Equations 11-2 through 11-5 are linearized by assuming that 



= + u, , (11-7) 



P = Po + Pi . (II-8) 



T = T + T, , (II-9) 



and 



P(p,T) = Po(PoJo) + p i - (l'- 1 0) 



where the subscripts and 1 refer to the average and perturbation values of the parameters, respectively. 

 The first-order equations are 



3P 



L + Po(V" u,) = , (11-11] 



3 t 



Po "^t = " VPl + ^(VUi) _ n 5 Vx(Vxu 1 ) , (11-12) 



PoC v -IJ 1 + Po (Cp rj ° v) v - "i - kV2T i = , (H-13) 



since cp v is of second order [45], and 



V a Po i T V 8To K 1 ^ ■ 



(11-14) 



It has been assumed that T„ = T 1vv = and that Yf = Y w = 1 - Hence, in fish flesh, equations 11-11 through 

 11-14 are 



^f + MV'U,) = , (H-15) 



o l 



|^ = - VP, + $V(V • Q,) n s ,V x (Vxu,) , 



Pof 3t = -VK"i + WV'Q,) - n.s ( VX(VX Ul ) , ( ||. 16) 



p " " (-T&), P ' ' " M7) 



These equations are the same in water, with the exception that the viscosity is set to zero. The isothermal 

 compressibility, K T , is defined as 



= _L / 9Po \ (H-18) 



K _ 1 / dpp \ 

 Kt " "p7 l~3P7~> )t 



and the adiabatic sound velocity, c, as 



02 = -9k- (IM9) 



Since Y = Yw = 1. equation 11-17 is 



P, (II-20) 



c, 

 and equation 11-15 is 



p, < c (2 



4^ + PcC^V-u,) = . (M-21) 



According to Helmholtz's theorem [47], the 0, vector field can be represented in terms of a vector 

 potential A and a scalar potential Q such that 



