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 solitai-y 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 wiU, satisfy the conditions are, 



' v=-b{e''y-e-"-l) cos^, 

 Q being = x—ct. 



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

 there does not appeal" to be any other foi-in capable of satisfying the necessary 

 conditions. 



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

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

 or increase the formulse, &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 /=0 and x= - j, u shall =0, and 

 v—Q: this gives 



and we get 



c=-6(e«2'_e-«2/) cosS. 



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



which we made in art. 8, that the form of the circular function is ^r •^~'' ''' 



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) (sinaxft—oosax(pl+G + c) 

 v=-(e''!>-e~"'y) {co6axft + &\na,z(pt+'H:) 

 where the circumstance that v=Q when ^=0 for aU values of ,r, gives the form of 



the exponentials; and the relation t~^ d~~^' S^^^^ *^^* °^ *^^ circular func- 

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



VOL. XIV. PART II. 4 Y 



