Topic 4

Waves & Oscillations

From guitar strings to the cosmic microwave background, waves are the universal carrier of energy and information — sound, light, water, and even the fabric of spacetime itself.

The wave equation
Superposition & interference
Standing waves & resonance
Sound waves
The Doppler effect

1. The Wave Equation

A wave is a disturbance that transfers energy without transporting matter. A wave is described by its amplitude $A$, wavelength $\lambda$, frequency $f$, and speed $v = f\lambda$:

$$\boxed{y(x,t) = A\sin(kx - \omega t + \phi)}$$

where $k = 2\pi/\lambda$ is the wave number and $\omega = 2\pi f$ is the angular frequency. For a string under tension $T$ with linear density $\mu$, the speed is $v = \sqrt{T/\mu}$. The general one-dimensional wave equation is:

$$\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$$

Transverse vs longitudinal waves. In a transverse wave, the displacement is perpendicular to propagation (like a wave on a string or light). In a longitudinal wave, displacement is parallel (like sound in air — compressions and rarefactions).

2. Superposition & Interference

When two waves meet, they superpose — their amplitudes add. This leads to interference: constructive (in phase, amplitudes add) where peaks align, and destructive (out of phase, amplitudes cancel) where a peak meets a trough.

$$y_{\text{total}} = y_1 + y_2 = 2A\cos\!\left(\tfrac{\Delta\phi}{2}\right)\sin\!\left(\bar{k}x - \bar{\omega}t + \bar{\phi}\right)$$

Thomas Young's double-slit experiment (1801) demonstrated light's wave nature: alternating bright and dark fringes appear because light from the two slits travels different distances to each point on the screen, creating varying path differences $\Delta L = d\sin\theta$.

Two-source interference. Two coherent wave sources emit circular waves. Where crests meet crests (constructive), bright bands appear; where crest meets trough (destructive), they cancel. Drag the sliders to change separation, frequency, and wave speed.

3. Standing Waves & Resonance

When a wave reflects and travels back, it interferes with itself. At resonant frequencies, the reflected wave reinforces the incoming one, creating standing waves with fixed nodes (zero displacement) and antinodes (maximum displacement). Only wavelengths $\lambda_n = 2L/n$ fit on a string of length $L$ fixed at both ends:

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}},\qquad n = 1, 2, 3, \dots$$

The fundamental ($n=1$) is the lowest note; higher harmonics ($n=2, 3, \ldots$) give the instrument its timbre. This is why a guitar and a violin playing the same note sound different — their overtone mixtures differ.

Bridge of physical intuition. The same math governs acoustic pipes, laser cavities, and even quantum particles in a box (see the Quantum section). Resonance is everywhere: the Tacoma Narrows Bridge collapse (1940), microwave ovens (water molecules resonate at 2.45 GHz), and the Schumann resonance (Earth's cavity rings at ~7.8 Hz).

4. Sound Waves

Sound is a longitudinal pressure wave in a medium. Its speed in air at 0°C is 331 m/s and increases ~0.6 m/s per °C: $v_{\text{sound}} \approx 331 + 0.6T_{\text{°C}}$. The decibel scale measures intensity logarithmically:

$$\beta = 10\log_{10}\!\left(\frac{I}{I_0}\right),\qquad I_0 = 10^{-12}\text{ W/m}^2$$

Human hearing spans roughly 20 Hz to 20 kHz and 0 to 120 dB (beyond which lies pain and then physical damage). Pitch maps to frequency, loudness to amplitude, and timbre to harmonic content.

5. The Doppler Effect

When a wave source moves relative to an observer, the observed frequency shifts: blueshift (approaching) raises pitch/frequency, redshift (receding) lowers it. For sound with a stationary observer and moving source:

$$\boxed{f' = f\frac{v_{\text{sound}}}{v_{\text{sound}} \mp v_s}}$$

The minus sign applies as the source approaches (frequency up), the plus as it recedes. The same principle applied to light gives astronomical redshift $z = \Delta\lambda/\lambda$, which Edwin Hubble used to discover the expansion of the universe: more distant galaxies recede faster, $v = H_0 d$.

The Doppler effect: wavefronts compress ahead of a moving source (higher frequency, blueshift) and stretch behind it (lower frequency, redshift).

Key Takeaways

Waves transfer energy without transporting matter; they are described by $y = A\sin(kx-\omega t+\phi)$ with speed $v = f\lambda$.
Superposition makes waves add — constructive and destructive interference create patterns seen in water ripples, sound, and light.
Standing waves arise from interference of reflected waves; boundary conditions pick natural frequencies (resonance) — the basis of all musical instruments.
Sound is a longitudinal pressure wave; the decibel scale compresses its enormous intensity range into a human-relevant scale.
The Doppler effect — frequency shifts from relative motion — connects a passing ambulance's siren to the expansion of the universe.

⚛️ Next: Quantum Mechanics →