The wave equation

What a wave is

A wave is a disturbance that travels through space and time, carrying energy and information. Water waves, sound waves, waves on strings, seismic waves, and electromagnetic waves all share this basic idea, even though their physical mechanisms differ.

A one-dimensional wave traveling along the x-axis can often be represented by a function

$y(x,t)$

where $y$ is the displacement of the medium at position $x$ and time $t$.

Traveling wave form

A wave moving to the right with speed $v$ can be written

$y(x,t)=f(x-vt)$

This expression means the shape $f$ moves in the positive x-direction without changing shape. At later times, the same feature appears at larger $x$.

A wave moving to the left can be written

$y(x,t)=f(x+vt)$

The sign tells the direction of travel.

Sinusoidal waves

A sinusoidal traveling wave moving to the right is often written

$y(x,t)=Acos(kx-omega t+phi)$

Here $A$ is amplitude, $k$ is wave number, $omega$ is angular frequency, and $phi$ is phase constant.

The wave number is related to wavelength:

k=rac{2pi}{lambda}

The angular frequency is related to period and frequency:

omega=rac{2pi}{T}=2pi f

The wave speed is

v=rac{omega}{k}=flambda

The one-dimensional wave equation

The ideal one-dimensional wave equation is

rac{partial^2 y}{partial x^2}=rac{1}{v^2}rac{partial^2 y}{partial t^2}

This partial differential equation relates curvature in space to acceleration in time. It says that the medium's local acceleration is tied to the shape of the disturbance.

A function of the form $f(x-vt)$ or $f(x+vt)$ satisfies this equation for wave speed $v$.

Why partial derivatives appear

A wave function depends on two variables: position and time. A partial derivative measures change with respect to one variable while holding the other fixed.

The derivative $partial y/partial x$ describes spatial slope at a fixed time. The derivative $partial y/partial t$ describes how the displacement changes at a fixed position.

This is different from a particle trajectory $x(t)$, which depends only on time.

Physical origin of the wave equation

The wave equation appears when neighboring parts of a system exert restoring effects on one another. For a string under tension, curved parts of the string experience net forces that accelerate them toward a straighter shape. For sound, pressure differences accelerate fluid elements. For electromagnetic waves, changing electric and magnetic fields regenerate one another.

The specific wave speed depends on the medium and restoring mechanism.

Phase

The phase of a sinusoidal wave is

$kx-omega t+phi$

Points with the same phase have the same state of oscillation. For a crest, the phase remains constant as the crest moves. Setting

$kx-omega t=constant$

and differentiating gives

krac{dx}{dt}-omega=0

so

rac{dx}{dt}=rac{omega}{k}=v

This is the phase velocity.

The big idea

The wave equation describes disturbances that propagate through space with speed $v$. Traveling wave solutions depend on $x-vt$ or $x+vt$, while sinusoidal waves use amplitude, wavelength, frequency, wave number, and phase. This mathematical structure is the foundation for wave physics across many systems.