Spherical Waves

To find spherical solutions to the wave equation it is natural to use spherical coordinates. x = r sin θ cos φ x = r sin θ cos φ y = r sin θ sin φ y = r sin θ sin φ z = r cos θ z = r cos θ

There is a very nice discussion of Spherical Coordinates at:

http://mathworld.wolfram.com/SphericalCoordinates.html

There is also a nice discussion of Cylindrical coordinates at the same site

http://mathworld.wolfram.com/CylindricalCoordinates.html

Beware the confusion about θ θ and φ φ . We are calling the polar angle θ . θ . All other mathematical disciplines get it wrong and call it φ . φ .

The Laplacian can be written in spherical coordinates, but where does that come from?looking at just the x x term ∂ ∂ x = ∂ r ∂ x ∂ ∂ r + ∂ θ ∂ x ∂ ∂ θ + ∂ φ ∂ x ∂ ∂ φ ∂ ∂ x = ∂ r ∂ x ∂ ∂ r + ∂ θ ∂ x ∂ ∂ θ + ∂ φ ∂ x ∂ ∂ φ Then you take the second derivative to get ∂ 2 ∂ x 2 ∂ 2 ∂ x 2 which as you can imagine is a tremendously boring and tedious thing to do.Since this isn't a vector calculus course lets just accept the solution.In the case of spherical waves it is not so difficult since the θ θ and φ φ derivative terms all go to 0 0 .

∇ 2 ψ ( r ) = 1 r 2 ∂ ∂ r ( r 2 ∂ ψ ∂ r ) = ∂ 2 ψ ∂ r 2 + 2 r ∂ ψ ∂ r = 1 r ∂ ∂ r ( ψ + r ∂ ψ ∂ r ) = 1 r ∂ 2 ( r ψ ) ∂ r 2 ∇ 2 ψ ( r ) = 1 r 2 ∂ ∂ r ( r 2 ∂ ψ ∂ r ) = ∂ 2 ψ ∂ r 2 + 2 r ∂ ψ ∂ r = 1 r ∂ ∂ r ( ψ + r ∂ ψ ∂ r ) = 1 r ∂ 2 ( r ψ ) ∂ r 2 Thus for spherical waves, we can write the wave equation: 1 r ∂ 2 ( r ψ ) ∂ r 2 = 1 v 2 ∂ 2 ψ ∂ t 2 1 r ∂ 2 ( r ψ ) ∂ r 2 = 1 v 2 ∂ 2 ψ ∂ t 2 Now we can multiply both sides by r r and since r r does not depend upon t t write ∂ 2 ( r ψ ) ∂ r 2 = 1 v 2 ∂ 2 ∂ t 2 ( r ψ ) ∂ 2 ( r ψ ) ∂ r 2 = 1 v 2 ∂ 2 ∂ t 2 ( r ψ ) This is just the one dimensional wave equation with a harmonic solution r ψ ( r , t ) = A e i k ( r ∓ v t ) r ψ ( r , t ) = A e i k ( r ∓ v t ) or ψ ( r , t ) = A r e i k ( r ∓ v t ) ψ ( r , t ) = A r e i k ( r ∓ v t )