# Dictionary:Wave equation

An equation that relates the spatial and time dependence of a disturbance which can propagate as a wave. In rectangular coordinates *x*, *y*, *z*, it is

^{2}ψ=(∂^{2}ψ)/(∂x^{2})+(∂^{2}ψ)/(∂y^{2})+(∂^{2}ψ)/(∂z^{2})=((1)/(V^{2}))(∂^{2}ψ)/(∂t^{2}),

where ψ represents wave displacement (pressure, rotation, dilatation, etc.) and *V* the velocity of the wave. Functions *f*(ℓ*x*+*my*+*nz*±*Vt*) are solutions to this equation. In spherical coordinates where *r* is the radius, θ the colatitude, and * the longitude, the wave equation becomes: *

^{2}))(∂

^{2}Ψ)/(∂t

^{2})=((1)/(r

^{2}))[((∂)/(∂r))(r

^{2}(∂Ψ)/(∂r))+((1)/(sinθ))((∂)/(∂θ))(sinθ(∂Ψ)/(∂θ))+((1)/(sin

^{2}θ))(∂

^{2}Ψ)/(∂

^{2})]

The foregoing are forms of the **scalar wave equation** These forms do not provide for the conversion of P-waves to S-waves nor vice-versa. The **vector wave equation** is more general; it is

*(''*ψ)–μ

*×(*×ψ)=ρ∂

^{2}ψ/∂

*t*

^{2},

which can be written in component form as

*x*

^{2}ψ_{}*+(μ+λ)(∂/∂*x

**'**)(∂ψ**x'**_{}*/∂*x

*+∂ψ*y

_{}*/∂*y

*+∂ψ*z

_{}*/∂*z

*)=ρ∂*t

^{2}ψ/∂

^{2}.

If divψ=0, this gives an S-wave; if curl ψ=0, a P-wave. The wave equation in polar anisotropic (transversely isotropic) media is given in Figure T-13.