Wave Equation — Physica

Equivalent forms

u_{tt} = c^2 u_{xx}

u(x,t) = F(x - ct) + G(x + ct)\quad(\text{d'Alembert})

\Box\, u = 0\quad(\Box = \partial_t^2/c^2 - \nabla^2)

Linear, so waves superpose; finite speed c, so cause precedes effect — the same hyperbolic form governs sound, light, strings, and gravitational waves.

Symbol	Name	SI unit	Dimension	Range
$u$	Displacementoutput Field value (e.g. string displacement)	1	1	-1 – 1
$c$	Wave speed Propagation speed of disturbances	m/s	L T^-1	0.1 – 5
$n$	Mode number Harmonic of the standing wave on a fixed string	1	1	1 – 6
$x$	Position Coordinate along the string of length 1	m	L	0 – 1

Unit systems

Symbol	SI	CGS	Imperial
$c$	m/s	cm/s	ft/s
$x$	m	cm	ft
$u$	1	1	1
$n$	1	1	1

Where it holds

Linear, non-dispersive media; small-amplitude (linearized) disturbances.

Dimensional analysis

u_tt \to [u]/[T^{2}]

c^{2} u_xx \to [L^{2}/T^{2}]\cdot [u]/[L^{2}] = [u]/[T^{2}]

The

c^{2}

exactly cancels the extra length

^{2}

from the two spatial derivatives, matching the two time derivatives.

\checkmark

Discovery

Jean le Rond d'Alembert · 1747

d'Alembert derived the vibrating-string equation in 1747 and gave its general solution as two waves traveling in opposite directions. Euler and Daniel Bernoulli then debated whether arbitrary shapes and sine-sums could represent the motion — a dispute that seeded Fourier analysis.

Try this

Pluck a string and a shape runs along it at a fixed speed, bounces off the ends, and comes back. One equation choreographs all of it.

Set the wave speed and mode and watch a standing wave breathe — the second time-derivative chasing the second space-derivative.

Research status: stable

Real-world applications

Acoustics and musical-instrument design
Electromagnetic wave propagation and antennas
Seismic wave modeling
Gravitational-wave physics (linearized GR)

Common misconceptions

Waves carry matter along — they transport energy and momentum, not the medium.
The wave equation is dispersive — the classical form is non-dispersive; all frequencies travel at the same c.
Standing waves don't move — they are superpositions of two oppositely traveling waves.

Experimental verification

Measured string and air-column frequencies match

f_n = nc/2L

; the speed of light

c = 1/\sqrt{\mu _{0}\varepsilon _{0}}

predicted by the EM wave equation agrees with direct measurement; LIGO detects gravitational waves obeying the same hyperbolic structure.

Derivation

For a string under tension T with linear density

\mu

, Newton's law on an element gives

\mu u_tt = T u_xx

, i.e.

u_tt = c^{2}u_xx

with

c = \sqrt{T/\mu }

.

The same form arises from Maxwell's equations

(c = 1/\sqrt{\mu _{0}\varepsilon _{0}})

and from linearized fluid dynamics for sound.

Limiting cases

Initial shape only, no velocity⟶ Splits into two half-amplitude traveling wavesd'Alembert:

u = \tfrac{1}{2}F(x-ct) + \tfrac{1}{2}F(x+ct)

.

Fixed ends⟶ Standing waves, frequencies

f_n = nc/2L

Boundary conditions quantize the allowed modes.

c \to \infty

⟶ Approaches a constraint (instantaneous equilibration)Infinite-speed limit loses the causal, propagating character.

What if…

What if I add damping?

A term $\gamma u_t$ makes waves decay; with a restoring term you get the Klein–Gordon equation, which is dispersive.

What if the medium varies?

c becomes c(x); waves refract, reflect, and can be trapped — the basis of waveguides and seismology.

1

Fundamental frequency of a string

Given ·

L:: 1
c:: 1

Find · f_1

Steps

Mode n has $f_n = nc/2L$
$n = 1$ , $c = 1$ , $L = 1$
$f_1 = 1/(2\cdot 1) = 0.5\,\mathrm{Hz}$ ; harmonics are integer multiples

Result ·

f_1 = c/2L = 0.5\,\mathrm{Hz}

2

Verify a traveling-wave solution

Given ·

u:: $\sin (x - ct)$

Find · Does it satisfy

u_tt = c^{2}u_xx

?

Steps

$u_tt = -c^{2}\sin (x-ct)$
$u_xx = -\sin (x-ct) \to c^{2}u_xx = -c^{2}\sin (x-ct)$
Both sides equal — $any F(x-ct)$ is a solution

Result ·

u_tt = -c^{2}\sin (x-ct)

,

c^{2}u_xx = -c^{2}\sin (x-ct) \checkmark

Laplacian→Heat Equation→Fourier Series→Dirac Delta Function→