PROFESSOR KELLAND ON THE THEORY OF WAVES. 533 



SECTION III. — SOLITARY WAVE MOTION. 



32. The subject for investigation in the ensuing section, is the transmission 

 of a sohtary wave. Waves of this kind are so generated, that, throughout the 

 whole length of the wave, the velocity parallel to x is positive. 



As to the vertical motion, it does not appear probable that any difference 

 would be caused in it by the horizontal motion ; we may then conceive all the 

 circumstances to remain the same as before, with the exception that the whole 

 wave has a transmission parallel to the axis of x. By art. 8, it appears that the 

 only functions which will satisfy the conditions are, 



u = b{e''-y-^e-^y) sin + 03^, 



v=-b{e''y-e-''y) GosO, 

 6 being = x—ct. 



We shall hereafter discuss the variation which these formulae admit of, but 

 there does not appear to be any other form capable of satisfying the necessary 

 conditions. 



We may remark that p is not now, as in the former case, a complete differen- 

 tial, except approximately ; it becomes, then, a question in what manner to vary 

 or increase the formulae, &c., to render it so accurately. 



This discussion will form a distinct branch of inquiry, into which I forbear 

 to enter at present. 



Let us return to our equations. 



The condition to be satisfied is, that, when t=0 and x=-t, u shall =0, and 

 t)=0: this gives 



and we get 



M = 6(e«2' + e-«2')(l + sin0) 



v=-b{e''y-e-''y)Go^e. 



33. Lest it should be thought that, in the case before ua, the assumption 

 which we made in art. 8, that the form of the circular function is *" —- x-ct, 



cos A 



is inapplicable here, I offer the following demonstration of the point. 

 Take the most general form involving only one circular function : 



u — {e'^y + e~'^y) (smaxft-GOSaa;(pt+G + c) 

 r=—(e''y — e~''-y) (cos ax f ( + sin ux(p ( + K) 



where the circumstance that v=0 when 3^=0 for all values of «, gives the form of 

 the exponentials; and the relation ^ + j~=^' gives that of the circular func- 

 tions. C and H are supposed to be functions of t. 



VOL. XIV. PART II. 4 Y 



