Waves
A pebble dropped in water creates ripples that spread outward.
Start with the simplest version: this lesson is about Waves. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
A pebble dropped in water creates ripples that spread outward. A vibrating tuning fork creates sound waves that travel through air and make your eardrum oscillate. Light waves travel from the sun, allowing us to see. Waves are everywhere—they carry energy and information without moving the medium itself. This chapter explores how waves form, how they propagate, and how they interact. Understanding waves reveals patterns governing sound, light, and seismic activity.
Transverse and Longitudinal Waves
A wave is a disturbance that propagates through space. Unlike oscillations (which occur at a fixed location), waves carry energy from one place to another.
Transverse waves: The medium oscillates perpendicular to the wave's direction of travel. A wave on a string moves forward while the string vibrates up and down. Light waves are transverse—electric and magnetic fields oscillate perpendicular to the light's propagation direction.
Longitudinal waves: The medium oscillates parallel to the wave's direction. Sound waves compress and rarefact the air along the propagation direction. P-waves (primary seismic waves) are longitudinal and propagate through Earth's interior, while S-waves (secondary) are transverse and cannot propagate through liquids.
Wave Properties
Wavelength (λ): Distance between successive crests (or troughs) of a wave. Longer wavelength means more spaced-out oscillations.
Frequency (f): Oscillations per unit time. Higher frequency means faster oscillation.
Period (T = 1/f): Time for one complete oscillation.
Amplitude (A): Maximum displacement of the medium from equilibrium. Larger amplitude means more intense wave (louder sound, brighter light).
Wave speed (v): Distance the wave travels per unit time. Crucially, wave speed depends on the medium, not on frequency or amplitude!
Relationship: v = fλ
This fundamental equation connects the wave's spatial extent (wavelength), temporal rate (frequency), and propagation speed. Doubling frequency halves wavelength if speed is constant.
Displacement Relation in a Traveling Wave
A harmonic wave (sinusoidal oscillation) traveling in the positive x-direction is described by:
y(x,t) = A sin(kx − ωt + φ)
where:
- A is amplitude
- k = 2π/λ is the wave number
- ω = 2πf is the angular frequency
- φ is the phase constant
This equation describes the displacement y at position x and time t. At a fixed position (x = constant), the medium oscillates sinusoidally with frequency f. At a fixed time (t = constant), the profile is sinusoidal with wavelength λ.
The speed is v = ω/k = fλ.
Speed of Waves in Different Media
Wave speed depends on the medium's properties:
Sound in air: v ≈ 340 m/s (depends on temperature; faster in warm air)
Sound in water: v ≈ 1480 m/s (faster because water is denser and less compressible)
Waves on a string: v = √(T/μ), where T is tension and μ is linear mass density. Tighter, lighter strings have faster waves.
Light in vacuum: c = 3 × 10⁸ m/s (constant everywhere)
Light in material: v < c (light slows in denser media, explaining refraction)
For waves on a string, doubling tension increases speed by √2, while doubling the string's mass per length decreases speed by √2. This explains why guitar strings are tuned by tension adjustment.
Interference of Waves
When two waves occupy the same space, they superpose—the displacement at each point is the sum of both waves' displacements.
Constructive interference: Waves align in phase, their amplitudes add, producing a wave of larger amplitude.
Destructive interference: Waves align opposite in phase (out of phase by π), their amplitudes cancel. If equal in amplitude, they cancel completely.
The interference pattern depends on the phase difference between waves:
- Phase difference = 0 (or multiples of 2π): Constructive
- Phase difference = π (or odd multiples): Destructive
- In between: Partial interference
This is why noise-cancelling headphones work: they measure ambient sound, generate an inverted wave, and superpose them to cancel noise through destructive interference.
Standing Waves and Resonance
A wave confined to a region (like a string tied at both ends) can reflect from boundaries. If the reflected wave has the right phase relationship with the incident wave, they interfere constructively, creating a standing wave—a stationary pattern of nodes (no displacement) and antinodes (maximum displacement).
For a string of length L, standing waves occur only at certain wavelengths:
λ_n = 2L/n (n = 1, 2, 3...)
The corresponding frequencies are:
f_n = nv/(2L) = n × f₁ (n = 1, 2, 3...)
where f₁ is the fundamental frequency and nf₁ are harmonics.
For a guitar string, the fundamental (n = 1) produces the note's pitch. Higher harmonics (n > 1) contribute the timbre (tone quality), which is why a guitar sounds different from a piano playing the same note—different harmonic content.
Standing waves explain why musical instruments have fixed pitches and why resonance excites them efficiently.
The Doppler Effect
When a source moves toward an observer, the observed frequency increases. When moving away, it decreases. This is the Doppler effect.
If the source moves at velocity v_s toward a stationary observer, the observed frequency is:
f' = f × [v/(v − v_s)]
where v is wave speed. As v_s approaches v, f' → ∞ (frequency shifts extremely).
If the observer moves toward a stationary source:
f' = f × [(v + v_o)/v]
The effects are different! Approaching a sound source and having a sound source approach you produce slightly different frequency shifts (relativistic effects, absent for sound, produce identical shifts).
Applications:
- Radar and radar guns: Measure vehicle speeds from Doppler shift
- Astronomy: Determine if stars approach or recede (redshift/blueshift)
- Medical ultrasound: Measure blood flow velocity from Doppler shift
- Ambulance sirens: Pitch rises as they approach, drops as they pass
Diffraction: Waves Bending Around Obstacles
Waves don't travel strictly in straight lines—they bend around obstacles, a phenomenon called diffraction. This is especially noticeable when wavelength is comparable to obstacle size.
Long-wavelength waves (like long-wavelength radio) diffract more easily around buildings than short-wavelength waves (like short-wavelength radio or light). This is why you can hear AM radio through obstacles but not see light through them.
Diffraction explains why sound from a speaker wraps around and reaches you even if you're not directly in front of it.
Refraction: Waves Changing Speed
When waves enter a medium where their speed changes, their wavelength changes, but frequency stays constant (determined by the source). This is refraction.
The refractive index (n) is the ratio of light speed in vacuum to speed in the medium: n = c/v.
Water has n ≈ 1.33, so light travels slower in water and bends when entering water from air. This is why objects underwater appear shallower than they actually are.
Refraction of light through different-density air layers (caused by temperature variations) creates mirages.
Socratic Questions
- Wave speed depends on the medium, but frequency is set by the source. If sound's frequency is fixed, how does wavelength change when sound travels from air into water (where sound is faster)?
- A guitar string and a piano string play the same note (same fundamental frequency). Yet they sound different. How does the concept of harmonics explain this timbral difference?
- Interference can cause two sound waves of equal intensity to produce either maximum intensity (constructive) or silence (destructive). How can adding waves sometimes produce zero amplitude?
- When an ambulance passes you, its siren frequency drops abruptly. Use the Doppler effect formula to explain why. What changes: the sound's actual frequency, the observed frequency, or the wavelength?
- Why can you hear someone calling from around a corner (diffraction) but not see them (minimal diffraction of visible light)? What property of waves and obstacles determines diffraction?