Dictionary:Wave equation

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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

2ψ=(∂2ψ)/(∂x2)+(∂2ψ)/(∂y2)+(∂2ψ)/(∂z2)=((1)/(V2))(∂2ψ)/(∂t2),


where ψ represents wave displacement (pressure, rotation, dilatation, etc.) and V the velocity of the wave. Functions f(ℓx+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:

((1)/(V2))(∂2Ψ)/(∂t2)=((1)/(r2))[((∂)/(∂r))(r2(∂Ψ)/(∂r))+((1)/(sinθ))((∂)/(∂θ))(sinθ(∂Ψ)/(∂θ))+((1)/(sin2θ))(∂2Ψ)/(∂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

(2μ+λ)(''ψ)–μ×(×ψ)=ρ∂2ψ/∂t2,


which can be written in component form as

μ2ψx+(μ+λ)(∂/∂'x')(∂ψx/∂x+∂ψy/∂y+∂ψz/∂z)=ρ∂2ψ/∂t2.


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.