The travelling-wave equation, phase and phase difference, and the formation and properties of stationary waves.

What this key area is asking

The SQA wants you to use the travelling-wave equation to describe a wave and extract its amplitude, frequency and wavelength, understand phase and phase difference, and explain the formation and properties of stationary (standing) waves by superposition.

The travelling-wave equation

This equation describes the displacement of any point in the medium at any time. By comparing a given equation with the standard form you read the amplitude straight off, equate the time coefficient to find the frequency, and equate the space coefficient to find the wavelength. The wave speed follows from $v = f\lambda$.

Phase and phase difference

Two points are in phase (moving identically) when their separation is a whole number of wavelengths, giving a phase difference of $0, 2\pi, 4\pi, \dots$. They are in antiphase (exactly opposite) when separated by an odd number of half-wavelengths, a phase difference of $\pi, 3\pi, \dots$. Phase difference is central to interference, where it decides whether waves reinforce or cancel.

Stationary waves

At a node the two waves always cancel, so the displacement is permanently zero; at an antinode they always reinforce, giving maximum amplitude. Nodes (and antinodes) are spaced half a wavelength apart, so measuring node spacing gives the wavelength. Stationary waves explain the resonant frequencies of strings and air columns: only wavelengths that fit the boundary conditions, with nodes at fixed ends, are allowed, producing the fundamental and its harmonics.

Examples in context

A guitar or violin string vibrates in stationary-wave patterns whose allowed wavelengths set the pitch, with the ends as nodes. An organ pipe or flute supports stationary sound waves in a column of air, with antinodes at open ends. Microwave ovens form stationary waves, which is why turntables are needed to even out the hot and cold spots at antinodes and nodes. Resonance in bridges and buildings, where a driving frequency matches a natural stationary-wave mode, is a design concern engineers must avoid.

Try this

Q1. State what the coefficient of $t$ in a travelling-wave equation $y = A\sin(2\pi f t - \frac{2\pi}{\lambda}x)$ represents. [1 mark]

• Cue. $2\pi f$, so it gives the frequency.

Q2. State the phase difference, in radians, between two points one wavelength apart on a wave. [1 mark]

• Cue. $2\pi$ (they are in phase).

Q3. State the distance between adjacent nodes on a stationary wave in terms of wavelength. [1 mark]

• Cue. Half a wavelength.