C.4 Standing waves and resonance

Standing waves arise from the superposition of waves travelling in opposite directions, while resonance and damping describe how oscillating systems respond to external driving forces.

Key ideas

Standing waves have fixed nodes and antinodes.
Boundary conditions determine allowed wavelengths and frequencies.
Resonance maximises energy transfer when a system is driven near a natural frequency.
Damping limits amplitude and alters the resonance response.

Important formulas

v = f\lambda

Relationship between wave speed, frequency and wavelength.

f_n = \dfrac{nv}{2L}

Allowed frequencies for a string fixed at both ends or a pipe open at both ends.

f_n = \dfrac{nv}{4L}\ (n = 1,3,5,\dots)

Allowed frequencies for a pipe closed at one end (odd harmonics only).

A(\omega)\ \text{max at } \omega \approx \omega_0

Resonance occurs when the driving frequency is close to the natural frequency.

Practice problems

Formation of a standing wave

Explain how a standing wave is formed on a stretched string fixed at both ends when driven at its fundamental frequency. Comment on energy transfer.

Nodes, antinodes and phase

Describe how displacement amplitude and phase vary between nodes and antinodes in a standing wave.

Frequencies of a stretched string

A string of length 0.80 m is fixed at both ends. The wave speed is 120\ \mathrm{m\,s^{-1}}. Calculate the fundamental frequency and the third harmonic.

Standing waves in a closed pipe

A pipe of length 0.65 m is closed at one end. The speed of sound is 340\ \mathrm{m\,s^{-1}}. Determine the fundamental frequency and the next two resonances.

Resonance curve

A mass–spring system is driven at different frequencies. Describe how the steady-state amplitude depends on driving frequency and explain resonance.

Effect of damping on resonance

Explain how increasing damping affects the amplitude, width of the resonance peak, and resonant frequency.

Types of damping

Compare light, critical and heavy damping in terms of motion and return to equilibrium.