Solution to the scalar wave equation in 3D In class, we showed that one particul
ID: 1768049 • Letter: S
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).