maxwell: light obeys the wave equation with no medium and a speed made of constants

\frac{\partial^2 E}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}, \qquad c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}

• Einstein at 16: run beside a light beam and see a frozen ripple? Maxwell forbids it

• 1905: take the equation at its word. Light moves at c for every observer. There is no surfer frame, and x' = x - vt must fail near c

The light clock: moving clocks run slow

\Delta t = \gamma\,\Delta t_0, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

• Same c, longer zigzag, slower tick
• Gold’s inner electrons: v/c \approx 0.6, which is why gold is yellow

Takeaway

A wave is any shape f(x \mp vt). Sine waves carry k = 2\pi/\lambda and \omega = 2\pi\nu with v = \omega/k, every wave obeys u_{xx} = u_{tt}/v^2 because acceleration follows curvature, and since that equation is linear, waves add. Light is the one wave nobody can surf.