240 DYNAMICAL THEORY OF SOUND 



we have just seen, is diminished by the lateral motion in the 

 ratio k' 2 a?. For a spherical surface with two nodal great circles 

 meeting at right angles the effect is much greater, the ratio 

 being -fakta*. And as we increase the number of compartments 

 into which the sphere is divided, the ratio, already very small, 

 decreases with enormous rapidity. 



For the sake of simplicity it has been assumed in the 

 preceding statements that the perimeter 2-n-a of the sphere is 

 small compared with X. The influence of lateral motion is 

 however not confined to this case, but will make itself felt 

 whenever the dimensions of the compartments referred to are 

 small compared with X. In the case of the oscillating sphere 

 there is no difficulty in working out the result without any 

 restriction to the value of ka, starting from the formula (13) of 

 77. 



Stokes has also investigated mathematically the case of a 

 cylinder vibrating at right angles to its length, where the same 

 cause is of course operative. In this way an estimate is 

 obtained of the direct effect of a vibrating string in generating 

 air-waves. This involves the ratio of the perimeter of the 

 cross-section of the string to the length of the air-waves, and 

 is in any practical case extraordinarily minute. As explained 

 in 24, almost the whole of the sound given out when a piano 

 string is struck comes from the sounding board. 



81. Scattering of Sound Waves by an Obstacle. 



We have next to consider the disturbance produced in a 

 train of sound waves by a rigid obstacle whose dimensions are 

 small compared with the wave-length. The scattered waves 

 which are sensible at a distance are due mainly to two causes. 

 If the obstacle were absent the space which it occupies would 

 be the seat of alternate condensations and rarefactions. The 

 effect of the obstacle in refusing to execute the corresponding 

 contractions and expansions of volume is, at a distance, 

 approximately the same as if in a medium otherwise at rest its 

 volume were to undergo a periodic change of just the opposite 

 character. The result is equivalent to a simple source. On 

 the disturbance thus produced there is superposed a second 



