= Harmonic sine mode in a half-strip
{title2=$\phi(x,y)=e^{-|n|\pi x/a}\sin(n\pi y/a)$}
For $a>0$, a <harmonic function> on $x\ge0$, $0\le y\le a$ with zero data on the horizontal sides, sine data $\sin(n\pi y/a)$ at the end, and decay at infinity is given by the displayed expression for every integer $n$. The second derivatives cancel since the decay exponent squared equals the transverse <wavenumber> squared. The absolute value is necessary for negative integers; $n=0$ gives the zero solution. The solution is bounded and decays uniformly in $y$.
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