Separation of Variables

Using the technique of separation of variables, let u(r,t) = v(r)w(t). Substituting into the wave equation,

Clear[v,w]
waveSeparable := D[v[r] w[t],{t,2}] == Div[c^2 Grad[v[r] w[t]]]
waveSeparable

Notice that we can factor w[t] out of the right hand side and separate the functions of r from the functions of t. waveSeparable2 = Simplify[waveSeparable] waveSeparable3 = waveSeparable2[[1]]/(v[r] c^2 w[t]) == waveSeparable2[[2]]/(v[r] c^2 w[t]) Since one side of the equation is a function of t and the other is a function of r, both must be a constant. Let's cheat and let that constant be -k^2. This will simplify the notation later.

timeDiffEqn = waveSeparable3[[1]] == -k^2
timeDiffEqn2 = 

   timeDiffEqn[[1]]*(c^2 w[t]) == timeDiffEqn[[2]]*(c^2 w[t])

radiusDiffEqn = waveSeparable3[[2]] == -k^2

radiusDiffEqn2 = 
   radiusDiffEqn[[1]]*(r v[r]) == radiusDiffEqn[[2]]*(r v[r])

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