What Are Waves?
A wave is a disturbance that transfers energy through a medium (or through space) without transferring matter. Throw a stone in a pond: the water surface ripples outward, but the water itself doesn't travel with the wave — it just bobs up and down.
This distinction — energy travels, matter stays put — is the defining feature of all wave motion. Sound is a pressure disturbance travelling through air. Light is an oscillating electromagnetic field propagating through space. Even a crowd "wave" in a sports stadium fits the definition: the wave moves around the stadium, but no one changes their seat.
Waves are everywhere. Understanding them unlocks electromagnetism, optics, acoustics, seismology, and quantum mechanics. All of these, at root, are wave theories.
Simple Harmonic Motion (SHM)
Simple harmonic motion is the oscillation that occurs when a restoring force is proportional to displacement from equilibrium and directed opposite to that displacement: F = −kx. It's the simplest and most important oscillatory motion in physics.
Hooke's Law — the foundation of SHM. k = spring constant (N/m) | x = displacement from equilibrium
The motion this produces is sinusoidal — displacement varies as x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is phase. The period of oscillation is T = 2π/ω, and for a spring-mass system: T = 2π√(m/k).
What's remarkable is how many systems behave like SHM — pendulums (for small angles), vibrating strings, LC circuits, molecular bonds. Any system with a stable equilibrium and a restoring force behaves like a harmonic oscillator near that equilibrium. This is why SHM is such a powerful concept: it's physics' simplest model for stability.
For deeper grounding on forces and equilibrium, see our Classical Mechanics guide.
Transverse vs Longitudinal Waves
All waves can be classified by how the medium's displacement relates to the wave's direction of travel.
| Property | Transverse Waves | Longitudinal Waves |
|---|---|---|
| Displacement direction | Perpendicular to travel | Parallel to travel |
| Can travel in vacuum? | Yes (EM waves) | No |
| Polarization possible? | Yes | No |
| Examples | Light, water surface waves, seismic S-waves | Sound, seismic P-waves, slinky compression |
| Visualization | Sine wave shape | Regions of compression/rarefaction |
In a longitudinal wave, the medium oscillates back and forth along the direction of propagation, creating alternating regions of compression (high pressure) and rarefaction (low pressure). Sound is the most familiar example — your eardrum detects these pressure variations.
Transverse waves oscillate perpendicular to travel direction. Light can be polarized — its electric field oscillation can be restricted to a single plane — precisely because it's transverse. Longitudinal waves cannot be polarized.
The Wave Equation: v = fλ
The fundamental wave equation states that wave speed equals the product of frequency and wavelength: v = fλ. This relationship holds for every wave in the universe — sound, light, water, seismic.
v = wave speed (m/s) | f = frequency (Hz) | λ = wavelength (m)
Frequency (f) measures how many complete oscillations pass a fixed point per second; wavelength (λ) measures the spatial distance between successive identical points on the wave (crest to crest, trough to trough). Their product gives the speed at which the waveform moves.
Here's the key insight: for a given medium, wave speed is fixed by properties of that medium — not by the source. Sound in air at 20°C always travels at 343 m/s. Changing frequency changes wavelength inversely, but not speed. This is why high-frequency sounds have shorter wavelengths than low-frequency sounds, even though both travel at the same speed through air.
Period and Angular Frequency
- Period T = 1/f — time for one complete oscillation
- Angular frequency ω = 2πf — radians per second
- Wave number k = 2π/λ — spatial frequency in radians per meter
- Full wave equation: y(x,t) = A sin(kx − ωt + φ)
Sound Waves
Sound is a longitudinal pressure wave that requires a mechanical medium — it cannot travel through a vacuum. The speed of sound depends on the medium's elasticity and density: v = √(B/ρ), where B is bulk modulus and ρ is density.
| Medium | Speed of Sound |
|---|---|
| Air (20°C) | 343 m/s |
| Water (25°C) | 1,480 m/s |
| Steel | 5,100 m/s |
| Diamond | 12,000 m/s |
Human hearing spans roughly 20 Hz to 20,000 Hz. Below 20 Hz is infrasound (produced by earthquakes and large animals); above 20,000 Hz is ultrasound (used in medical imaging and sonar). The speed of sound in air increases with temperature: approximately 0.6 m/s per degree Celsius above 0°C.
Sound intensity is measured in decibels (dB): L = 10 log₁₀(I/I₀), where I₀ = 10⁻¹² W/m² is the threshold of human hearing. Every 10 dB increase represents a tenfold increase in intensity — 60 dB (normal conversation) is 10⁶ times more intense than the threshold of hearing.
Electromagnetic Waves
Electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of travel. They require no medium and travel through vacuum at c = 299,792,458 m/s.
Maxwell's equations predict that any accelerating charge produces electromagnetic radiation — a revolutionary insight from the 1860s that unified electricity, magnetism, and optics. All electromagnetic waves travel at the same speed in vacuum, differentiated only by frequency (and thus wavelength).
| Type | Frequency Range | Wavelength | Application |
|---|---|---|---|
| Radio waves | 3 Hz – 300 MHz | 1 mm – 100 km | Broadcasting, WiFi |
| Microwaves | 300 MHz – 300 GHz | 1 mm – 1 m | Radar, microwave ovens |
| Infrared | 300 GHz – 400 THz | 700 nm – 1 mm | Thermal imaging, remote controls |
| Visible light | 400 – 700 THz | 400 – 700 nm | Vision, photography |
| Ultraviolet | 700 THz – 30 PHz | 10 – 400 nm | Sterilization, sunburn |
| X-rays | 30 PHz – 30 EHz | 0.01 – 10 nm | Medical imaging |
| Gamma rays | > 30 EHz | < 0.01 nm | Nuclear medicine, sterilization |
For the full theory behind EM waves, see our Electromagnetism guide covering Maxwell's equations.
Superposition and Interference
When two or more waves occupy the same region of space simultaneously, the total displacement at any point is the algebraic sum of the individual displacements. This is the principle of superposition — the most powerful tool in wave analysis.
Superposition leads to two critical phenomena:
- Constructive interference: When two waves are in phase (crests align), amplitudes add. The combined wave is larger.
- Destructive interference: When two waves are 180° out of phase (crest meets trough), amplitudes cancel. The combined wave is smaller — or zero.
Young's double-slit experiment (1801) demonstrated that light interferes — producing alternating bright and dark fringes on a screen. This was definitive evidence for the wave nature of light, centuries before quantum mechanics complicated the picture. (Source: Young, 1801)
Noise-cancelling headphones use destructive interference actively — a microphone samples incoming sound, and speakers emit an inverse waveform that cancels it. It's superposition as consumer technology.
Resonance and Standing Waves
Resonance occurs when a system is driven at its natural frequency, causing it to absorb maximum energy and oscillate with maximum amplitude. Every physical system has natural frequencies determined by its size, shape, and material properties.
When two waves of the same frequency travel in opposite directions through the same medium, they superpose to produce a standing wave — a pattern of fixed nodes (zero displacement) and antinodes (maximum displacement) that appears not to travel.
For a string of length L fixed at both ends, standing waves form at frequencies:
n = 1, 2, 3, ... (harmonic number) | v = wave speed in string | L = string length
The fundamental frequency (n=1) is the first harmonic. Music is built on these harmonics — a guitar string vibrates primarily at its fundamental, but also at overtones (n=2, 3, 4...) that give instruments their distinctive timbre.
Resonance can be destructive. The Tacoma Narrows Bridge collapsed in 1940 when wind-induced oscillations matched a structural resonance frequency, driving amplitudes until the bridge failed. (Source: Billah & Scanlan, 1991)
The Doppler Effect
The Doppler effect is the change in observed frequency of a wave when the source and observer are in relative motion. A source moving toward you emits crests more frequently (higher observed frequency); moving away, less frequently (lower observed frequency).
Use + in numerator when observer moves toward source; − when moving away. Opposite signs for denominator.
The Doppler effect explains the falling pitch of a passing ambulance siren, the redshift of receding galaxies, and the workings of radar speed guns, weather Doppler radar, and medical ultrasound. Astronomers use the Doppler shift of light from distant galaxies to measure how fast they're receding — evidence for the expansion of the universe. (Source: Hubble, 1929)
Waves vs Particles: The Fundamental Tension
| Property | Waves | Particles |
|---|---|---|
| Energy transfer | Spread over wavefront | Localized |
| Interference | Yes — waves superpose | No — particles don't interfere |
| Diffraction | Yes — bends around obstacles | No |
| Medium required | For mechanical waves; no for EM | No |
| Quantum picture | Both — wave-particle duality | Both — wave-particle duality |
Quantum mechanics revealed that this distinction breaks down at small scales. Light (clearly wave-like) also behaves as particles (photons) in the photoelectric effect. Electrons (clearly particle-like) also produce diffraction patterns. The de Broglie relation λ = h/p assigns a wavelength to every particle. Explore this in Modern Physics.
Common Misconceptions About Waves
- "Waves carry matter." They don't — they carry energy. Ocean waves move water up and down, not forward. The wave moves forward; the water circles back to its original position.
- "Louder sound travels faster." Sound amplitude (loudness) doesn't affect its speed — speed is determined by the medium's properties, not the wave's energy.
- "Higher pitch = faster sound." Frequency and speed are independent for sound in a given medium. Higher pitch means shorter wavelength, same speed.
- "Resonance always breaks things." Resonance is also useful — it's essential to MRI machines, musical instruments, radio tuners, and molecular spectroscopy.
- "Interference destroys energy." In destructive interference, energy isn't destroyed — it's redistributed to other locations where constructive interference occurs.
Real-World Applications of Wave Physics
- Medical ultrasound: High-frequency (1–18 MHz) sound waves reflect off tissue boundaries at different rates. Computers reconstruct the time-of-flight data into real-time images — no radiation required.
- Noise cancellation: Active noise-cancelling headphones sample ambient sound and emit inverse waveforms, using destructive interference to reduce noise by 20–30 dB.
- Musical acoustics: The shape of a violin body creates resonance chambers that amplify specific harmonics, giving the instrument its tonal character.
- Earthquake seismology: P-waves (longitudinal) travel through Earth's liquid outer core; S-waves (transverse) don't. Seismograph data reveals Earth's internal structure without drilling a single hole.
- Fiber optics: Light travels by total internal reflection through glass fibers, carrying internet traffic at the speed of light — roughly 200,000 km/s in glass.
Frequently Asked Questions
Summary & Next Steps
Wave physics rests on a few core ideas: the wave equation v = fλ, the distinction between transverse and longitudinal waves, superposition and interference, resonance, and the Doppler effect. Together they explain the behavior of sound, light, seismic activity, and quantum particles.
Waves aren't just a physics topic — they're the medium through which energy propagates in nature. Every time you listen to music, use a phone, or see sunlight, you're experiencing wave physics in action.
Continue Learning
- Classical Mechanics — Forces, energy, and the foundations that underpin oscillatory motion
- Energy & Thermodynamics — How wave energy connects to heat and entropy
- Electromagnetism — Maxwell's equations that describe light as an EM wave
- Optics — Reflection, refraction, lenses, and light wave phenomena
- Modern Physics — Quantum waves, photons, and wave-particle duality
References: [1] Young, T. (1802). On the theory of light and colours. Philosophical Transactions of the Royal Society. [2] Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. PNAS. [3] Billah, K.Y. & Scanlan, R.H. (1991). Resonance, Tacoma Narrows Bridge Failure. American Journal of Physics.