The diffraction of a plane wave through a grating

The problem of diffraction and scattering of a normally incident plane wave of sound by an infinite plane grating consisting of infinitely thin, coplanar, equally spaced (b apart) strips with parallel edges is solved. The potentials on the two sides of the screen are written as Fourier expansions in terms of the velocity in the aperture, and an integral equation for this velocity is determined. An impedance parameter $Z$ whose real part is the transmission coefficient, is defined, and it is shown that the real and imaginary parts of the reciprocal of this parameter may both be specified by variational expressions, which are absolute minima for the solution to the aforementioned integral equation. An alternative formulation, in terms of the pressure discontinuity across the screen, is given, leading to an integral equation and to variational expressions for the real and imaginary parts of ${\left ( {1 - Z} \right )^{ - 1}}$ A solution to the integral equation is given which reduces the problem to the solution of an infinite number of simultaneous equations. It is shown that solving only one of these equations gives a solution which is essentially a solution to Laplace’s equation, while if $N$ equations are solved the terms neglected are of the order ${N^{ - 1}}\left [ {{{\left ( {1 - 4{b^2}/{N^2}{\lambda ^2}} \right )}^{ - 1/2}} - 1} \right ]$ or less, where $\lambda$ is the wave length. The solution is extended to the case of a vertically polarized electromagnetic incident wave by direct analogy and to the case of a horizontally polarized electromagnetic wave by a transformation which is a special case of Babinet’s principle. The integral equations for an aperture of finite thickness are set up, and an approximate solution including only first order terms in thickness/wavelength is given. Results are given in the form of curves for the transmission coefficient vs. aperture opening for several ratios of grating spacing to wave length less than unity. For the special case of a half open grating the transmission coefficient is plotted out to that value of spacing/wavelength at which the results agree with Kirchhoff’s theory.