308 



Lord Rayleigh on the Propagation of 



This equation expresses the dependence of u upon y. The 

 dependence on t and x is given by the factors already con- 



sidered, viz 



That (44) is complex indicates that the phase varies with y. 

 The realized expression will be 



-{ 



W-fWnh 



. e 



X cos J nt — m" 



# + 



(yi—y*)jn* 



2a W 2 



}, • (45) 



from which we infer (i.) that the intensity is least in the 

 middle where y = 0, and increases towards the walls until 

 the frictional layer is approached ; (ii.) that, as y 2 increases 

 from the centre, constancy of phase demands a diminishing x, 

 or, in other words, that the wave-surface is convex towards 

 + x and the wave divergent. 



We have now to trace the solution of (34) when the right- 

 hand member becomes large. Writing it in the form 



z tan z r /ri*t/j 



(46) 



we have to find such a complex value of z, say f +^, that 

 the function on the left is real. Initially, when y 1 is small, 



£ + i v =p (cos0 + isin0) = /3(cos67i c + isin67i°) ; 



and 



If we retain the angle 67^° and increase p, we find, calcu- 

 lating by means of (31), that i~% z . tan z becomes complex 

 with imaginary part positive. Thus if p— 1, we get 



i'-** . ians=-80 (cos 9° 54'+isin 9° 54'). 



This is a sign that must be reduced. If we take p = l, 

 0=60°, we find 



*— *s. tan - = -83 (cos 2° V-i sin 2° 7'). 



If /o = r5, while = 60% we get in detail 



z=%+i<n='75 + ix 1*299 ; 



sin2f=sin85° 57 / =«998, sinh2^ = 6'695, 



cos2f=cos85° 57 / = -071, cosh 2*7 = 6*769, 



