foundation29 min studyLesson 8 in Physics

Waves, phase, and interference

Follow a disturbance through space and understand interference as addition of amplitudes under a linear model.

Before you begin

On this page

The question for today

What you’ll understand

Separate the motion of a wave pattern from local oscillation.
Relate wavelength, frequency, and phase speed.
Predict constructive and destructive interference with stated coherence assumptions.

A vibrating string carries a disturbance away from the point you shake. A marked piece of the string moves locally; it does not travel along with a crest. A wave is therefore a function of position and time, rather than one particle's trajectory. This is our first concrete step toward fields, where a dynamical quantity is assigned at every location.

A moving pattern and an oscillating point are different

A snapshot of a sinusoidal wave repeats after one wavelength in space. At a fixed position, the displacement repeats after one period in time. To follow a particular crest, keep the phase constant as time increases. Its resulting speed is the phase speed. The local transverse velocity of the string is a time derivative of displacement and is not that crest speed.

The sign between the spatial and temporal phase terms determines propagation direction once axes are chosen. For positive wave number and angular frequency, a phase of kx minus omega t travels toward increasing x. Its amplitude determines local displacement, not the distance the wave travels. A wave can transmit energy while the medium returns repeatedly around its original equilibrium position.

y(x,t)=A\cos(kx-\omega t+\phi),\quad k=\frac{2\pi}{\lambda},\quad f=\frac{\omega}{2\pi},\quad v_{\mathrm{ph}}=\frac{\omega}{k}=f\lambda

This single harmonic traveling wave uses wavelength λ in metres and frequency f in hertz. It has a definite phase speed. More general dispersive media can give different phase and group velocities; fλ is not automatically a signal or information speed.↔ Scroll wide equations horizontally.

The linear approximation makes superposition possible

For a stretched string with small slopes and approximately constant tension, the transverse displacement obeys a linear wave equation. If two functions solve that same linear equation with compatible conditions, their sum does too. This is superposition. Real water waves, elastic materials, and electromagnetic media can become nonlinear; visually adding two shapes does not prove that a nonlinear system evolves by their sum.

Interference describes the local addition of overlapping disturbances. In-phase equal-amplitude waves reinforce; opposite-phase equal-amplitude waves cancel at a given place and time. Energy is not generally proportional to the signed displacement. Destructive interference at one region therefore does not imply that total energy has been destroyed: the complete field, its gradients, boundary conditions, and energy transport must be considered.

\frac{\partial^2y}{\partial t^2}=v^2\frac{\partial^2y}{\partial x^2},\qquad v=\sqrt{\frac{\mathcal T}{\mu}}

For a taut string, tension 𝒯 has units newtons and linear mass density μ has units kg/m. The model neglects bending stiffness, damping, and large slopes. The partial derivatives describe change with one variable while holding the other fixed.↔ Scroll wide equations horizontally.

Relative phase controls the sum

Two coherent waves of the same frequency and wave number have a stable phase difference. Their resultant amplitude depends on that difference and on both original amplitudes. Equal-amplitude waves yield complete cancellation only at a phase offset of an odd multiple of pi. If amplitudes differ, opposite phase leaves a residual rather than a perfectly flat result.

Oppositely traveling waves can form standing waves with fixed nodes, while different nearby frequencies produce beats. Our lab uses co-propagating waves of the same frequency instead, so its changing interference should not be called a standing-wave or beat simulation. Quantum theory will also add amplitudes, but its amplitudes are generally complex probability amplitudes, not literal string displacements.

A_{\mathrm{sum}}^2=A_1^2+A_2^2+2A_1A_2\cos\delta

For two coherent sinusoidal waves sharing k and ω, δ is their relative phase. This formula gives the squared amplitude of the sum, not a universal energy density. For A₁ = A₂ = A it becomes 4A²cos²(δ/2).↔ Scroll wide equations horizontally.

Work it through

Two drives, one resulting wave

Two coherent waves have equal amplitude $2.0\,\mathrm{mm}$ , frequency $5.0\,\mathrm{Hz}$ , wavelength $0.60\,\mathrm{m}$ , and relative phase $120^\circ$ . They travel in the same direction in a linear medium. Find phase speed and resultant amplitude.

Each wave's phase speed is $f\lambda=(5.0)(0.60)=3.0\,\mathrm{m\,s^{-1}}$ . This uses crest motion, not the transverse speed of a medium element.
Convert the phase difference to $2\pi/3$ radians, for which $\cos\delta=-1/2$ . The amplitude formula gives $A_{\mathrm{sum}}^2=2A^2+2A^2(-1/2)=A^2$ .
Take the nonnegative square root: $A_{\mathrm{sum}}=A=2.0\,\mathrm{mm}$ . The result is smaller than the in-phase amplitude of $4.0\,\mathrm{mm}$ , but does not vanish because the phase difference is not $180^\circ$ .

The result

The sum travels with phase speed $3.0\,\mathrm{m\,s^{-1}}$ and has amplitude $2.0\,\mathrm{mm}$ , assuming equal frequencies and a stable phase difference.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Add waves before interpreting the sum

Change phase difference and amplitude ratio a for sin(2x − t) + a sin(2x − t + phase). Compare both components with the sum. Coordinates are normalized; these equal-frequency co-propagating waves are neither a standing-wave system nor a dispersion solver.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forOpposite phase cancels completely only when the two amplitudes are equal.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Opposite-phase displacements subtract. The larger amplitude exceeds the smaller by 2 mm, leaving that residual. Complete destructive cancellation would require equal amplitudes as well as opposite phase.

Your turn

Try it yourself

A string has tension $36\,\mathrm{N}$ and linear density $0.010\,\mathrm{kg\,m^{-1}}$ . A source oscillates at $20\,\mathrm{Hz}$ . Find the wave speed and wavelength in the ideal-string approximation.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First use the material and tension to find speed; only then use v = fλ.

Compare with the solution

The speed is $v=\sqrt{36/0.010}=60\,\mathrm{m\,s^{-1}}$ . Therefore $\lambda=v/f=60/20=3.0\,\mathrm{m}$ . Changing source frequency at the same tension and density changes wavelength in this nondispersive model, not the propagation speed. A real stiff or dispersive string can require a more detailed relation.

Keep the idea

Take it with you

A traveling pattern's phase speed differs from the medium's local motion.
Superposition follows from a linear model and can fail outside its approximation.
Interference depends on relative phase and amplitude, not phase alone.

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