Wave Equation — Physica

Equivalent forms

\Box y = 0

\nabla^2 y = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}

One linear PDE describes every wave the universe has ever made.

Symbol	Name	SI unit	Dimension	Range
$y$	Wave displacementoutput Displacement of the medium from equilibrium	m	L	-1 – 1
$v$	Wave speed Phase velocity of the wave through the medium	m/s	L·T⁻¹	1 – 500
$x$	Position Spatial coordinate along the direction of propagation	m	L	0 – 10
$t$	Time Time elapsed since reference instant	s	T	0 – 10

Unit systems

Symbol	SI	CGS	Imperial
$y$	m	cm	ft
$v$	m/s	cm/s	ft/s
$x$	m	cm	ft
$t$	s	s	s

Where it holds

Valid for small-amplitude linear waves in non-dispersive media. Fails for shock waves, nonlinear regimes, and dispersive media (where v depends on frequency).

Dimensional analysis

\partial ^{2}y/\partial t^{2} \to [L]/[T^{2}] = L\cdot T^{-2}

v^{2}\cdot \partial ^{2}y/\partial x^{2} \to [L^{2}\cdot T^{-2}]\cdot [L/L^{2}] = L\cdot T^{-2}

Both sides have units of acceleration.

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Discovery

Jean le Rond d'Alembert · 1747

D'Alembert derived it while studying vibrating strings; Euler and Daniel Bernoulli extended it. It became the prototype of every wave equation in physics — sound, light, gravity.

Try this

Why do guitar strings, ocean swells, and light beams all obey the same equation?

A taut string vibrates with wave speed v = 200 m/s. Verify that y(x,t) = A sin(kx - ωt) satisfies the wave equation when ω/k = v.

Research status: stable

Real-world applications

Seismology: P- and S-waves through Earth's interior follow the wave equation.
Music: standing waves on strings and air columns set instrument pitches.
Telecommunications: signals on coaxial cables and optical fibers.
Medical imaging: ultrasound pulse propagation in tissue.

Common misconceptions

The wave equation describes the medium's motion, not the wave itself — the wave is a pattern, the medium oscillates.
It only works for sinusoids — it admits any twice-differentiable $f(x\pm vt)$ .
Both directions of propagation are required — single-direction waves are also valid solutions.

Experimental verification

Verified on stretched strings, water tanks, and electromagnetic transmission lines. D'Alembert's general solution

y = f(x-vt) + g(x+vt)

is observed directly in pulse-propagation experiments.

Derivation

From Newton's second law on a string element: tension T provides restoring force

T\cdot \partial ^{2}y/\partial x^{2}\cdot dx

, mass

is \mu \cdot dx

, acceleration

is \partial ^{2}y/\partial t^{2}

.

Combining:

\mu \cdot \partial ^{2}y/\partial t^{2} = T\cdot \partial ^{2}y/\partial x^{2}

, giving

v^{2} = T/\mu

.

The same form emerges from Maxwell's equations (with

v=c)

and from acoustic pressure equations.

Limiting cases

\partial ^{2}y/\partial x^{2} = 0

⟶

\partial ^{2}y/\partial t^{2} = 0

Flat profile means no acceleration — wave is uniform or static.

v \to \infty

⟶ Instantaneous propagationUnphysical limit corresponding to rigid-body motion.

v \to 0

⟶ Static deformationNo wave propagation — medium becomes a frozen displacement field.

What if…

What if the medium is dispersive?

Wave speed depends on wavelength, so pulses spread out as they travel — like white light through a prism.

What if amplitude is huge?

Nonlinear terms appear (e.g., KdV, Burgers' equation) and solitons or shocks may form instead of clean waves.

What if we add damping?

An extra term $\gamma \cdot \partial y/\partial t$ makes waves decay exponentially — modeling viscous fluids and lossy cables.

1

Verify a traveling sine wave

Given ·

v:: 200

Find · Confirm

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

satisfies the equation

Steps

Compute $\partial y/\partial t$ = - $A\omega \cos (kx - \omega t)$
Compute $\partial ^{2}y/\partial t^{2}$ = - $A\omega ^{2} \sin (kx - \omega t)$
Compute $\partial y/\partial x = Ak \cos (kx - \omega t)$
Compute $\partial ^{2}y/\partial x^{2} = -Ak^{2} \sin (kx - \omega t)$
Substitute: - $\omega ^{2} = -v^{2}k^{2} \Rightarrow \omega = vk \checkmark$

Result ·

\partial ^{2}y/\partial t^{2}

= -

A\omega ^{2}\sin (kx-\omega t)

;

\partial ^{2}y/\partial x^{2} = -Ak^{2}\sin (kx-\omega t)

. Equation requires

\omega ^{2} = v^{2}k^{2}

, i.e.

v = \omega /k

.

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2

Find wave speed on a string

Given ·

T tension:: 80
mu linear density:: 0.002

Find · v

Steps

Write $v^{2} = T/\mu$ from the wave-equation derivation
Substitute $T = 80\,\mathrm{N}$ , $\mu = 0.002\,\mathrm{kg/m}$
$v^{2} = 40000\,\mathrm{m}^{2}/s^{2}$
$v = 200\,\mathrm{m/s}$

Result ·

v = \sqrt{T/\mu } = \sqrt{80 / 0.002} = \sqrt{40000} = 200\,\mathrm{m/s}

Wave Speed Equation→Standing Wave Frequencies→Snell's Law→