Response of a Vibrating Plate in a Fluid 





Vk^ + (m2 - 1)^2 _ 2kMa - p' 



In the present case, p. is a small number (~ 0.0015) so that approxi- 

 mate values of the zeros may be found, expressed as a power series 

 in |Ji. To the second order these zeros are 



±2 2 ,2. 



where 



"^ ^ ^ " 2 7[k^ + (M^ - DP^KA^^ + 2A:¥ + P^ - yV 



± _ kM ± Vk^ + (M^- l)fi ^ 

 A r 



M^ - 1 



°'(^)^^)^~ '^(±),{±)2 



./ [. . (^^Wh) ] 



*\^),wf'^U,i.),^ ^' + (^' - 1 ) A^.) , (.,,- ^^^\.) , ,.),-p'] 



1/2 



where 



The last four zeros exist on both sheets. The location of the first 

 two zeros may be distinguished into three possibilities: when 

 \ < A" then a' and a^ are respectively on the unphysical and 

 physical sheets; as y is increased such that Ag < "y < Ag, then cj 

 moves off the unphysical sheet and crosses over to the physical 

 sheet and o^s remains unchanged; as y is further increased such 

 that Ag < "Y then a^ crosses over to the unphysical sheet and Wg 

 remains unchanged, A typical configuration for the poles and branch 

 points is shown in Fig. 6, 



493 



