SIMPLE-HARMONIC WAVES. DIFFRACTION 241 



wave-system, which is due to the immobility of the obstacle. 

 If the latter were freely movable, and if it had moreover the 

 same density as the surrounding air, it would swing to and fro 

 with the air-particles, and the second wave-system would be 

 absent. This system is accordingly the same as would be 

 produced if the obstacle were constrained to oscillate with 

 a motion exactly equal and opposite to that of the air in the 

 primary waves when undisturbed. The effect is, as we have 

 seen in 79, that of a double source. It might appear, at first 

 sight, that the former of these disturbing influences would be 

 much less important than the second, but in its effect at a 

 distance it becomes comparable, owing to the greater attenuation 

 by lateral motion of the waves proceeding from a double source. 

 If Q be the volume of the obstacle, the strength of the 

 simple source due to the first cause is 



where s, <f> refer to the primary waves. In the case of a system 

 of plane waves 



incident on a small obstacle at 0, this gives a velocity-potential 



As regards the second cause, we will assume for simplicity 

 that the obstacle has the degree of symmetry postulated in 

 79 with respect to the direction (Ox) of the vibration in the 

 air-waves. If the wave-system (2) were undisturbed, the 

 velocity of the air-particles at would be represented 

 symbolically by ikC, and the strength of the double source due 

 to the obstacle moving with this velocity reversed would be 

 ik(Q + Q') (7, in the notation of 79. The scattered waves at 

 a distance, due to the immobility, are therefore represented by 



(4) 



by 76 (24). The complete result is given by < = fa + < 2 . 

 It follows that the amplitude of the scattered waves at any 

 L. 16 



