Theme C — Wave Behaviour
C.4.4 Harmonics in Strings and Pipes — IB Physics HL Revision Notes & Topical Questions
5 min read 11 parts Lesson 30 of 37 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
C.4.4 Harmonics in Strings and Pipes — IB PHYSICS SL notes: Calculate the frequencies of the harmonics for strings and for open and closed pipes and relate them to the fundamental.Fundamental (first harmonic) — the standing wave with the longest wavelength that satisfies the boun…
- Check first
- Wave speed
- Instruments
- Guitar fret
- Shared energy
- Centre pluck
Section 1 of 11Fundamental and harmonics
1Fundamental (first harmonic)
- Fundamental (first harmonic) the standing wave with the longest wavelength that satisfies the boundary conditions, with the lowest frequency f1
2Harmonics
- Harmonics the higher-frequency standing waves that also fit the boundary conditions
- They are the only frequencies and wavelengths that form standing waves in the medium.
3Frequency
- Frequency in a given medium f = v/λ, so each allowed wavelength gives an allowed frequency
4Timbre
- Timbre the mixture of harmonics determines the sound quality of an instrument
5Check first
- Check first look at the boundary conditions before any calculation, because they set the wavelengths
Section 2 of 11Strings fixed at both ends
Condition
- Condition nodes at both ends mean L = nλn/2
Wavelength
- λn = 2L/n
Frequency
- Frequency fn = nv/2L = nf1, with n = 1, 2, 3 …
- Every integer multiple of the fundamental occurs.
Fundamental
- Fundamental one loop made of two nodes (the fixed ends) and one antinode, with λ1 = 2L and f1 = v/2L
Pattern
- Pattern the n-th harmonic has n antinodes and n + 1 nodes including the ends
- The second harmonic has three nodes and two antinodes.
- The third harmonic has four nodes and three antinodes.
Wave speed
- Wave speed v = √(T/μ) for tension T and mass per unit length μ
- Tightening a string raises its pitch.
- Thicker strings sound lower.
Section 3 of 11Pipe open at both ends
Condition
- Condition antinodes at both ends, so L = nλn/2 as for a string
Frequency
- Frequency fn = nv/2L with n = 1, 2, 3 … and a complete series of harmonics
Pattern
- Pattern the n-th harmonic has n nodes and n + 1 antinodes
- The fundamental has a single node in the middle and two antinodes.
- The second harmonic has two nodes and three antinodes.
Instruments
- Instruments flutes and open organ pipes produce the full series
Section 4 of 11Pipe closed at one end
Condition
- Condition a node at the closed end and an antinode at the open end, so L is an odd number of quarter-wavelengths
Wavelength
- λn = 4L/n with n = 1, 3, 5 …
Frequency
- Frequency fn = nv/4L, so the frequencies are f1, 3f1, 5f1 …
Pattern
- Pattern the lowest harmonic is a half-loop with one node and one antinode
- The next pattern has two nodes and two antinodes and is the third harmonic, not the second.
Octave
- Octave the fundamental v/4L is half that of an open pipe of the same length, so it sounds an octave lower
Tone
- Tone the missing even harmonics give a clarinet-like tone
Section 5 of 11Harmonics and overtones
Overtone
- Overtone any resonant frequency above the fundamental
- The first overtone is the second harmonic in a string or open pipe but the third harmonic in a closed pipe.
Real instruments
- Real instruments they produce a mixture of harmonics, and adding them gives the characteristic waveform
Section 6 of 11Changing the pitch
Guitar fret
- Guitar fret pressing a finger on a fret shortens L, so f1 = v/2L rises
Flute
- Flute opening holes changes the effective length of the pipe
Slide instruments
- Slide instruments a trombone alters L continuously
Ratios
- Ratios frequency ratios depend only on length ratios when the speed is unchanged
Section 7 of 11Plucking position
Shared energy
- Shared energy a plucked string stores energy in several harmonics at once
Centre pluck
- Centre pluck it excites odd harmonics strongly and suppresses even harmonics, which have a node there, giving a mellow tone
Bridge pluck
- Bridge pluck it adds many higher harmonics and gives a brighter tone
Section 8 of 11Resonance tube
1
Lower a tube into water while a tuning fork of known frequency sounds above it.
2
Raise or lower the tube until a loud sound occurs, which is resonance.
3
Note each air-column length at which resonance occurs.
4
Resonance happens at lengths of about λ/4, 3λ/4 and so on.
5
Use the difference between successive resonance lengths, which is λ/2 and cancels the end correction.
- Speed of sound
- v = fλWhat the symbols mean: v = wave speed (m s−1) · f = frequency (Hz) · λ = wavelength (m)
- v = fλ
Section 9 of 11More from the specification
1
Harmonic standing wave phase
- Harmonic standing wave phase particles situated between adjacent wave nodes vibrate simultaneously in phase,
- while reversing phase across each nodal point.
2
Natural Frequency and Free Oscillations
- When an oscillating mechanical system is displaced and released, it vibrates freely at its characteristic natural frequency without any periodic external driving force.
3
Forced Vibrations and Resonance
- When a periodic external driving force acts on an oscillator, the system executes forced oscillations at the driving frequency;
- if the driving frequency matches the natural frequency, the amplitude reaches a dramatic maximum known as resonance.
4
Mechanical Damping
- Damping forces dissipate mechanical energy from an oscillating system as thermal energy,
- reducing the oscillation amplitude over time; damping is categorized as light, critical,
- or heavy (overdamping).
5
Critical Damping Applications
- In measuring instruments such as analogue weighing scales and vehicle shock absorbers, critical damping allows the displaced indicator to return to its equilibrium position in the shortest possible time without oscillating.
6
Driving frequency
- The frequency of the periodic external force
7
Forced oscillation
- Oscillation caused by a periodic driving force
8
Progressive wave
- Wave that transfers energy without transferring matter.
9
Driving force
- Force applied by an engine or motor to produce motion.
10
Heavy damping
- Damping greater than critical, giving a slow return without oscillation
11
External force
- Force from the surroundings.
12
Resistive force
- Friction and drag opposing motion.
13
Damping
- Energy loss to friction that reduces amplitude.
14
Oscillation
- Repeated back-and-forth motion.
15
Amplitude
- Maximum displacement from equilibrium.
16
Equilibrium
- State with zero net force.
17
Stationary
- Not moving.
18
Force
- A push or pull measured in newtons.
19
Position
- Location relative to a reference point.
20
Natural frequency
- The frequency at which a system oscillates freely
Section 10 of 11Key terms
| Term | Meaning |
|---|---|
| Fundamental | Lowest-frequency standing wave. |
| Harmonic | Standing wave with a frequency that is a multiple of the fundamental. |
| Overtone | Any resonant frequency above the fundamental. |
| Open pipe | Open at both ends, antinodes at each end. |
| Closed pipe | Closed at one end, node there and antinode at the other. |
Section 11 of 11Exam technique
1
Believing closed pipes have all harmonics — only odd ones occur.
2
Using L = λ/2 for the fundamental of a closed pipe.
3
Calling the first overtone the first harmonic.
4
Using the speed of light or the wrong medium's speed.
5
Forgetting that tightening a string changes its wave speed and therefore its frequency.
Marked practiceQuizzes for this lesson
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