282 Lord Rayleigh's Acoustical Observations. 



of this beautiful phenomenon ; but my efforts were without 

 success until a few months since. The difficulties to be over- 

 come are entirely different in optics and in acoustics, on 

 account of the immense disproportion of wave-lengths. In 

 the former case the disk must be small and accurately shaped, 

 and the source of light must be of very small angular magni- 

 tude — in practice, an image of the sun formed by a distant lens 

 of short focus. In the latter case the difficulty is to arrange 

 the experiment on a scale that shall be adequate in comparison 

 with the wave-length of the sound. 



The best way of considering the subject theoretically is with 

 the use of Huyghens's zones. The plane of the disk is divided 

 into zones by its intersection with spheres whose centres are 

 at the point under consideration, and whose radii form an 

 arithmetical progression with common difference J\. The 

 vibrations due to these zones are at first nearly equal, but gra- 

 dually diminish to nothing, unless the outer boundary of the 

 aperture is circular ; and thus the aggregate effect is repre- 

 sented by a series of which the terms are of opposite sign and 

 of slowly diminishing magnitude. Now the sum of such a 

 series is equal approximately to half its first term ; so that the 

 whole effect of the aperture outside the disk is independent 

 of the disk's diameter — that is to say, is the same as if no 

 obstacle at all were present. 



This way of regarding the matter shows at once what 

 degree of accuracy is required in the figure of the circumfe- 

 rence, which must not sensibly encroach upon the first exterior 

 zone. If x be the radius of a circle in the plane of the disk, 

 b the distance of the point under consideration, and r 2 = x" + b 2 , 

 dx = rdr \ x; so that if dr = ^\, dx=r\ | 2x. If, therefore, 

 x be the radius of the disk, the radial curve should be a small 

 fraction of rX | 2x. 



In like manner, we may form an estimate of the size of 

 the bright spot, a subject which has been treated analytically 

 by Airy*. If the disk be moved laterally through the width 

 of one zone, it is clear that the effect at the old point will be 

 materially changed. Hence the diameter of the bright spot 

 is comparable with r\ \ x, and its apparent magnitude as seen 

 from the disk is comparable with X | x. For the full success 

 of the experiment, the apparent magnitude of the luminous 

 source should be of the same order. 



When we pass to the analogous experiment in acoustics, it 



is of course impossible to retain any approximation to optical 



conditions. Instead of a ratio of X : x, equal to, say, yq o o o> 



we must be satisfied with some such value as ^ l -. In order to 



* Phil, Mag. May 1841. . 



