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We use the transfer matrix formulation of scattering theory in two-dimensionsto treat the scattering problem for a potential of the form$v(x,y)=\zeta\,\delta(ax+by)g(bx-ay)$ where $\zeta,a$, and $b$ are constants,$\delta(x)$ is the Dirac $\delta$ function, and $g$ is a real- orcomplex-valued function. We map this problem to that of$v(x,y)=\zeta\,\delta(x)g(y)$ and give its exact and analytic solution for thefollowing choices of $g(y)$: i) A linear combination of $\delta$-functions, inwhich case $v(x,y)$ is a finite linear array of two-dimensional$\delta$-functions; ii) A linear combination of $e^{i\alpha_n y}$ with$\alpha_n$ real; iii) A general periodic function that has the form of acomplex Fourier series. In particular we solve the scattering problem for apotential consisting of an infinite linear periodic array of two-dimensional$\delta$-functions. We also prove a general theorem that gives a sufficientcondition for different choices of $g(y)$ to produce the same scatteringamplitude within specific ranges of values of the wavelength $\lambda$. Forexample, we show that for arbitrary real and complex parameters, $a$ and$\mathfrak{z}$, the potentials $ \mathfrak{z}\sum_{n=-\infty}^\infty\delta(x)\delta(y-an)$ and$a^{-1}\mathfrak{z}\delta(x)[1+2\cos(2\pi y/a)]$ have the same scatteringamplitude for $a< \lambda\leq 2a$.

Exact Solution of the Two-Dimensional Scattering Problem for a Class of $\delta$-Function Potentials Supported on Subsets of a Line