Physics › Waves › Stationary waves
Stationary waves
Two waves travelling in opposite directions can lock into a pattern that stays where it is. Fix both ends of a string and only certain wavelengths survive, so the string ends up with its own set of natural frequencies.
Builds on Progressive waves.
IN THIS TOPIC
- Describe how two travelling waves superpose to form a stationary wave, and define node and antinode.
- Find the allowed wavelengths of a string fixed at both ends, and use the first-harmonic frequency equation.
- State the amplitude and phase relationships that separate stationary waves from progressive ones.
WHAT YOU PROBABLY THINK
A stationary wave is a wave that has stopped moving.
Two waves, one string
Send a single pulse along a rope tied to a wall and it reflects and comes back. Drive the rope continuously instead, and the reflected wave has to travel back through the incoming one. The rope now carries two waves at the same time. They have the same frequency and the same speed, their amplitudes are similar, and they move in opposite directions.
Wherever two waves overlap, the displacement of the rope is the sum of the displacements each wave would produce on its own. This is the principle of superposition, and it is the only new idea this topic needs. At most driving frequencies the sum is a shifting muddle. At certain frequencies, though, the two waves combine into something with a striking property: the pattern stays where it is. This is a stationary wave, also called a standing wave.
Be careful with the name. The rope itself is still moving, in places more violently than either travelling wave would move it alone. What has stopped is the pattern. The crests no longer travel along the rope. Instead, each point of the rope oscillates about its rest position with a fixed amplitude of its own.
Nodes and antinodes
At certain points the two waves always arrive in antiphase, so they cancel and the rope there never moves at all. These points are called nodes. Midway between them the waves always arrive in phase and reinforce, so the rope swings with the largest amplitude found anywhere on the pattern. These points are antinodes.
Two distances are worth learning. Adjacent nodes are λ/2 apart, because one complete wavelength contains two loops of the pattern, not one. The distance from a node to the nearest antinode is λ/4.
A stationary wave also transfers no energy along the rope. Each section keeps its own energy, which passes back and forth between kinetic and potential forms as the rope oscillates. Contrast this with a progressive wave, whose whole job is to carry energy from one place to another.
Harmonics on a stretched string
A string fixed at both ends must have a node at each end, because a fixed point cannot move. That single condition restricts the wavelengths the string can support: a pattern only fits if it places a whole number of loops between the two ends.
Each loop is half a wavelength, so a pattern of n loops on a string of length L has
The simplest pattern, n = 1, is the first harmonic. Many books call it the fundamental, but AQA papers say first harmonic, and it is sensible to use their term. It has the longest wavelength the string allows, 2L, and therefore the lowest frequency. The wave speed is fixed by the string itself, so v = fλ makes the harmonic frequencies whole-number multiples of the first: fn = nf1.
The frequency of the first harmonic is
where T is the tension in the string and μ is its mass per unit length, measured in kg m−1. Raising the tension or using a lighter string raises the frequency. A guitarist uses both: the tuning pegs change T, and the thicker, heavier strings play the lower notes.
Phase along a stationary wave
On a progressive wave, every point oscillates with the same amplitude, and there is a steady phase difference between one point and the next. A stationary wave differs on both counts. The amplitude depends on position, running from zero at a node to a maximum at an antinode, and only two phase relationships exist anywhere on the pattern.
All the points between one pair of adjacent nodes move in phase. They reach their maximum displacements at the same instant, though each has its own amplitude. Points on opposite sides of a node move in antiphase, a phase difference of π rad or 180°. Cross two nodes and the motion is back in phase.
THE EXAM BIT
- Describing formation: say that two waves of the same frequency, travelling in opposite directions, superpose. Answers about the wave “bouncing back and interfering with itself” usually drop a mark.
- Adjacent nodes are λ/2 apart, not λ. If in doubt, sketch two loops and read the wavelength off your own diagram.
- Use the term first harmonic on AQA papers rather than “fundamental”.
- In the first-harmonic equation, μ is mass per unit length. If the mass is given in grams, convert to kilograms before dividing by the length; the unit kg m−1 is the reminder.
- A phase-difference question about a stationary wave has only two possible answers: 0 or π rad.
CHECK YOURSELF
A string 1.20 m long is fixed at both ends and vibrates in its third harmonic. (a) What is the wavelength? (b) At the same tension, how does its frequency compare with the first harmonic?
Show a hint
How many half-wavelengths fit into the 1.20 m?
Show the answer
(a) The third harmonic has three loops. Each loop is half a wavelength, so L = 3λ/2, which gives λ = 2L/3 = 2 × 1.20 / 3 = 0.80 m.
(b) The wave speed depends only on the tension and the mass per unit length, so it has not changed. In the first harmonic the wavelength is 2L = 2.40 m. The wavelength has fallen by a factor of three, so by v = fλ the frequency has risen by the same factor: the third harmonic is three times the frequency of the first.
Notice what decided the answer. The fixed ends select the wavelengths that can exist, and the fixed wave speed then turns each wavelength into a frequency. The oscillator driving the string does not choose the frequencies; the string does.
Nothing travels along the string.
Only certain wavelengths are allowed.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.