Harmonic sine mode in a half-strip ($\varphi (x,y)=e^{-|n|\pi x/a}\sin (n\pi y/a)$)

For $a>0$, a harmonic function on $x\ge 0$, $0\le y\le a$ 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 $y$.