Separation of Variables

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

In[13]:=
Clear[v,w]

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

Out[15]=
                2               2

               c  w[t] v'[r] + c  r w[t] v''[r]

v[r] w''[t] == --------------------------------

                              r

Notice that we can factor w[t] out of the right hand side and separate the functions of r from the functions of t.

In[16]:=
waveSeparable2 = Simplify[waveSeparable]

Out[16]=
                2

               c  w[t] (v'[r] + r v''[r])

v[r] w''[t] == --------------------------
                           r

In[17]:=
waveSeparable3 = waveSeparable2[[1]]/(v[r] c^2 w[t]) ==
				 waveSeparable2[[2]]/(v[r] c^2 w[t])

Out[17]=
w''[t]     v'[r] + r v''[r]

------- == ----------------

 2              r v[r]

c  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.

In[18]:=
timeDiffEqn = waveSeparable3[[1]] == -k^2

Out[18]=
w''[t]       2

------- == -k

 2
c  w[t]

In[19]:=
timeDiffEqn2 = 
   timeDiffEqn[[1]]*(c^2 w[t]) == timeDiffEqn[[2]]*(c^2 w[t])

Out[19]=
             2  2
w''[t] == -(c  k  w[t])

In[20]:=
radiusDiffEqn = waveSeparable3[[2]] == -k^2

Out[20]=
v'[r] + r v''[r]      2

---------------- == -k
     r v[r]

In[21]:=
radiusDiffEqn2 = 
   radiusDiffEqn[[1]]*(r v[r]) == radiusDiffEqn[[2]]*(r v[r])

Out[21]=
                       2
v'[r] + r v''[r] == -(k  r v[r])

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