2.4. Standing Waves and Resonance (C4)

Standing waves are formed by the superposition of two travelling waves of the same frequency and amplitude moving in opposite directions, leading to a pattern of nodes and antinodes.

Travelling Waves

• Energy moves through the medium

• Wave propagates in space

• All points oscillate with different phases

• Common in: sound propagation, water ripples

Standing Waves

• Energy is not transferred — it stays within the system

• Result of two identical waves travelling in opposite directions interfering

• Certain points (nodes) remain at rest; others (antinodes) oscillate with maximum amplitude

• Common in: musical strings, air columns, microwave cavities

The shape (or allowed modes) of standing waves depends on how the ends of the system are constrained:

Fixed ends (e.g. string tied at both ends):

• Nodes at both ends.

• Only specific wavelengths allowed: λn=2L/n, n=1,2,3

✅ Open ends (e.g. open air column):

• Antinodes at both ends.

• Same equation for allowed wavelengths: λn=2L/n, n=1,2,3

✅ One fixed end and one open end (e.g. closed pipe):

• Node at closed end, antinode at open end.

• Only odd harmonics form: λn=4L/n, n=1,3,5,…

These boundary conditions determine the resonant frequencies, or natural modes, that the system can sustain.

Resonance occurs when an external force is applied at a frequency equal to a natural frequency of the system.

The system absorbs energy efficiently, leading to maximum amplitude oscillations.

Examples:

• Pushing a swing at just the right frequency.

• Tuning forks driving resonance in other objects.

• Sound waves reinforcing specific harmonics in a musical instrument.

In resonance:

• The driving frequency matches the system’s natural frequency.

• Energy builds up because it is added in phase with the motion.

• Leads to large amplitudes — if not damped, this can even lead to structural failure (like the Tacoma Narrows Bridge!).