C.4 Standing waves and resonanceIB Physics SL: Revision notes
Section 1
Formation of standing waves
A standing wave forms when two identical waves (same frequency, wavelength, speed and similar amplitude) travel in opposite directions and superpose, usually a wave and its reflection. The wave profile does not move along; instead each point oscillates with its own amplitude. Unlike a travelling wave, a standing wave has no net transfer of energy along it.
Section 2
Nodes, antinodes, amplitude and phase
Nodes are points of zero amplitude, where the two waves always cancel. Antinodes are points of maximum amplitude. Adjacent nodes (or antinodes) are λ/2 apart; a node and the nearest antinode are λ/4 apart. All points between two adjacent nodes oscillate in phase with different amplitudes; points in neighbouring segments are in antiphase (π rad).
In a standing wave, amplitude varies with position and phase is either 0 or π. In a travelling wave it is the other way round: constant amplitude, phase varying continuously.
Section 3
Strings and pipes
String fixed at both ends: nodes at both ends; L = nλ/2, so fₙ = nv/2L, all harmonics n = 1, 2, 3 …
Pipe open at both ends: displacement antinodes at both ends; also L = nλ/2, all harmonics.
Pipe closed at one end: node at the closed end, antinode at the open end; L = λ/4, 3λ/4, 5λ/4 …, so only odd harmonics (f₁, 3f₁, 5f₁ …) with f₁ = v/4L. For the same length, its fundamental is half that of an open pipe.
Sketch the pattern in words first: count how many quarter- or half-wavelengths fit in L, then use v = fλ.
Section 4
Resonance
Every oscillating system has a natural frequency, the frequency at which it oscillates when displaced and released. A driving (forcing) frequency is imposed by an external periodic force. When the driving frequency equals the natural frequency, resonance occurs: energy is transferred at the greatest rate and the amplitude becomes maximum. The amplitude grows until the energy lost to damping each cycle equals the energy supplied. Far from the natural frequency the amplitude is small.
Section 5
Damping and resonance curves
Damping is the loss of energy from an oscillating system by resistive forces, reducing its amplitude. Increasing damping lowers the maximum amplitude, makes the resonance peak broader (less sharp) and shifts the resonant frequency slightly lower than the natural frequency.
Section 6
Light, critical and heavy damping
Light (under-) damping: the system oscillates many times with gradually decreasing amplitude (a pendulum in air).
Critical damping: the system returns to equilibrium in the shortest possible time without oscillating (car suspension, door closers).
Heavy (over-) damping: the system returns to equilibrium without oscillating but more slowly than with critical damping (a pendulum in thick oil).
Heavy damping does not return a system to equilibrium fastest; critical damping does.
That's the notes covered.
Carry on to the next subtopic.