Standing Waves and Resonance

A standing (stationary) wave forms when two waves of equal frequency and amplitude travel through each other in opposite directions and superpose. In practice the second wave is almost always the reflection of the first: a wave sent along a string reflects from the far end and overlaps with the waves still arriving. At certain frequencies the superposition settles into a stable pattern that appears to stand still rather than travel.

Standing waves rely entirely on the phase change that occurs when a wave reflects at a boundary. When a transverse wave on a string meets a fixed boundary, it undergoes a phase change of $\pi\ \text{rad}$ ($180^\circ$); the incoming and reflected waves interfere destructively at that point, establishing a permanent node.

Conversely, when a wave meets a free boundary, or a sound wave reaches the open end of a pipe, it reflects with zero phase change, so the interference there is constructive and an antinode forms.

Unlike a travelling wave, a standing wave transfers no net energy along the medium: the energy is trapped within each loop, cycling between kinetic and potential forms.

You must be able to explain nodes and antinodes in terms of superposition and state the formation conditions:

• same frequency
• similar amplitude
• opposite directions

Nodes are points of permanently zero displacement, produced by continuous destructive interference; antinodes are points of maximum amplitude, produced by continuous constructive interference.

The pattern has a strict geometry: adjacent nodes are $\frac{\lambda}{2}$ apart, and a node and its neighbouring antinode are $\frac{\lambda}{4}$ apart. This is how you extract a wavelength from a sketch or photograph of a standing wave.

Two features separate standing waves from travelling waves. First, amplitude depends on position: a point at an antinode swings widely while a point near a node barely moves, whereas on a travelling wave every point eventually oscillates with the same amplitude.

Second, phase comes in only two values: all points within one loop oscillate in phase ($0$ phase difference), while points in adjacent loops, separated by an odd number of nodes, are $\pi\ \text{rad}$ out of phase.

Because the boundaries fix where nodes and antinodes must sit, only certain wavelengths fit on the system, so only a discrete set of frequencies, the harmonics, can be sustained. For a guitar string fixed at both ends, the second harmonic has a node at the midpoint and twice the fundamental frequency.

Be ready to compare the amplitude and phase of two labelled points on a diagram.

Boundary conditions determine which harmonics exist. A fixed end of a string, or the closed end of a pipe, forces a displacement node; a free end, or the open end of a pipe, forces a displacement antinode.

For a string fixed at both ends, whole numbers of half-wavelengths must fit in the length $L$, so $\lambda_n = \frac{2L}{n}$ and the harmonic frequencies are given by:

$$f_n = \frac{nv}{2L}, \quad n = 1, 2, 3...$$

This gives every harmonic. A pipe open at both ends has antinodes at both ends and produces exactly the same harmonic series.

A pipe closed at one end must have a node at the closed end and an antinode at the open end, so only odd numbers of quarter-wavelengths fit:

$$f_n = \frac{nv}{4L}, \quad n = 1, 3, 5...$$

Only odd harmonics exist. This is why a clarinet, which behaves like a closed--open pipe, sounds roughly an octave below a flute of similar length, which behaves like an open--open pipe.

For sound, remember that a displacement node is a pressure antinode and vice versa. You must be able to:

• sketch the first few harmonics for each system
• label nodes ($N$) and antinodes ($A$)
• calculate harmonic frequencies and wavelengths from $L$ and $v$

Every oscillating system has a natural frequency: the frequency at which it vibrates freely after a single disturbance, set by its physical properties such as mass, stiffness or length. If instead a periodic external force drives the system, it is forced to oscillate at the driving frequency, usually with small amplitude.

Resonance occurs when the driving frequency matches the natural frequency. Energy then transfers from the driver to the system most efficiently, and the amplitude builds up to a maximum, limited only by damping.

A child's swing illustrates this: small pushes timed once per swing cycle produce a large amplitude, while mistimed pushes achieve little. Other examples include:

• a wine glass driven loudly at its natural frequency
• parts of a vehicle that buzz at particular engine speeds
• structures that must be designed so wind or footsteps cannot drive them near a natural frequency

Standing waves are resonance phenomena: a string or air column responds strongly only when driven at one of its harmonic frequencies, which are its natural frequencies.

You should be able to sketch a resonance curve of amplitude against driving frequency, showing a sharp peak at the natural frequency, and describe what happens to the amplitude on either side of the peak.

Damping is the removal of energy from an oscillating system by resistive forces such as friction, air resistance or viscous drag. In a freely oscillating damped system the amplitude decays over time while the period stays almost constant.

Qualitatively:

• light damping gives many oscillations of gradually shrinking amplitude
• critical damping returns the system to equilibrium in the shortest time without oscillating
• heavy damping also prevents oscillation but the return is slow

Car shock absorbers are designed close to critical damping so the car settles quickly after a bump.

Damping does not merely reduce the amplitude of free oscillations over time; it fundamentally alters the system's resonance response curve. Under forced oscillation, increasing the level of damping:

• lowers the maximum peak amplitude of the resonance curve
• broadens the frequency range over which a significant response occurs
• shifts the peak resonant frequency slightly below the undamped natural frequency

This is why engineers add dampers to bridges and tall buildings: even if wind drives the structure near a natural frequency, the resonant amplitude stays small.

You must be able to sketch resonance curves for different amounts of damping on the same axes and describe these effects qualitatively; no damping calculations are required.