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8.1.3Waves and superposition

Wavelength of Stationary Waves

5 min read 8 parts Lesson 14 of 21 1 quiz

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Lesson summary

8.1.3 Wavelength of Stationary Waves — AS PHYSICS 9702 notes: Find the wavelengths and frequencies of harmonics on strings and in air columns.Frequency — stationary waves have different patterns depending on the frequency of vibration.Situation — the pattern also depends on how t…

  • Resonant frequencies
  • Fundamental mode
  • Higher harmonics
  • nth harmonic
  • Higher modes
  • Harmonics present

Section 1 of 8Patterns depend on the situation

Frequency
  • Frequency stationary waves have different patterns depending on the frequency of vibration.
Situation
  • Situation the pattern also depends on how the wave is created, such as a string or an air column.
Resonant frequencies
  • Resonant frequencies the possible frequencies depend on the length L and the wave speed v.

Section 2 of 8String fixed at both ends

Fundamental mode
  • Fundamental mode the simplest pattern is a single loop of two nodes and one antinode, also called the first harmonic.
Wavelength
  • Wavelength only half a wavelength fits in the string, so λ1 = 2L.
Higher harmonics
  • Higher harmonics they appear as the frequency increases.
nth harmonic
  • nth harmonic it has n antinodes and n + 1 nodes, with wavelength λn = 2L/n.
  • The second harmonic has three nodes and two antinodes, with λ = L.
  • The third harmonic has four nodes and three antinodes, with λ = 2L/3.
Frequency
  • Frequency found from the wave equation, fn = v/λn = nv/2L.

Section 3 of 8Air column closed at one end

Ends
  • Ends a node forms at the closed end and an antinode at the open end.
Fundamental
  • Fundamental a quarter wavelength with one node and one antinode, so L = λ/4.
Higher modes
  • Higher modes each extra harmonic adds one node and one antinode.
Harmonics present
  • Harmonics present only odd harmonics form.
General length
  • L = nλ/4 with n odd, for example L = 3λ/4 for the third harmonic.

Section 4 of 8Air column open at both ends

Ends
  • Ends an antinode forms at each open end.
Fundamental
  • Fundamental half a wavelength with one node and two antinodes, so L = λ/2.
Higher modes
  • Higher modes each extra harmonic adds one node and one antinode.
Harmonics present
  • Harmonics present odd and even harmonics both form.
General length
  • L = nλ/2.

Section 5 of 8Harmonic numbering

Harmonic numbering
Fundamental
  • Fundamental it is the first harmonic, n = 1, the lowest frequency.
Both ends fixed or both open
  • Both ends fixed or both open a full wavelength is the second harmonic.
Matching
  • Matching match the wavelength to the harmonic named in the question, for example the second lowest note in a closed pipe is the third harmonic.

Section 6 of 8More from the specification

1Reflected ultrasonic wave pulses
  • Reflected ultrasonic wave pulses transmitting periodic wave pulses along boundaries creates interfering counter-propagating waves forming stable stationary harmonics.
2Displacement
  • Change in position.
3Amplitude
  • Maximum displacement from equilibrium.
4Resultant
  • Vector sum of two or more vectors.
5Pulse
  • Single disturbance travelling through a medium.
6Loudspeaker
  • Device that converts an electrical signal into sound by the motor effect.
7Standing wave
  • A pattern formed by two identical waves travelling in opposite directions, with fixed nodes and antinodes.
8Origin
  • Origin a stationary wave arises from superposition of a wave and its reflection;
    • adjacent nodes, or adjacent antinodes, are half a wavelength apart.

Section 7 of 8Key terms

TermMeaning
Fundamental modeSimplest stationary wave pattern with the lowest resonant frequency, the first harmonic.
HarmonicResonant mode of a stationary wave, with frequency a whole-number multiple of the fundamental where allowed.
Resonant frequencyFrequency at which a stationary wave forms in a string or air column.
Closed pipePipe closed at one end with a node there and an antinode at the open end.
Open pipePipe open at both ends with an antinode at each end.

Section 8 of 8Exam technique

1
Using L = λ for the fundamental of a string — the fundamental has L = λ/2.
2
Allowing even harmonics in a pipe closed at one end — only odd harmonics occur.
3
Putting an antinode at a closed end — a closed end is a node.
4
Naming the second lowest note in a closed pipe the second harmonic — it is the third harmonic.
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