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PhysicsDifficulty 2-4

Waves

A wave is a disturbance that travels from place to place, carrying energy without carrying the material along with it. Mechanical waves such as sound need a medium, while electromagnetic waves such as light can cross empty space. Every wave obeys v = fλ, and when waves overlap they add together (superposition), producing constructive and destructive interference.

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The Linear Wave Equation

A one-dimensional sinusoidal travelling wave can be written y(x, t) = A sin(kx − ωt), with wave number k = 2π/λ and angular frequency ω = 2πf, so v = ω/k = fλ. Any function of (x − vt) solves the linear wave equation below.

∂2y∂x2=1v2∂2y∂t2\dfrac{\partial^2 y}{\partial x^2} = \dfrac{1}{v^2}\dfrac{\partial^2 y}{\partial t^2}

The linear wave equation for a disturbance y(x, t) travelling at speed v.

Because the equation is linear, the sum of two solutions is another solution — this is the mathematical content of superposition. Adding A sin(kx − ωt + φ) and A sin(kx − ωt) with the identity sin u + sin v = 2 sin((u+v)/2) cos((u−v)/2) gives the result below.

yR(x,t)=[2Acos⁡ ⁣(φ2)]sin⁡ ⁣(kx−ωt+φ2)y_R(x,t) = \left[2A\cos\!\left(\tfrac{\varphi}{2}\right)\right]\sin\!\left(kx - \omega t + \tfrac{\varphi}{2}\right)

Superposition of two equal-amplitude waves differing only by phase φ.

For unequal amplitudes the resultant amplitude is √(A₁² + A₂² + 2A₁A₂ cos Δφ) (the phasor sum), ranging from |A₁ − A₂| (Δφ = π) to A₁ + A₂ (Δφ = 0). With A₁ = 3 and A₂ = 4 at Δφ = π/2 it is √(9 + 16) = 5. Counter-propagating identical waves superpose into a standing wave with fixed nodes; with both ends fixed, λₙ = 2L/n and fₙ = nv/(2L).

Info: Wave speed comes from the medium: on a string v = √(F/μ), where F is the tension and μ the mass per unit length; in a fluid v = √(B/ρ), with B the bulk modulus and ρ the density. Stiffer media raise v; denser media lower it. Liquids and solids are far denser than air but also far harder to compress, and the stiffness wins: sound in water or steel is much faster than in air.
Full explanation — the complete reference version every reading depth is based on

What a wave is

Drop a stone into a pond and ripples spread outwards, yet a floating leaf only bobs up and down: the disturbance travels, the water mostly stays where it is. That is the defining idea of a wave: a pattern that moves energy (and momentum) from one place to another without moving the material itself along the way.

  • Mechanical waves — water waves, sound and seismic waves — need a medium, a material whose particles push back when disturbed.
  • Electromagnetic waves — radio, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays — need no medium and cross a vacuum at the speed of light.
  • In a transverse wave the medium moves at right angles to the direction of travel (a wave on a rope); in a longitudinal wave it moves back and forth along the direction of travel.
  • Sound in air is longitudinal: a vibrating source makes regions of higher pressure (compressions) and lower pressure (rarefactions) that travel outwards.

Describing a wave

  • Wavelength (λ): the distance between two matching points on neighbouring cycles, such as crest to crest, in metres.
  • Period (T): the time for one complete cycle to pass a point, in seconds.
  • Frequency (f): the number of cycles passing a point each second, in hertz (1 Hz = 1 cycle per second). Frequency is set by the source.
  • Amplitude (A): the largest displacement from the undisturbed position. It is independent of how fast the wave travels.
v=fλ=λT,f=1Tv = f\lambda = \dfrac{\lambda}{T}, \qquad f = \dfrac{1}{T}

The wave equation: speed equals frequency times wavelength. It holds for every type of wave.

The speed itself is set by the medium, not by the source. Sound travels at about 331 m/s in air at 0 °C and about 343 m/s at 20 °C, and much faster in liquids and solids, which are far harder to compress — about 1480 m/s in fresh water at 20 °C.

Worked example: a 440 Hz note

A tuning fork sounds the note A at 440 Hz in air at 20 °C, where sound travels at 343 m/s. Rearranging v = fλ gives λ = v / f = 343 m/s ÷ 440 Hz ≈ 0.78 m, so each compression is a little under 80 cm from the next. These are the default settings of the ScienceVerse waves simulation, which computes the same result (0.7795… m). In water the same note's wavelength is about 1480 ÷ 440 ≈ 3.4 m (our calculation): the frequency stays the same, but the faster speed stretches each wave.

Superposition and interference

When two waves meet in the same place, the medium's displacement is simply the sum of the two separate displacements — the principle of superposition. The waves then carry on unchanged, as if they had never met.

  • Constructive interference: identical waves arriving in phase (crest on crest) add to twice the amplitude.
  • Destructive interference: identical waves arriving half a cycle (180°, or π radians) out of phase cancel completely.
  • Any phase difference in between gives a partial result, which is why sound from two speakers is louder in some spots of a room than others.
AR=2Acos⁡ ⁣(φ2)A_R = 2A\cos\!\left(\dfrac{\varphi}{2}\right)

Resultant amplitude of two equal-amplitude waves with phase difference φ (its size is what matters; a negative value means the resultant is inverted).

AR=A12+A22+2A1A2cos⁡ΔφA_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\Delta\varphi}

The general phasor sum for two waves of the same frequency but different amplitudes — the formula the waves simulation uses. With A₁ = A₂ = A it gives the size of 2A cos(φ/2).

When identical waves travel in opposite directions along a string fixed at both ends, superposition produces a standing wave: points called nodes never move, and antinodes between them swing with the largest amplitude. Only wavelengths λₙ = 2L/n fit on a string of length L, which is why a guitar string sounds a definite note and its overtones.

Where waves connect

Waves build on Motion (each bit of the medium oscillates) and Energy (waves transport it). Light is an electromagnetic wave, so reflection, refraction and interference all reappear there, and the same v = fλ links a radio station's frequency to its wavelength.

How we know

Each part of this picture is observable. You see a distant firework's flash before you hear its bang, direct evidence that sound travels at a finite speed much slower than light. A bat times its echoes to judge distance. Loud and quiet spots near two speakers, and the fixed nodes of a vibrating string, only make sense if displacements add. The same linear wave equation describes waves on strings and sound, and electromagnetic waves obey superposition too.

Common misconception: Misconception: a wave carries the water (or air) along with it. It does not — for small waves the medium mostly oscillates about where it started. What travels is the pattern, and with it energy and momentum. A floating cork mostly rises and falls in place as small waves pass, rather than travelling with them.
Common misconception: Misconception: a higher-pitched sound travels faster. In a given medium all frequencies of sound travel at essentially the same speed; a higher frequency simply has a shorter wavelength, keeping v = fλ the same.

Assumptions and limits

Info: v = fλ holds for every type of wave, but superposition and the simple sinusoidal wave model hold only for linear waves. For mechanical waves that means amplitudes small compared with the wavelength; push the medium too far and its restoring force stops being linear. Ocean waves are nonlinear, and these simple models do not fully explain them. The ScienceVerse waves simulation also assumes two sources with exactly the same frequency and a fixed phase difference, and ignores the way real waves weaken as they spread out.

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Content status: published 1 October 2026.

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