514 PROFESSOR CHRYSTAL 



15. Example 2. — Consider the effect of a steady (i.e. non -progressive) harmonic 

 disturbance of pressure on the y-nodal seiche during a single period of that seiche. 

 If a be the range of the pressure disturbance, we may put in (46) 



f(w, t) -- U sin (mi - $)/{«■) ..... (48), 



where 2*7r/m is the period of the pressure disturbance, and (p/m its phase. We then 

 get, putting t„ = for convenience, 



'+i 

 dk v = l(2v+ l)a I dwQ,' v (w) I dt ?i v sin n v t sin (mt- 4>)f(w) . . . (49), 



P = ±(2„ + l)a \dwQ\(w)f(w), .... (50), 



where T = 27r/n„ 

 Hence, if 



a quantity independent of t or T, we have 



3/.V/P = U cos <£ - V sin <£ ; 



where 



| dt n v sin n v t sin mt 



i 



V = I dt n v sin n v t cos mt 



(51). 



i 



If 6 = m/n v , we find 

 AVhence 



U= -2 sin ttO cos tt0/(1 - 6 2 ) , 

 V = 2 sin 7T0 sin 7r0/(l - s ) 



(52). 



2 sin irt 



dk = -V{^™) C os(Tr6-4>) . (53). 



So far as <£ is concerned, the numerical value of dk v is a maximum when <p = ird, or 

 <£ = 7r(l-0). 



The maximum disturbance possible is therefore produced when 6 is so chosen that 

 f(0) = 2 sin tt6>/( 1 — O 2 ) is a maximum, i.e. when 0=-838 approximately, which gives 

 f(0) = 3*273. It will be seen, however, from the graph of/(0) (fig. 24) that this function 

 varies very slowly indeed near its maximum. We have in fact f('7) = 3*173, 

 f(l) = 7r= 3*142. Hence, between = *7 and 0=1 the divergence from the maximum 

 value of/(0) is only about 4 per cent.* 



If we take the special case of the uninodal seiche, and suppose f{w) = «•*, we find 

 P = ^a ; and for the maximum possible value of 3/q 



3^ = 1-180, 



* This result may seem at first sight to be in contradiction with the ordinary theory of forced vibration ; but it 

 is not really so. In the ordinary theory we consider a jjractically infinite number of oscillations, and take into 

 account the viscosity of the system. In the present case we consider only one oscillation, and neglect the viscosity. 

 It is obvious that this latter supposition is nearer the truth in the case of lake oscillations, because the disturbance! 

 of pressure are always transient, and usually periodic only for a very few oscillations. 





