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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#12, Rev. A, due Wednesday, 2024-04-28 }
\end{center}

\vspace{.3 in}

{\bf 1.} Solve Laplace's equation $\nabla^{2}V=(\partial_{x}^{2}+\partial_{y}^{2})V=0$ for the {\bf tent potential}
$V(x,y)$ defined on the region $-a{<}x{<}a$ and $-b{<}y{<}b$ with boundary conditions $V(x,\pm b) = 0$ and $V(\pm a, y) = V_0 (1-|y/b|)$ by following these steps:

{\bf \quad a)} Substitute the eigenfunctions $X(x)$, $Y(x)$ and eigenvalues $-k_{x}^{2}$, $-k_{y}^{2}$ of the two differential operators  $\partial_{x}^{2}$, $\partial_{y}^{2}$ of Laplace's equation to get the {\it disperson relation} between the two eigenvalues.

{\bf \quad b)} Apply boundary conditions at $y=\pm b$ to quantize $k_{n}$, $Y_{n}(y)$ and therefore $X_{n}(x)$, and form a linear combination of all eigenfunctions to obtain the general solution $V(x,y)=\sum_{n}c_{n}X_{n}(x)Y_{n}(y)$.

{\bf \quad c)} Apply boundary conditions at $x=\pm a$ to solve the coefficients $c_{n}$ of the general solution.

{\bf \quad d)} Sketch the solution and its first three Fourier components in Mathematica.


{\bf 2. Drumhead waves} are described by the PDE $\big(\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla_{\perp}^{2}\big)\eta(\rho,\phi,t)=0$, where the wave velocity $v=\sqrt{\gamma/\sigma}$ depends on the surface tension $\gamma$ and the mass density $\sigma$ of the drumhead.

{\bf \quad a)} Use $\partial_{t}e^{-i\omega t}=-i\omega\, e^{-i\omega t}$ to obtain the Helmholtz equation $(\nabla_{\perp}^{2}+k^{2})\eta=0$ by replacing $\partial_{t}$ with its {\it eigenvalue}.  Determine the \href{https://en.wikipedia.org/wiki/Dispersion_relation}{\it dispersion relation} between spatial $k$ and temporal $\omega$ frequencies.

{\bf \quad b)} Expand $\nabla_{\perp}^{2}\eta$ in cylindrical coordinates and show the radial part has the equivalent forms
\[\textstyle{
  \frac{\partial^2}{\partial\rho^2} +
  \frac{1}{\rho}\frac{\partial}{\partial\rho}
  ~~=~~ \frac{1}{\rho}\frac{\partial}{\partial\rho}
  \rho\frac{\partial}{\partial\rho}
  ~~=~~ \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial\rho^2}\sqrt{\rho} +
  \frac{1}{4\rho^2}
}\]

{\bf \quad c)} Use the eigenvalue equation $\partial_{\phi}\Phi_{m}(\phi)=im\,\Phi_{m}(\phi)$ to factor out the $\phi$ dependence in the Laplacian and obtain the \href{https://en.wikipedia.org/wiki/Bessel_function}{\it Bessel equation}. %Jackson 2ed (3.75,75) p103
Plot the first three {\it Bessel functions} $J_{0}(x)$, $J_{1}(x)$, and $J_{2}(x)$, where $x=k\rho$.  Find the lowest-order Taylor approximation of each function as $x\to0$ and the asymptotic approximation as $x\to\infty$.  The energy, which is proportional to the amplitude $\eta$ squared, spreads out as the circular wavefront expands.

{\bf \quad d)} Use the boundary conditions $\eta(\rho,0)=\eta(\rho,2\pi)$ and $\eta_{,\phi}(\rho,0)=\eta_{,\phi}(\rho,2\pi)$ to show that $m$ must be an integer.  Use the linearity of $\partial_{\phi}$ on $\Phi_{m}(\phi)\pm\Phi_{-m}(\phi)$ to show that $\cos(m\phi)$ and $\sin(m\phi)$ are also eigenfunctions of $\partial_{\phi}^{2}$ (but not $\partial_{\phi}$---why?) and determine the eigenvalues.  Apply the boundary condition $\eta(a,\phi)=0$ to find the possible values of $k$, in terms of $x_{nm}$, the $n^{\mathrm{th}}$ \href{https://mathworld.wolfram.com/BesselFunctionZeros.html}{zero of the Bessel function} $J_{m}(x)$.  For each combination of $m,n$ plot the {\it node lines} where $\eta_{mn}(\rho,\phi)=0$ and find the vibrational frequency $\omega_{mn}$ of this mode.

{\bf \quad e)} [{\it bonus}: how could this solution be modified to solve the three-dimensional wave equation $(\partial_{t}^{2}/v^{2}-\nabla^{2})\Psi(\rho,\phi,z,t)=0$ with boundary conditions $\Psi(a,\phi,z,t)=\Psi(\rho,\phi,\pm b,t)=0$?]

%bonus: DCTC


{\bf 3. Harmonics} are the common {\bf multipole} angular solutions of any PDE \mbox{involving the Laplacian.}

{\bf \quad a)}  In the long wavelenth limit $k\to 0$, the Helmholtz equation becomes the Laplace equation $\nabla_{\perp}^{2}\eta=0$. The eignfunctions of $\partial_{\phi}$ are still the {\it cylindrical harmonics} $\Phi_m(\phi)=e^{im\phi}$, but for the radial solution, $\lim_{k\to 0}J_{m}(k\rho)$ transforms each Bessel function to its lowest order Taylor approximation.  Put the ${\it ansatz}$ $R_{m}(\rho)=\rho^{\alpha}$, into Laplace's equation and solve for $\alpha$ to find two independent solutions $R_{m}(\rho)$.  One which is finite at the origin $\rho=0$ and the other as $\rho\to\infty$.  Show that for $m=0$, $R(\rho)=\ln(\rho)$ is a second independent solution.  Express the {\it planar harmonics} $R_{m}(\rho)e^{im\phi}$ which are finite at origin in the form $(x+iy)^{m}=A_{m}+iB_{m}$ and expand $A_{m}$, $B_{m}$ as polynomial solutions to Laplace's equation.  Explicitly verify these solutions up to $m=3$.

{\bf \quad b)} Expand $\nabla^{2}V(r,\theta,\phi)$ in spherical coordinates and show the radial part has the forms
\[\textstyle{
  \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}
  ~=~ \frac{1}{r^2}\frac{\partial}{\partial r} r^2\frac{\partial}{\partial r}
  ~=~ \frac{1}{r}\frac{\partial^2}{\partial r^2}r
}\]

{\bf \quad c)} Factor out the $\phi$-dependence as in \#1c) and identify the $\theta$-operator $L^{2}=\frac{-d}{dx}(1-x^{2})\frac{d}{dx}+ \frac{m^{2}}{1-x^{2}}$, %Jackson 2ed (3.9) p85
where $x=\cos\theta$ [distinct from the Cartesian coordinate $x$ and from $x=k\rho$ of \#2c)!], to obtain the \href{https://en.wikipedia.org/wiki/Associated_Legendre_polynomials}{\it{general Legendre equation}}, for polar waves.  Restriction to $m=0$ yields the \href{https://en.wikipedia.org/wiki/Legendre_polynomials}{\it{Legendre polynomials}} of H07\#2c.  Continuity at the poles $\theta=0,\pi$ requires that $\ell=|m|,|m+1|,|m+2|,\ldots,\infty$.  Find all ten polar eigenfunctions $P_{\ell}^{|m|}(x)$ up to $\ell=3$.  Pick two harmonics and show that they are eigenfunctions of $L^{2}$.  The combined \href{https://en.wikipedia.org/wiki/Spherical_harmonics}{\it{spherical harmonics}} $Y_{lm}=(-1)^{m}\sqrt{\frac{(2\ell+1)}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}P_{l}^{m}(\cos\theta)e^{im\phi}$ are normalized eigenfunctions of the operator $L^{2}(\theta,\phi)$ with eigenvalues $\lambda=\ell(\ell+1)$.  They represent the atomic $s,p,d,f$ \href{https://en.wikipedia.org/wiki/Atomic_orbital}{{\it orbitals}} for $\ell=0,1,2,3$.  Draw the node lines of each of these $\ell,m$-modes on a sphere.

{\bf \quad e)} Write $\nabla^{2}$ using the third form of part 3b) and $L^{2}$.  Factor out the angular dependence and solve the radial Laplace equation for the two eigenfunctions $R_{l}(r)$ for each $\ell$ as in part a) to obtain the \href{https://en.wikipedia.org/wiki/Solid_harmonics}{\it solid harmonics} $R_{\ell}^{m}(\bs r)$ and $I_{l}^{m}(\bs r)$.  Expand $R_{\ell}^{m}(\bs r)$ in Cartesian coordinates, factoring out the planar harmonic in each.  These multinomials in $x,y,z$ are used to label the sub-orbitals.

{\bf \quad f)} [{\it bonus}: Solve the Laplace boundary value problem in all space with a point flux source at $\bs r'=(r',\theta',\phi')$ to obtain the potential $V(\bs r)=\sum_{lm}{R_{l}^{m}}^{*}(\bs r')I_{l}^{m}(\bs r)$ if $r<r'$ or $\sum_{lm}{R_{l}^{m}}^{*}(\bs r')I_{l}^{m}(\bs r)$ if $r>r'$. $Q_{lm}=\int dq' {R_{l}^{m}}^{*}(\bs r')$ is the interior [or ${I_{l}^{m}}^{*}(\bs r')$ for the exterior] {\it multipole moment} of the charge distribution, and $I_{l}^{m}(\bs r)$ [or $R_{l}^{m}(\bs r)$] is its corresponding potential.  Compare with the point potential Green's function $V(\bs r)=\frac{1}{4\pi\rcurs}$ to obtain the {\it addition theorem} %Jackson 2ed (3.38) p92, (3.62) p101, (3.70) p102
$\frac{1}{\rcurs}=\sum_{\ell=0}^{\infty}\frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}P_{\ell}(\cos\gamma)$, where $\gamma$ is the angle between $\bs r$ and $\bs r'$ and $P_{\ell}(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}Y_{lm}^{*}(\theta',\phi')Y_{lm}(\theta,\phi)$, and $r_{<}$, $r_{>}$ are the lesser and greater values of $r$, $r'$, respectively.]

{\bf \quad g)} [{\it bonus}: Show that the spherical solution of the Helmholtz equation $(\nabla^{2}+k^{2})j_{l}(kr)Y_{lm}(\theta,\phi)$ is similar to cylindrical with $m\to\ell+\frac12$, and thus the solutions are the \href{https://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions}{\it spherical Bessel functions} $j_{l}(kr)=\sqrt{\frac{\pi}{2kr}}J_{\ell+1/2}(kr)$.  The same principle holds for all wave equations in different dimensions.  Calculate and illustrate the modes of a spherical wave confined to $r<a$ \`a la \#2d).]

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