242 DYNAMICAL THEORY OF SOUND 



distant point is, for similar forms, directly proportional to the 

 volume of the obstacle and inversely proportional to the 

 square of the wave-length. This latter particular might have 

 been foreseen without calculation. The ratio to the original 

 amplitude must necessarily vary directly as the volume Q, 

 and inversely as the distance r, and in order that the result 

 may come out a pure number we must divide by X 2 , since X 

 is the only other linear magnitude involved. The emission 

 of energy, being proportional to the square of the amplitude, 

 will therefore vary as X~ 4 . This law of the inverse fourth 

 power holds also in optics, and for a similar reason, with respect 

 to the scattering of light by particles whose dimensions are 

 small compared with the dimensions of the light-waves. The 

 blue of the sky, for instance, is attributed to the relative 

 preponderance of the shorter waves in the light scattered by 

 the molecules of air, and possibly by other particles ; in the 

 transmitted light, on the other hand, the longer waves pre- 

 dominate. The theory is due to Lord Rayleigh, who has also 

 pointed to an acoustic illustration in what are called " harmonic 

 echoes." If a composite musical note, consisting of a funda- 

 mental tone with its octave, &c., be sounded near a grove 

 of trees, for example, the ratio of the intensity of the octave to 

 that of the fundamental will in the scattered sound be 16 times 

 what it was in the original note. The scattered sound may 

 therefore appear to be raised in pitch by an octave. 



The actual scattering of energy is found by adding the 

 results due to the simple and the double source. This may be 

 proved by calculating the work done at the surface of a sphere 

 of large radius r. The terms due to the combined action of the 

 two sources contain a factor cos 9, and so disappear when 

 integrated over the surface. Hence, by 76 (15), (26), 



The energy-flux in the primary waves being %pk 2 cC 2 , by 

 76 (8), the ratio which the energy scattered per second bears 

 to this is 



(6) 



