Bow Waves and other Non-Linear Ship Wave Problems 



free-surface becomes unstable when the normal acceleration 

 vanishes. In our case this occurs when the centrifugal effect related 

 to the free-surface curvature offsets the gravity acceleration. Since 

 we expect the free-surface to become steep as Fr-j- increases, 

 there must be a critical Fr^ characterizing instability. 



The gravity free -surface problem is, however, nonlinear. 

 To linearize it we consider a small Fr_ perturbation expansion, 

 i.e. and expansion for a state near rest. Referring the variables 

 to T' (Fig. 2) and (gT')''^^ and expanding as follows: 



F(Z) = ^ + i^ = Fr^F, (Z) + Fr^VglZ) + . . . (1) 



W(Z) = U - iV = Fr^VV, (Z) + Fr^^W^{Z) + . . . (2) 



N(X) = Fr^^N,(X) + Fr/NgiX) + ... (3) 



we obtain from the exact free surface and body boundary conditions 

 the following equations: 



at first order (Fig. 2b) 



^, = (ASBA) (4) 



W, = 1 (X ^ - oo) (5) 



i.e. a flow beneath a rigid wall replacing the free-surface at its 

 unperturbed elevation. In addition 



N =1 (1 - U,^) (X < 0, Y = 0) 



second order 



>i'2=-U,N, (AS, X<0, Y = 0) (7) 



*2 = (SBA, X > 0, Y = H(x)) (8) 



i.e. a flow generated by a source distribution along the degenerated 



free-surface, and 



N2= - U,U2 (X< 0, Y = 0) (9) 



It is easy to ascertain that ^2 ^s zero at infinity such that 



6il 



