OF FLUIDS ON THE MOTION OF PENDULUMS. [21] 



The first of these equations gives p = a constant, for it evidently cannot be a function of t, 

 since the effect of the motion vanishes at an infinite distance from the plane ; and if we include 

 this constant in II, we shall have p = 0. Let V be the velocity of the plane itself, and suppose 



V=csmnt (9) 



Putting in the second of equations (8) 



v = Xi sin nt + X s cos nt, (10) 



we get 



_ ,*X t ,*X X ^d l X 2 



da? dor n dx 



The last of these equations gives 



-v^r* . / n / n Vt~- x ,~ .In / n „ 



X^e 2 * (Asm \/ —,x + Bcos\/ —x)+e 2fi (C sin V — ;«> + D cos V — >*)• 



2/x 2(oi 2(j. 2/x 



Since X 2 must not become infinite when x = oo , we must have C = 0, D = 0. Obtaining Jf, 

 from the first of equations (11), and substituting in (10), we get 



__ */ -p / Yt /ft 



v= e 2/i ' {-Asin(nt-\/ — ,») + J? cos (nt - \/ — ,w)\. 



Now by the equations of conditions assumed in Art. S, we must have v = V when w = 0, 

 whence 



"V — ,i / n 



2 " sin(w#-\/ — ,w) . (12) 



v = ce 



To find the normal and tangential components of the pressure of the fluid on the plane, we 

 must substitute the above value of v in the formulae (4), (5), and after differentiation put 

 x = 0. P lf T 3 will then be the components of the pressure of the solid on the fluid, and 

 therefore - P u — T 3 , those of the pressure of the fluid on the solid. We get 



^1 = 0, T 3 



. /«»* , . x Am' /„ 1 dV\ 



VV -Y^nt+cosnt) = pV-y[ r+- -ft)- • 03) 



The force expressed by the first of these terms tends to diminish the amplitude of the 

 oscillations of the plane. The force expressed by the second has the same effect as increasing 

 the inertia of the plane. 



8. The equation (12) shews that a given phase of vibration is propagated from the plane 

 into the fluid with a velocity -y/(8ja'»), while the amplitude of oscillation decreases in geometric 

 progression as the distance from the plane increases in arithmetic. If we suppose the time of 

 oscillation from rest to rest to be one second, n = -k ; and if we suppose -y/m'= ill6 inch, which, 

 as will presently be seen, is about its value in the case of air, we get for the velocity of propa- 

 gation .2908 inch per second nearly. If we enquire the distance from the plane at which the 



amplitude of oscillation is reduced to one half, we have only to put \J — t x = log, 2, which 



2fx 



gives, on the same suppositions as before respecting numerical values, x = .06415 inch nearly. 



