302-303] MODULUS OF DECAY. 549 



the gravity of the fluid (which is proportional to gp) and the 

 viscosity (/u,), the influence of inertia being insensible. It appears 

 from (7) and (15) that m = k, nearly, so that the motion is ap 

 proximately irrotational. 



The type of motion corresponding to (23), on the other hand, 

 depends, as to its persistence, on the relation between the inertia 

 (p) and the viscosity (/a), the effect of gravity being unimportant. 

 It dies out very rapidly. 



The above investigation gives the most important of the normal modes, of 

 the prescribed wave-length, of which the system is capable. We know a priori 

 that there must be an infinity of others. These correspond to pure-imaginary 

 values of m, and are of a less persistent character. If in place of (6) we 

 assume 



((7cos m y + D sin m y} &quot; 

 with m 2 = W-ajv .................................. (iv), 



and carry out the investigation as before, we find 



(a 2 + 2i^ 2 a +gk + T &} A - i(gk + T k*} C- Zivkm aD = 01 



2ik*A + (tf-m *)C=0) ......... W 



Any real value of m is admissible, these equations determining the ratios 

 A : C : D ; and the corresponding value of a is 



a =- v (k* + m *) ................................ (vi). 



In any one of these modes the plane xy is divided horizontally and 

 vertically into a series of quasi-rectangular compartments, within each of 

 which the fluid circulates, gradually coming to rest as the original momentum 

 is spent against viscosity. 



By a proper synthesis of the various normal modes it must be possible to 

 represent the decay of any arbitrary initial disturbance. 



303. The equations (12) and (13) of the preceding Art. 

 enable us to examine a related question of some interest, viz. the 

 generation and maintenance of waves against viscosity, by suit 

 able forces applied to the surface. 



If the external forces p yyi p xy be given multiples of e ikx+at , 

 where k and a are prescribed, the equations in question deter 

 mine A and C, and thence, by (9), the value of rj. Thus we find 



p yy (a 2 + 2vk*QL + o- 2 ) A - i (o- 2 + Zvkma) C 



gk(A-iC) 



p xy = a. ^ 2ivk*A + (a + 2^ 2 ) 

 #/077 ^ J. iC 



