104 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



To obtain the pressure at the surface, we must write z instead of y. 

 48. Call P the pressure at the surface corresponding to a point, whose co- 

 ordinates are x and z ; then P + x + &c. is the pressure at another point of the 



surface where the co-ordinates are x + Sz, % + -? 8z+ &c. 



dx 



But the pressure at the surface is constant ; consequently - = 0. 



Let u,, v,, -jfr &c., represent the values of u, v, -? &c. at the surface, or when 



d I u t 



x is substituted for y. 



., ^P_ dz t du, du,dz dv, 



111 Gil ~ Q ~ """* I U . ~f~ U . 1- D '- 



dx dx \ dx dz dx dx 



dv, dz\ d d<p t d d(j), dz 

 'dz dx) dx' dt dz dt ' dx' 



TT / du, dv, d dd>\dz du. dv, d d d). . 



Hence (5'+ M /-r- + ,-r- + ;r- p \ ~r + u , T 2 + , + T- r-=0; 

 V dz ' dz dz dt/dx ' dx ' dx dx dt 



an equation evidently coinciding with equation (7) Art. 9. 



49. We have still another mode of obtaining the same result. 



In fact, let us consider the pressure as a function of t, the time. If then we 

 admit that the solution of the problem is that given in equations (1), (2), and (3), 



we must have -^-b(e Ky + e~"' y ) sin 6, and consequently 



U X 



K 





cos 6 + ((-) . . . (10.) 



But as < enters merely as a symbolical abbreviation of a certain quantity, 

 we can take it of such a value that F (t)=0. 



With this value of 0, it is evident that the value of P, given by equation (9), 

 will become P=f(t)gh+ a circular function ofxct. 



But the pressure at the surface must of necessity be constant ; we obtain, 



therefore, f(i)ffh + F ( sm (a; c^)) = a constant, for all values of x and t. Now 



I COS 



this cannot be the case unless/ (t) is a constant quantity; for by giving to x such 

 values that xct shall remain constant or differ by multiples of 2 IT, we may 

 render all the expression except f(t) constant for all values of t. Hence for such 

 values of x the expression assumes the form f(t)+ a const. = a const, for all values 

 of t. But this requires that/() be constant for all values of t. 



We have therefore P = C -g s - 1 ( M , 2 + O - -?-' . 



* at 



