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Traveling Waves

Source lecture(s): SC133 Lec 22

Intuition

Flick a rope: a bump races along it, yet no piece of rope travels with the bump — each piece only bobs in place, briefly, handing the motion to its neighbor. A wave transports energy and information without transporting matter. Stadium waves, sound, light, earthquakes: the pattern moves; the medium (if any) stays home.

Describing a wave

A sinusoidal wave moving in \(+x\):

\[y(x, t) = A\sin(kx - \omega t + \phi)\]
Symbol Name Meaning
\(A\) amplitude maximum displacement
\(k = 2\pi/\lambda\) wave number radians per metre
\(\omega = 2\pi f\) angular frequency radians per second
\(\lambda\) wavelength spatial period
\(T = 1/f\) period temporal period

The pattern repeats in space every \(\lambda\) and in time every \(T\); a crest advances at the phase speed

\[\boxed{\,v = \lambda f = \frac{\omega}{k}\,}\]

Sign flip (\(kx + \omega t\)) reverses the direction. Transverse waves displace ⊥ to travel (rope, light); longitudinal waves displace along it (sound).

What sets the speed: the medium

For a stretched string with tension \(F_T\) and linear density \(\mu\) (kg/m):

\[v = \sqrt{\frac{F_T}{\mu}}\]

The pattern is universal: \(v = \sqrt{\text{restoring property}/\text{inertia property}}\) — tension/density here, stiffness/density for sound, gravity/depth for water waves. Speed belongs to the medium; frequency belongs to the source; wavelength adjusts: \(\lambda = v/f\).

The wave equation

Newton's second law applied to a string element yields

\[\frac{\partial^2 y}{\partial t^2} = v^2\,\frac{\partial^2 y}{\partial x^2}\]

— the archetypal wave equation. Any function of \((x \mp vt)\) solves it: shape in, shape out, moving at \(v\). This same PDE returns for sound, light (SC134), and gets solved numerically in PHY653's FDTD.

Energy transport

Each element does SHM with the wave's frequency; the average power carried by a sinusoidal string wave is

\[P = \tfrac12 \mu \omega^2 A^2 v\]

Energy flow \(\propto A^2\) and \(\propto \omega^2\) — amplitude-squared scaling is a theme from SHM through optics.

Worked example: guitar string

A 65 cm guitar string (\(\mu = 6\times10^{-3}\) kg/m) is tuned to 110 Hz (A2) fundamental. Required tension? (Fundamental: \(\lambda = 2L\); see standing waves.)

\(v = \lambda f = 1.3\times110 = 143\,\text{m/s}\); \(F_T = \mu v^2 = 6\times10^{-3}\times(143)^2 \approx 123\,\text{N}\) — about 12.5 kg of pull. Six strings ⇒ nearly 70 kg on the neck, which is why guitars have truss rods.

Try it live

The wave studio animates traveling waves and their superposition — drag frequency and speed, then switch to standing-wave and beats modes.

Common mistakes

  • The medium travels with the wave. No — watch a floating leaf as ripples pass: it circles in place.
  • "Higher frequency means faster wave." In a non-dispersive medium, \(v\) is fixed; raising \(f\) just shortens \(\lambda\).
  • Confusing wave speed with particle speed. The string element's speed (\(\partial y/\partial t\), max \(A\omega\)) is unrelated to the pattern speed \(v\).
  • Mixing \(k\) and \(\omega\) roles — \(k\) counts radians per metre, \(\omega\) per second; their ratio, not either alone, is the speed.

Knowledge graph position

Prerequisites: SHM. Leads to: Superposition, Sound, EM waves (SC134), plasma waves (PC368).

Quiz

Q1 (computational). A wave \(y = 0.02\sin(4\pi x - 200\pi t)\) (SI). Find \(\lambda\), \(f\), and \(v\).

Answer

\(k = 4\pi \Rightarrow \lambda = 0.5\,\text{m}\); \(\omega = 200\pi \Rightarrow f = 100\,\text{Hz}\); \(v = \omega/k = 50\,\text{m/s}\) (moving in \(+x\)).

Q2 (conceptual). You double the tension in a string while the attached oscillator keeps the same frequency. What happens to \(v\) and \(\lambda\)?

Answer

\(v = \sqrt{F_T/\mu}\) grows by \(\sqrt2\); frequency is fixed by the source, so \(\lambda = v/f\) also grows by \(\sqrt2\).

Q3 (multiple choice). A wave pulse travels down a rope. The rope's material moves: (a) along with the pulse (b) perpendicular to the rope, briefly (c) not at all

Answer

(b). Transverse motion only — each element bobs and settles as the pulse passes through it.