Sound Waves in the Atmosphere. 105 



(e. g. when successive approximation can be resorted to, or 

 when the type of wind is given), but it does not seem worth 

 while reproducing them here. 



It should be noticed that (25) displays the correction as the 

 sum of three terms : the first represents the effect of the wind 

 in refracting the sound, the second its effect in convecting or 

 bodily displacing the sound, and the third the effect of tem- 

 perature variation. The last term can be exhibited in a 

 compact form, due to Hill, as follows : If T is the tempera- 

 ture, then a = a (T/T )2 ; assuming a linear temperature lapse, 

 we can write 



T = T (1-2.VC), a=a„(l-*/C), . . (26) 



approximately, the value of the constant C for the usual 

 gradient of 1° F. in 300 feet being 300,000 feet. In this 



case 



^cot0 o =ijcot0 o =!g, . . . (27) 



approximately. Thus the correction due to temperature varia- 

 tion is proportional to the horizontal distance of the source. 

 For a linear temperature fall the sound-rays are arcs of 

 circles ; for from (26) the velocity of sound at any point is 

 proportional to the depth of the point below the horizontal 

 plane £ = C, and hence any plane wave moves as though 

 rotating as a rigid plane * about the line of its intersection 

 with the plane z = C ; its angular velocity in any position is 

 (a cos#)/C, 6 being its inclination to the vertical. The 

 exact expression for the correction in this case is found to be 



§ 6. Total reflexion and the range of audibility. 



It may occur that for certain values of z the general 

 refraction equation 



a sec 6 — a sec O = W — W 



gives values of sec 6 numerically less than unity. In this 

 •case, as is well known, the wave-front considered is swung 

 round so as to be unable to penetrate the neighbouring 



* This remark is also due to Hill. 



