GENERAL THEORY OF SOUND WAVES 221 



There is therefore total reflection, at the stratum to which the 

 accents refer, of all wave-fronts whose initial inclination (0) to 

 the vertical falls short of a certain limit. 

 Along any one ray we have 



sec < H = const., (5) 



c 



or, by (2), sec i/r + sec 2 ^ = const., ....(6) 



provided ^ be not too great. If we differentiate this with 

 respect to the arc s, and put d-^/ds = 1/R, dy/ds = sin i|r, we 



find 



1 /, 20" .\ IdU /lyx 



r . 1 + -- sec \Ir =__ (7) 



E\ c r J c dy 



The ray is therefore curved downwards or upwards, according 

 as dU/dy is positive or negative, i.e. according as the ray is 

 travelling with or against the wind. If the gradient dU/dy 

 be uniform, the rays have 



all the same uniform curva- ri 



ture, approximately, owing 



to the smallness of U/c, 



unless indeed the inclination Fig 71 



vjr becomes considerable. It 



will be noticed that in this problem the path of a ray is 



not reversible. 



This is a convenient place for a reference to what is 

 known as " Doppler's principle " *. Suppose, for instance, 

 that a periodic source of sound is approaching a stationary 

 observer. The number of maxima of (say) the condensation s 

 which strike the ear of the latter in a second is increased, and 

 the pitch is therefore raised. The diminution in the period is 

 to the period when the source is at rest in the ratio of the 

 velocity of approach to the velocity of sound. When the 

 source recedes from the observer, this ratio is negative, and 

 the pitch is lowered. When the motion of the source is 

 oblique to the rays by which the sound is heard, the com- 



* Christian Doppler (1803 54), an Austrian mathematician, professor of 

 physics at Vienna 1851. 



