## Abstract

In previous papers, solutions and numerical results were obtained for two-dimensional sound pulses incident on (i) an infinite uniform strip (Fox 1948) and (ii) an infinite uniform slit in a perfectly reflecting screen (Fox 1949). A general method given in the latter paper is now used to obtain an exact solution, corresponding to successive diffraction waves, for the case of a plane sharp-fronted pulse of constant unit pressure incident normally on a regular grating of perfectly reflecting strips. An asymptotic solution is also obtained for use in the later stages when numerical evaluation of the exact solution becomes unmanageable owing to the large number of diffraction waves involved. Numerical calculations have been carried out for $\beta $ = $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{4}$, where $\beta $ is the aperture area expressed as a fraction of the total grating area. It is found that the average pressures on the two faces of the grating are relatively independent of $\beta $ provided $\beta \geq \frac{1}{2}$, i.e. for gratings having 50% or more aperture area; for such gratings, the pressure is first equalized at time t = 2b/c, where 2b is the width of a strip and c is the velocity of sound, and subsequently the average pressure on the back of a strip overshoots the incident pressure by a maximum of about 12%, while the average pressure on the front drops to a minimum of 12% below the incident pressure. For gratings with smaller aperture areas in the range $\frac{1}{2}$ > $\beta \geq \frac{1}{3}$, the time to first equalization is only delayed slightly beyond time t = 2b/c, but the magnitude of the overshooting is appreciably reduced as the fractional aperture area decreases. For much smaller values of $\beta $ the overshooting becomes negligible and the equalization of pressure on the two faces of the grating becomes asymptotic rather than oscillatory in character. The transmitted pressures at some distance to the rear of the grating will form an effectively plane pulse which will exhibit a sharp front, of magnitude $\beta $, followed by a pressure rise which may overshoot the incident pressure of unity. This overshooting of transmitted pressure attains its greatest value, of magnitude 6%, when $\beta $ = $\frac{1}{2}$ and is small for both large apertures ($\beta \rightarrow $ 1) and small apertures ($\beta \rightarrow $ 0).. Correspondingly, the reflected pulse at some distance to the front of the grating will exhibit a sharp front of magnitude 1 - $\beta $ and may contain a negative phase of magnitude up to 6% of the incident unit pressure. The asymptotic solution becomes more accurate as $\beta $ decreases and should be useful for gratings with small fractional aperture area; especially as in these cases the exact solution would become extremely laborious to evaluate at a relatively early stage when pressures to the rear of the grating are still small.