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Wave equation

The 2D wave equation is

∂2ψ∂t2=vp2∂2ψ∂x2. \frac{\partial^2 \psi}{\partial t^2} = v_p^2 \frac{\partial^2\psi}{\partial x^2}.

The 3D wave equation is

∂2ψ∂t2=vp2∇2ψ. \frac{\partial^2 \psi}{\partial t^2} = v_p^2 \nabla^2 \psi.

These equations appear pretty much everywhere. The same equation describes physical waves, sound waves, and electromagnetic waves.

Standing wave

A standing wave is a solution to the wave equation. It has the form

ψ(x,t)=Asin⁡(Kx+α)sin⁡(ωt+β). \psi(x,t) = A \sin(\mathcal K x + \alpha) \sin(\omega t + \beta).

Because the wave equation is linear, we can use superposition to find more solutions. In particular the superposition of infinite standing waves can describe any initial conditions.

The parameters K\mathcal K and α\alpha are determined by boundary conditions, while AA and β\beta are determined by initial conditions. We incorporate the boundary conditions by setting ψ\psi or ψ′\psi' equal to zero at boundary, depending on if it is closed or open, respectively.

We can use a Fourier series to describe any (reasonable) initial conditions in a standing wave. See for example in the case of massive string.

Traveling wave

Any function of the form f(x+vt)f(x+vt) is a solution to the wave equation. Let ☺︎:=x+vt☺︎ := x + vt.

∂f∂x=∂f∂☺︎∂f∂t=−v∂f∂☺︎∂2f∂x2=∂2f∂☺︎2∂2f∂t2=v2∂2f∂☺︎2 \begin{align*} \frac{\partial f}{\partial x} &= \frac{\partial f}{\partial ☺︎} &\quad \frac{\partial f}{\partial t} &= -v \frac{\partial f}{\partial ☺︎} \\ \frac{\partial^2 f}{\partial x^2} &= \frac{\partial^2 f}{\partial ☺︎^2} &\quad \frac{\partial^2 f}{\partial t^2} &= v^2 \frac{\partial^2 f}{\partial ☺︎^2} \end{align*} ∂2f∂t2=v2∂2f∂x2. \frac{\partial^2 f}{\partial t^2} = v^2 \frac{\partial^2 f}{\partial x^2}.

If the xx and vtvt terms have the same sign, the wave is traveling in the −x-x direction, otherwise, the +x+x direction.

Initial conditions are easiest to specify with a traveling wave, but boundary conditions can be more difficult to express. In the particular case where there is one boundary (for example, at the boundary of two strings), we can represent the boundary behavior as a superposition of incident, reflected, and transmitted waves.