Separation of Variables
Separation of variables is a technique for solving certain partial differential equations. It assumes that the
overall solution (a function of several variables) is simply a product of single variable functions, described mathmatically
below
$$
\psi(x,y,...,z)=\mathbb{X}(x)\mathbb{Y}(y)... \mathbb{Z}(z)
$$
The whole goal of assuming an overall function of the above form is to separate, or isolate, all terms dependant
on only one independant variable on one side of the equality. This allows us to exploit
the theorem that states that equalities between differential relations of different independant variables requires
the relations equal some constant or zero; effectively 'separating' the PDE into the corresponding
number of ODEs (which are generally easier to solve analytically).
It's generally favored by physicists as a choice technique for solving several common PDEs.
While it's criteria for applicability seems dense, one can always attempt a solution of the form
described above, and if it works, it works.
Helmholtz Equation
The Helmholtz equation for real values of \(k\) is given below.
$$
\nabla^2\psi+k^2\hspace{.05cm}\psi=0
$$
Cylindrical Coordinates (Circular)
In cylindrical coordinates, the Helmholtz equation takes the form
$$
\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \psi}{\partial r}\right)
+\frac{1}{r^2}\frac{\partial^2 \psi}{\partial \phi^2}+\frac{\partial^2 \psi}{\partial z^2}+k^2\psi=0
$$
The first assumption we always make in applying the technique of separation of variables is
to write out the overall function \(\psi\) as a product of functions solely dependant only on one variable.
In the context of cyclindrical coordinates then, we have the following,
$$
\psi(z,\phi,r)=\mathcal{Z}(z)\Phi(\phi)R(r)
$$
Substituting this into the defining differential equation and dividing through by \(\psi\) we get the following:
$$
\frac{1}{rR}\frac{\partial }{\partial r}\left(r\frac{\partial R}{\partial r}\right)
+\frac{1}{r^2\Phi}\frac{\partial^2 \Phi}{\partial \phi^2}+\frac{1}{\mathcal{Z}}\frac{\partial^2 \mathcal{Z}}{\partial z^2}+k^2=0.
$$
The key now is to, one by one, single out all dependencies on one variable to one side of the equality. We can then
leverage the fact that all the coordinates are independent of each other. Thus, any isolated expression dependent solely
on one independent variable must equal some constant, else the choice of value for the parameter is no longer arbitrary and the
expression is not satisfied. The three separated ODE's for each variable are now shown below.
$$\begin{split}
\frac{1}{\mathcal{Z}}\frac{\partial^2 \mathcal{Z}}{\partial z^2}&=-k^2-\frac{1}{rR}\frac{\partial }{\partial r}\left(r\frac{\partial R}{\partial r}\right)
-\frac{1}{r^2\Phi}\frac{\partial^2 \Phi}{\partial \phi^2}\\
& = l^2\\
\Rightarrow \ \ \mathcal{Z}(z)&=Ae^{\pm zl}\quad \text{(Exponential)}
\end{split}$$
$$\begin{split}
\frac{1}{\Phi}\frac{\partial^2\Phi}{\partial \phi^2}& =-\left[\frac{r}{R}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r} \right)+r^2(l^2+k^2)\right]\\
& = -m^2\\
\Rightarrow \ \ \Phi(\phi)&=Ae^{\pm iml}\quad \text{(Sinusoidal)}
\end{split}$$
$$\begin{split}
\frac{r}{R}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r} \right)+r^2n^2-m^2& = 0\\
\Rightarrow \ \ R(r)&=J_n(kr)\quad \text{(Cylindrical Bessel)}
\end{split}
$$
Notice here the sign of \(m^2\) and \(l^2\) are aribtrary. Here, assuming real constants, the conventional
signs are utlilized as they most often correspond to relevant phyiscal systems of interest.
Spherical Coordinates
In spherical coordinates (with \(\phi\) set as the azimuthal angle and \(\theta\) the co-latitude)
the Helmholtz equation takes the form
$$
\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right)
+\frac{1}{r^2\sin\theta }\frac{\partial}{\partial \theta}\left( \sin\theta\frac{\partial \psi}{\partial \theta}\right)
+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 \psi}{\partial \phi^2}+k^2\psi=0
$$
Again, we write out the overall function \(\psi\) as a product of single variable functions.
In the context of spherical coordinates, we have the following:
$$
\psi(r,\phi,\theta)=R(r)\Theta(\theta)\Phi(\phi).
$$
Substituing this into the spherical Helmholtz equation and then dividing by \(\psi\) results in the relation
below.
$$
\frac{1}{Rr^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)
+\frac{1}{\Theta r^2\sin\theta }\frac{\partial}{\partial \theta}\left( \sin\theta\frac{\partial \Theta}{\partial \theta}\right)
+\frac{1}{\Phi r^2\sin^2\theta}\frac{\partial^2 \Phi}{\partial \phi^2}+k^2\psi=0
$$
Below are the separated equations for the spherical Helmholtz equation.
$$
\begin{align}
\frac{1}{\Phi}\frac{\partial^2\Phi}{\partial \phi^2}&=-\left[ \frac{\sin^2\theta}{R}\frac{\partial}{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)
+\frac{\sin\theta}{\Theta }\frac{\partial}{\partial \theta}\left( \sin\theta\frac{\partial \Theta}{\partial \theta}\right)+kr^2\sin^2\theta\right]\\
& = -l^2\\
\Rightarrow \ \ \Phi(\phi)& =e^{il\phi}\quad\text{(Complex Exponential)}\\
\\
\\
\frac{1}{R}\frac{\partial}{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)+k^2r^2 & = -\left[
\frac{1}{\Theta\sin\theta }\frac{\partial}{\partial \theta}\left( \sin\theta\frac{\partial \Theta}{\partial \theta}\right)
+\frac{1}{\Phi \sin^2\theta}\frac{\partial^2 \Phi}{\partial \phi^2}\right]\\
\frac{r^2}{R}\frac{\partial^2 R}{\partial r^2}+\frac{2r}{R}\frac{\partial R}{\partial r}+k^2r^2& = n^2\\
\Rightarrow \ \ R(r)& =J_n(kr) \quad\text{(Spherical Bessel)}\\
\end{align}$$
$$\begin{align}
\frac{\sin\theta}{\Theta }\frac{\partial}{\partial \theta}\left( \sin\theta\frac{\partial \Theta}{\partial \theta}\right)
+ n^2\sin^2\theta
& = -\left[\frac{1}{\Phi}\frac{\partial^2 \Phi}{\partial \phi^2}
\right] \\
& = l^2\\
\Rightarrow \ \ \Theta(\theta)& = P^m_{l}(\cos\theta ) \quad\text{(Associated Legendre Polynomials)}\\
\end{align}
$$
Cartesian Coordinates
In \(d\)-dimensional Cartesian coordinates, the Helmholtz equation takes the form
$$
\sum_{i=1}^{d}\frac{\partial^2\psi}{\partial x_i^2}+k^2\psi=0
$$
Our assumed solution form is then
$$
\psi(x_1,...x_d)=X_1(x_1)X_2(x_2)...X_d(x_d).
$$
Hence, all coordinates can be easily and similarly decomposed into
$$\begin{split}
\frac{\partial^2X_i}{\partial x_i^2}&=l_i^2X_i\\
\Rightarrow\ \ X_{i}(x_i)& =e^{\pm l_ix_i} \quad \text{(Complex Exponential)}\\
\end{split}\\
$$
Where \(\sum_dl_d^2=-k^2\).
Resources
Separation of variables is covered in Arfken, Weber, and Harris' Mathematical Methods for Physicist's.
