24 Jun 2025
Laplace Equation in Rectangular Geometry
Separation of variables for two- and three-dimensional rectangular boundary-value problems.
In a two-dimensional rectangle $0<x<a$, $0<y<b$, Laplace’s equation is
\[\boxed{\frac{\partial^2u}{\partial x^2} +\frac{\partial^2u}{\partial y^2}=0}.\]Take the separated form $u(x,y)=X(x)Y(y)$. Substitution gives
\[X''Y+XY''=0.\]After division by $XY$,
\[\frac{X''}{X}=-\frac{Y''}{Y}.\]The left side depends only on $x$ and the right side only on $y$; because they are equal for all $(x,y)$, both must equal a constant. Choose $-k^2$ so that zero conditions at $x=0,a$ have nonzero solutions:
\[X''+k^2X=0, \qquad Y''-k^2Y=0.\]Suppose
\[u(0,y)=u(a,y)=u(x,0)=0, \qquad u(x,b)=f(x).\]The first two conditions require
\[X_n(x)=\sin\frac{n\pi x}{a}, \qquad k_n=\frac{n\pi}{a}, \qquad n=1,2,\ldots.\]The condition at $y=0$ removes the $\cosh(k_ny)$ term, leaving $Y_n\propto\sinh(k_ny)$. Superposition gives
\[u(x,y)=\sum_{n=1}^{\infty}A_n \sin\frac{n\pi x}{a}\sinh\frac{n\pi y}{a}.\]At $y=b$,
\[f(x)=\sum_{n=1}^{\infty}A_n \sinh\frac{n\pi b}{a}\sin\frac{n\pi x}{a}.\]Multiply by $\sin(m\pi x/a)$ and integrate from $0$ to $a$. Orthogonality,
\[\int_0^a\sin\frac{n\pi x}{a}\sin\frac{m\pi x}{a}dx =\frac a2\delta_{mn},\]gives
\[A_n=\frac{2}{a\sinh(n\pi b/a)} \int_0^a f(x)\sin\frac{n\pi x}{a}dx.\]Hence
\[\boxed{ u(x,y)=\sum_{n=1}^{\infty}B_n \frac{\sinh(n\pi y/a)}{\sinh(n\pi b/a)} \sin\frac{n\pi x}{a}},\]where
\[\boxed{B_n=\frac2a\int_0^a f(x) \sin\frac{n\pi x}{a}dx}.\]For $f(x)=V_0\sin(\pi x/a)$, only $B_1=V_0$ is nonzero:
\[\boxed{ u=V_0\sin\frac{\pi x}{a} \frac{\sinh(\pi y/a)}{\sinh(\pi b/a)}}.\]
Three independent variables
For a rectangular box, $u=XYZ$ in $u_{xx}+u_{yy}+u_{zz}=0$ gives
\[\frac{X''}{X}+\frac{Y''}{Y}+\frac{Z''}{Z}=0.\]If $u=0$ at $x=0,a$ and $y=0,b$, choose
\[X_m=\sin\frac{m\pi x}{a}, \qquad Y_n=\sin\frac{n\pi y}{b}.\]Then $Z$ obeys
\[Z''-\gamma_{mn}^2Z=0, \qquad \gamma_{mn}^2=\left(\frac{m\pi}{a}\right)^2 +\left(\frac{n\pi}{b}\right)^2.\]For each $(m,n)$,
\[Z_{mn}=C_{mn}\cosh(\gamma_{mn}z) +D_{mn}\sinh(\gamma_{mn}z).\]The remaining boundary data determine $C_{mn}$ and $D_{mn}$ in the double sum. If the additional face $z=0$ is held at zero, $C_{mn}=0$ and only the $\sinh(\gamma_{mn}z)$ term remains. The separation step is the same; two independent transverse boundary conditions select the two integer labels.
The two-dimensional mode and its four boundary residuals are checked symbolically in the Unit II Maxima worksheet.
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