160 



Dr. E. H. Barton on the 



expression giving 



the 



(2) 



(3) 



supplementing by another 



direction of the ray. 



Total reflexion cannot occur if the wave-front is initially 



horizontal. 



In a region where the horizontal wind increases 



uniformly as we ascend, the rays instead of forming a 

 catenary describe a more complicated curve which,, 

 however, reduces to a parabola in the special case of 

 rays whose wave-fronts are horizontal. 

 Relation between Direction of Propagation and Wave- 

 Front. — Let a region be imagined in all parts of which the 

 wind is horizontal of speed u; let a plane wave-front be 

 inclined to the horizontal, and let the direction of propaga- 

 tion of this wave of sound be inclined (j> to the vertical. It is- 

 required to find <p in terms of 0, u, and v, the velocity of sound. 

 Let AH in fig. 1 represent the wave-front at a certain 

 instant, and let CD represent it after the lapse of a short time 

 Fig. 1. — Drift of Sound-Kays in a Wind. 

 V 



u- 



denoted by t. Then the Huyghens' wavelet whose origin 

 is A may be conceived as radiating from A in every direction 

 at speed v compounded with the horizontal velocity?/. Hence 

 the wavelet from A after any time t is a circle whose radius 

 is vt but whose centre is transferred a distance ut horizontally 

 in the direction of the wind. Thus, lay off horizontally 

 KA! = ut, then from A / describe with radius vt the arc EOF, 

 and we have the wavelet required. Similarly we get the 

 wavelets originating at B and at any other points along AB. 

 The new wave-front is the envelope CD of these wavelets, and 

 is obviously parallel to AB. Also the direction of propaga- 

 tion of the wave is AC, making the angle VAC = <£ with the 

 vertical ; whereas A'C is perpendicular to the wave-front 



