Wave Displacement Equation

A progressive wave — sometimes called a travelling wave — carries energy from one place to another without permanently moving the medium it passes through. Think of a Mexican wave in a football stadium: each fan momentarily stands and sits, yet the wave itself travels all the way around the ground. Every point in the medium executes simple harmonic motion, oscillating about its equilibrium position. The wave displacement equation is the single mathematical expression that captures all of this behaviour: it tells you the displacement of any particle at any position and at any moment in time. The displacement equation combines two independent ideas — oscillation in time and variation in space — into one compact formula. The time part describes how a single particle bobs up and down with period T; the space part describes how the pattern of crests and troughs is spread out with wavelength λ. When you write the equation, you must also specify the direction of travel. A wave moving in the positive x-direction has a different sign convention from one moving in the negative direction, and mixing these up is one of the most common errors students make. Mastering this equation unlocks almost every other topic in waves: superposition, standing waves, the Doppler effect, and electromagnetic wave theory all build directly on it. The equation is elegant precisely because it encodes frequency, wavelength, amplitude, and direction of travel in a single line of mathematics. Understanding each term physically — not just symbolically — is the key to using it confidently.

Key takeaways

Start with a single oscillating particle
Introduce spatial variation using wave number k
Combine time and space into one equation
Connect wave speed, frequency, and wavelength
Physical interpretation of the phase term

Start with a single oscillating particle

Imagine fixing your eyes on one particle in the medium — say, a water molecule on the surface of a pond. It oscillates vertically in simple harmonic motion: y = A sin(ωt), where ω = 2π/T is the angular frequency and A is the amplitude. This describes time variation only; it does not yet say anything about neighbouring particles.

Introduce spatial variation using wave number k

Particles at different positions along the wave are out of phase with each other. A particle a distance x from the source lags behind (for a wave travelling in the +x direction) by a phase angle of kx, where k = 2π/λ is the wave number (also called angular wave number), measured in rad m⁻¹. Physically, k tells you how many radians of phase are packed into each metre of the wave.

Combine time and space into one equation

Combining the oscillation in time with the spatial phase lag gives the full displacement equation: y(x, t) = A sin(ωt − kx) for a wave travelling in the +x direction, or y(x, t) = A sin(ωt + kx) for a wave travelling in the −x direction. The minus sign is crucial: it means that as x increases, the phase decreases, which corresponds to the crest pattern moving forward.

Connect wave speed, frequency, and wavelength

The speed of the wave v is the rate at which a crest (a point of constant phase) moves. Setting ωt − kx = constant and differentiating gives v = dx/dt = ω/k. Substituting ω = 2πf and k = 2π/λ recovers the familiar relation v = fλ. This shows that the wave equation is internally consistent with every other wave formula you already know.

Physical interpretation of the phase term

The quantity (ωt − kx) is called the phase of the wave at position x and time t. Two points are in phase if their phase difference Δφ = kΔx is a multiple of 2π, which means Δx is a multiple of λ. Two points are exactly out of phase (antiphase) if Δφ = π, meaning they are separated by half a wavelength. The phase concept is essential for understanding interference and superposition.

The Key Equation

y(x,\,t) = A\sin(\omega t - kx)

y — Displacement (m): The distance a particle has moved from its equilibrium position at a given x and t.
A — Amplitude (m): The maximum displacement from equilibrium; determines the energy carried by the wave.
\omega — Angular frequency (rad s⁻¹): Rate of oscillation in radians per second; ω = 2πf = 2π/T.
t — Time (s): The moment at which the displacement is evaluated.
k — Wave number (rad m⁻¹): Spatial frequency; k = 2π/λ. It measures how many radians of phase occur per metre.
x — Position (m): The distance of the particle from the chosen origin along the direction of wave propagation.

Worked Example

Problem. A transverse wave on a string is described by y(x, t) = 0.04 sin(20πt − 5πx), where x and y are in metres and t is in seconds. Find: (a) the amplitude, (b) the frequency, (c) the wavelength, and (d) the wave speed. Then find the displacement of the particle at x = 0.1 m when t = 0.05 s.

y(0.1,\,0.05) = 0.04\sin\!\left(\tfrac{\pi}{2}\right) = 0.04 \text{ m}

Answer. Amplitude = 0.04 m; frequency = 10 Hz; wavelength = 0.4 m; wave speed = 4.0 m s⁻¹; displacement at the given point and time = 0.04 m (the particle is at maximum positive displacement).

Figure

Diagram

Origins · 1 minJean le Rond d'Alembert and how this was discovered

Jean le Rond d'Alembert

1717–1783 · French

D'Alembert was the first to derive and solve the one-dimensional wave equation in 1747, while studying the vibrations of a stretched string. He showed that the general solution is a superposition of two functions travelling in opposite directions — a result now called d'Alembert's solution — which is the mathematical ancestor of the displacement equation used in every wave course today. D'Alembert was abandoned as a baby on the steps of the Saint-Jean-le-Rond church in Paris, from which he took his name. Despite this inauspicious start, he became one of the most celebrated mathematicians of the French Enlightenment and co-edited the famous Encyclopédie with Diderot.

Wave Speed, Frequency and Wavelength — The relation v = fλ is embedded inside the wave displacement equation through the identity v = ω/k, so these concepts are inseparable.
Amplitude and Intensity — The amplitude A in the displacement equation determines the energy and intensity of the wave, linking the mathematical form to observable power.
Superposition of Waves — Superposition involves adding displacement equations for two or more waves, making fluency with y(x,t) essential before tackling interference.
Standing Waves — Standing waves arise when y = A sin(ωt − kx) and y = A sin(ωt + kx) are superposed, showing why understanding both signs of the displacement equation matters.
Simple Harmonic Motion — Each particle in a progressive wave undergoes SHM; the time-dependence A sin(ωt) in the displacement equation is exactly the SHM equation for a single oscillator.