C.4 Standing waves and resonance

UNIT SYLLABUS

SL/HL 4 hours

Trap a wave between two boundaries and it interferes with its own reflection: the result is a standing wave, with nodes that never move and antinodes that oscillate hardest. Unlike travelling waves, standing waves transfer no energy — they store it. Only certain wavelengths fit the boundary conditions, which is why a guitar string or organ pipe sounds definite notes: the harmonics. The same physics of natural frequencies leads to resonance — drive any oscillator at its natural frequency and the amplitude grows dramatically, for better (musical instruments, MRI) or worse (bridges, buildings in earthquakes). Damping tames the response.

Guiding Questions

? How do standing waves arise from the superposition of travelling waves?
? Under what conditions does resonance occur, and how is it controlled?

First three harmonics on a string fixed at both ends: each mode fits a whole number of half-wavelengths between the fixed nodes.

✓ What the IB expects you to master

Explain standing wave formation as superposition of two identical waves travelling in opposite directions.
Identify nodes and antinodes, and compare amplitudes and phases of points along a standing wave (all points between adjacent nodes in phase; opposite sides of a node in antiphase).
Determine the standing wave patterns and harmonic frequencies for strings (fixed/free ends) and for open and closed pipes.
Contrast standing and travelling waves: energy storage vs transfer, amplitude varying with position vs constant.
Describe resonance: amplitude response driven at the natural frequency.
Explain how damping reduces maximum amplitude and slightly lowers the resonant frequency; distinguish light, critical and heavy damping.

1 Key Formulas

Harmonics (string / open-open pipe)

f_{n} = \frac{nv}{2L}

Harmonics (closed-open pipe)

f_{n} = \frac{nv}{4L},\quad n = 1, 3, 5, \ldots

2 Exam Preparation & Topic Explanations

Drawing the pattern before the algebra

Sketch the boundary conditions first: fixed string end or closed pipe end = node; free end or open pipe end = antinode. Fit the simplest pattern, then count half- or quarter-wavelengths to relate $\lambda$ to $L$ . The formula follows from the sketch — never the other way round.

Remember closed–open pipes skip even harmonics; strings and open–open pipes have them all.

Pro Exam Strategy

The fundamental of a closed–open pipe fits a quarter wavelength: $\lambda_1 = 4L$ .
In pipes the wave is longitudinal — the sketched curve shows displacement amplitude envelope, not string shape.
Resonance graph: amplitude against driving frequency peaks at natural frequency; damping lowers and broadens the peak.
End corrections and overtones vs harmonics wording: the $n$ th harmonic has frequency $nf_1$ — the IB counts harmonics, not overtones.

3 MCQ Practice

Q1. At a node of a standing wave:

The amplitude is maximum and energy is transferred
The displacement is always zero
The particles oscillate with maximum speed
The wave changes phase by 90°

Q2. A pipe closed at one end resonates at a fundamental frequency of 85 Hz. Its next resonant frequency is:

170 Hz
255 Hz
340 Hz
425 Hz

Q3. Two points on a standing wave separated by one node oscillate:

In phase with equal amplitudes
In antiphase, generally with different amplitudes
90° out of phase
In phase, generally with different amplitudes

4 Short Answer Questions

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