Sound Waves in the Atmosphere. 101 



which integrates in the form 



a/l-a Jl =u -u, 

 or, putting l = cos6, l = cosd Q , 



asec0 — a o sec O = u — u . 



This is the general expression of the law of refraction when 

 both the refractive index and the velocity of the medium are 

 variable; the particular cases, u — u — 0, and a=-a , are, of 

 course, the ordinary forms. It must be emphasized that 

 6 specifies the direction of the normal, not the direction of 

 propagation. It may be remarked here that the path of a 

 sound-ray in a moving medium is not in general reversible. 



§ 5. Application to a point source in the atmosphere. 



In the atmosphere vertical air-currents can usually be 

 neglected, but the components of wind-velocity change with 

 the height, frequently rapidly and irregularly ; the velocity 

 of: sound, being dependent on the temperature, is also a 

 function of the height, usually decreasing upwards. The 

 atmosphere can therefore be considered as a s( ratified 

 medium, as defined in the last section, and the equations 

 there found are immediately applicable when what we may 

 call the "wind-structure" and " temperature-structure" of 

 the atmosphere are known. The effects of atmospheric wind 

 and wind-variations, and temperature variations in dis- 

 placing and refracting sound-waves were first noticed by 

 Stokes * and Osborne Reynolds f ; the latter accounted (by 

 consideration of refraction) for the fact that a sound is heard 

 better to leeward of its source than to windward. Examples 

 of the calculation of paths of rays have been given by 

 Lord Rayleigh J and Barton §, The question of the varia- 

 tion of the direction of the normal to a sound-wave along a 

 ray is, however, separate from the calculation of the ray 

 itself. 



In this section we shall be concerned with the practical 

 problem of an observer on the ground who, making observa- 

 tions on the sound received from an unseen aerial source, 

 wishes to correct the direction so obtained for the effects of 

 transmission through the atmosphere. We suppose that the 

 instrument he uses determines the direction of the normal to 

 the incident wave-front. 



* Brit. Assoc. Report, 1857. + Proc. Roy. Soe. 1874. 



X ' Treatise on Sound,' vol. ii. pp. 132-4. 



§ Phil. Mag. Jan. 1901, p. 159; 'Text-Book on Sound.' pp. 99 L06. 



