Solution to the scalar wave equation in 3D In class, we showed that one particul

Question

Solution to the scalar wave equation in 3D

In class, we showed that one particular solution to the scalar wave equation in 3D was a plane wave, for which the wave took an argument of the form r*u plusminus vt . It was called a plane wave because, at a fixed time, the function U(r*u plusminus vt) has the same value over a plane specified by r.u = constant. In this problem, you will find a different type of solution under spherical symmetry. Assume we are looking for a spherically symmetric solution to the scalar wave equation of the form U(r,t)/r, where r is the radial position coordinate in spherical coordinates. Show that if U(r,t)/r satisfies the scalar wave equation in 3D, U(r,t) satisfies a ID scalar wave equation. What, therefore is the solution to the 3D scalar wave equation under spherical symmetry (i.e. what is the form of U(r,t)/r)? (This type of solution is called a spherical wave). For a fixed time, what are the surfaces over which this spherical wave is constant? How do these surfaces propagate as time advances?

Explanation / Answer

Let f(r,t) = U(r,t) / r

The 3D wave equation is

d2f/dt2 = v^2*laplacian (f)

or equivalent

d2f/dt2 = v^2*(grad)^2 (f)

In spherical coordinates the laplacian is written as

laplacian = (grad)^2 = d2/dr2 +(2/r)*d/dr

(http://en.wikipedia.org/wiki/Wave_equation#Spherical_waves)

In our case

df/dr = d/dr (U/r) = 1/r*dU/dr -U/r^2

d2f/dr2 = d/dr (1/r*dU/dr -U/r^2) = (1/r)*d2U/dr2 -(1/r^2)*dU/dr -(1/r^2)*dU/dr -2U/r^3

therefore  

laplacian(f) = d2f/dr2 +(2/r)*df/dr =

=(1/r)*d2U/dr2 -(2/r^2)*dU/dr -2U/r^3 + (2/r)*[1/r*dU/dr -U/r^2] =(1/r)*d2U/dr2

But

d2f/dt2 =(1/r)*d2U/dt2

And therefore the 3D wave equation becomes

(1/r)*d2U/dt2 =v^2*(1/r)*d2U/dr2

or equivalent

d2U/dt2 =v^2*d2U/dr2

WHICH IS THE 1D WAVE EQUATION that has as solution U(r,t)

b)

Solutions of the form

f(r,t) = U(r,t) / r

have constant surfaces SPHERICAL (or said simpler spheres). As the wave propagates these surfaces become of bigger and bigger radius (the spheres become bigger and bigger with time).

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