Superposition, standing waves and resonance

The principle of superposition

When two or more waves pass through the same point at the same time, the resulting displacement is the sum of the displacements each wave would cause on its own:

$y_{\text{total}} = y_1 + y_2$

Displacements are signed, so a crest (positive) and a trough (negative) partly or completely cancel. After overlapping, each wave carries on unchanged, as if the other had never been there. Waves pass through each other; they do not bounce off each other.

• Constructive interference: waves meet in phase (crest on crest), so the amplitudes add and the result is larger.
• Destructive interference: waves meet out of phase (crest on trough), so the result is smaller. Equal amplitudes cancel completely, for that instant.

Destructive interference does not destroy energy. At the moment two equal and opposite pulses cancel, the string is flat but every part of it is moving, so the energy is all kinetic for that instant. Noise-cancelling headphones use the idea: a microphone picks up incoming noise and the speaker plays the inverted wave so the two superpose to nearly zero at your ear.

Standing waves

A wave that travels along, carrying energy from place to place, is a progressive (travelling) wave. If two identical waves travel in opposite directions through the same medium, usually a wave and its reflection, their superposition is a standing wave: a pattern that oscillates in place but does not travel.

• Nodes are points that never move, because the two waves always cancel there.
• Antinodes are points of maximum oscillation, halfway between nodes, with an amplitude of twice that of each travelling wave.
• Neighbouring nodes (or antinodes) are $\lambda/2$ apart, where $\lambda$ is the wavelength of the travelling waves.
• All points between two neighbouring nodes move in phase; points in the next section move in antiphase.

A string fixed at both ends can only hold standing waves that have a node at each end, so only certain wavelengths "fit". Those allowed patterns are the harmonics that give musical instruments their notes; page 3F works through them.

Resonance in mechanical systems

Pull a mass on a spring down and let go: it oscillates at its own natural frequency $f_0$, set by the system (the mass and the stiffness of the spring), not by how far you pulled it. Every object that can vibrate has one or more natural frequencies.

Now drive the system with a periodic force at a driving frequency $f_d$. After a short time it settles into oscillating at $f_d$. How big the oscillation is depends on how close $f_d$ is to $f_0$:

• Far from $f_0$, the pushes are often against the motion, so little energy is transferred and the amplitude stays small.
• When $f_d = f_0$, every push is in step with the motion, so energy is transferred to the system on every cycle and the amplitude grows until the energy lost each cycle (to friction, air resistance and so on) equals the energy put in. This is resonance: the largest amplitude occurs when the driving frequency matches the natural frequency.

Damping is anything that removes energy from the oscillation. Heavier damping gives a lower, broader peak; very light damping gives a tall, sharp one. Within the system, energy is continually transformed between kinetic energy and potential energy (elastic or gravitational), and damping transforms some into thermal energy each cycle.

Resonance is useful in a swing pushed in time, the body of a guitar, the quartz crystal that keeps time in a watch, and MRI scanners. It is dangerous in bridges, buildings in earthquakes, and car parts that rattle at particular engine speeds. Engineers either shift the natural frequency away from likely driving frequencies or add damping, such as the tuned mass dampers in tall buildings.