Wu 



y = h(x,t) , (-1 <x < 1 , t >0) (1') 



h again being arbitrary and assumed to be always small. The motion starts at 

 t = from a uniform state; the free-stream velocity u(t) may depend on t. 

 Let u and v again denote, respectively, the x and y components of the per- 

 turbation velocity. We introduce the Prandtl acceleration potential 



0(x,y,t) = {p^-p)/p , (16) 



where p^ is the pressure at infinity, and p is the fluid density. An harmonic 

 function 0(x,y,t) conjugate to 4> may be defined by 4>^ = 4>y, 4>y = -0^, where 

 the subscripts x and y denote differentiations. By virtue of the incompressi- 

 bility and irrotationality, the complex acceleration potential f = + ii/; and 

 the complex velocity w = u - iv are analytic functions of the complex variable 

 z = x + iy for all real t. (We borrow the notation w and z for this different 

 purpose in this section.) By neglecting the nonlinear terms of all the small 

 quantities, Euler's equation of motion is linearized to give 



'' ■ " Bf 3w Bw " ■ ^ - .^„. 



-= -+ U(t) — • •.. ■ (17) 



dz at oz 



The linearized boundary conditions are: 



v(x,0^,t) = V(x,t) = hj + Uh^ , (-1 < x < 1) (18) 



Here, condition (19) is obtained by applying (18) to the imaginary part of (17); 

 condition (20) follows from that ^ is even, and hence is odd in y; (21) is the 

 Kutta condition for the flow at the trailing edge z = 1. Condition (22) for w 

 may also be specified as I z| — 00, |arg z| > 0, i.e., as z ^ 00 in the region ex- 

 cluding the trailing vortex sheet. 



Integration of (17) to obtain the boundary value of ^ on the plate can be 

 done by using the method of characteristics. However, with variable u(t), it 

 is more convenient to make use of the Laplace-transform method. We first 

 introduce the variable 



T(t) 



= J U(t) dt (23) 



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