DPST1021 Chap.13 Travelling Waves, Superposition and Standing Waves
Travelling Waves, Superposition and Standing Waves
Week 11 of Physics 1A moves from a single oscillator to a disturbance that travels. Lecture 27 separates transverse waves, where the medium moves across the direction of travel, from longitudinal waves, where it moves along it, and introduces the sinusoidal wave function y(x, t) = A sin(kx − ωt + φ).
You are expected to name every symbol in that expression, compute amplitude, wavelength, frequency and speed from a given equation, and sketch the wave together with the motion of its particles.
Lecture 28 then supplies two string results without requiring their derivations: the speed set by tension and linear density, and the average power carried by a sinusoidal wave.
It also covers what happens at a boundary, where part of a wave can reflect and part can pass on, and the principle of superposition, under which overlapping displacements simply add.
Two waves that differ only in phase combine into one wave of modified amplitude, a result the lecture asks you to derive.
Lecture 29 combines waves running in opposite directions to produce a standing wave, with nodes that never move and antinodes that swing hardest. Clamping both ends of a string restricts the allowed wavelengths, which leads to the harmonic series and the frequency formula on the data sheet.
The material is assessed in Homework Quiz 9 and Test 2, and Lectures 30 to 32 carry the same ideas into sound and air columns.
What this chapter covers
- 01
Wave types: distinguish transverse from longitudinal motion of the medium and give an everyday example of each, such as a string and sound in air.
- 02
Wave function terms: read amplitude, wave number, angular frequency and phase constant from y(x, t), and convert them to wavelength, frequency and speed.
- 03
Direction and particle motion: decide the travel direction from the sign between kx and omega t, and find the peak transverse speed as omega A.
- 04
String speed and power: calculate speed from tension and linear density, then the transmitted power, noting its dependence on amplitude squared and omega squared.
- 05
Boundaries: sketch reflected and transmitted pulses at fixed ends, free ends and joins between light and heavy strings, keeping frequency unchanged across the join.
- 06
Superposition and phase: add two equal waves differing in phase to get amplitude 2A cos(phi/2), with constructive and destructive limits.
- 07
Standing waves and harmonics: locate nodes and antinodes, and list allowed wavelengths and frequencies for a string fixed at both ends.
Worked example · free
Recovering a phase difference from a resultant amplitude
- 1Superposition of equal waves differing in phase by φ gives amplitude 2A cos(φ/2), so cos(φ/2) = 6.0/8.0 = 0.75.
- 1Inverse cosine: φ/2 = 41.4°, so φ = 82.8°, or 1.45 rad.
- 1A full wavelength corresponds to 2π of phase, so the offset is Δx = (1.445/2π) × 0.50 m = 0.12 m.
- 1Check the limits: a zero phase difference would give 8.0 mm and a difference of π would give zero, so 6.0 mm correctly needs a value between them.
Key terms
- Transverse wave
- A wave in which each element of the medium is displaced at right angles to the propagation direction, for example on a guitar string.
- Longitudinal wave
- A disturbance in which the medium moves parallel to the direction of travel, producing compressions and rarefactions, as in sound.
- Wave number
- The quantity k equal to 2 pi divided by the wavelength, measured in radians per metre.
- Linear density
- The mass per unit length of a string, mu, in kilograms per metre, which together with tension sets the wave speed.
- Node
- A point on a standing wave with zero displacement at all times, located half a wavelength from its neighbours.
- Fundamental
- The lowest-frequency standing wave a string can support, also called the first harmonic, with wavelength twice the string length.
Travelling Waves, Superposition and Standing Waves FAQ
Is the speed of a string element the same as the speed of the wave?
No. The wave speed depends only on tension and linear density, while each element moves up and down with a speed that changes through its cycle and peaks at omega times A. A high-frequency, large-amplitude wave can have elements moving faster than the wave itself, and a gentle wave can have them moving far slower.
What happens to the wavelength when a wave moves onto a heavier string?
The source still sets the frequency, but the heavier string has a larger linear density and so a lower speed. Since wavelength equals speed divided by frequency, it shortens by the same factor as the speed. A string four times as heavy per metre halves both the speed and the wavelength.
How can I tell the travel direction if the equation is written differently?
Rewrite it so that x and t appear together as kx minus omega t or kx plus omega t, even if the sine is written as sin(omega t minus kx), since that form equals minus sin(kx minus omega t) and still moves toward positive x. Follow a crest: if keeping the phase fixed needs x to grow with time, the wave moves forward.
Exam move
Treat the wave function as a decoding exercise first. Take several equations with different signs and phase constants, and for each one state amplitude, wavelength, frequency, speed and direction before doing anything else. Then sketch the snapshot at t = 0 and mark which elements are rising, because Lecture 27 expects both the drawing and the link to simple harmonic motion.
For string speed and power, write mu in kilograms per metre before substituting, since grams per metre is the most frequent unit error. Practise superposition with phase differences of 0, pi over 2 and pi until the amplitude results feel automatic. Finally, draw the first four harmonics on a fixed string from memory, count nodes and antinodes, and match each to its wavelength and frequency.
Homework Quiz 9 spans all three lectures, and its questions are the pool Test 2 draws on, so treat it as the main rehearsal.
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