Tuak and Taylor 



ct)(x,y,z) = cf5(x,y,-h) - i(z +hf V>(x,y , -h) + ... (3.7) 



where V^ = [d'^/bx^) + (a^/8y^) . On substitution in (3,4) we obtain 

 immediately to leading order in h the equation 



- ghV^(j)(x,y,-h) + U2())jx,y,-h) = 0, 



2 2 



[(l-F^)-|^+-^]4,(x,y,-h) = 0, (3.8) 



where F = U/Vgh . 



Equation (3.8) is formally identical to the equation describing 

 linearized aerodynamics in a two-dimensional flow of a compressible 

 fluid, with the Froude number F playing the role of the Mach number 

 (see e.g. Sedov [ 1965]). Indeed, the problem of solving (3,8) subject 

 to (3.5) is identical to that for subsonic (F < 1) or supersonic (F > 1) 

 flow over a non-lifting wing of thickness b(x) , and we may use 

 directly the results obtained in aerodynamics. Of course Michell was 

 not so fortunate, and we should say that aerodynamicists could have 

 used Michell's results, the first solution of any boundary-value prob- 

 lem for a non-trivial general boundary. 



The character of Eq. (3.8) is different according as F < 1 

 when it is elliptic and F > 1 when it is hyperbolic, and different 

 mathematical properties and solution techniques apply in these two 

 cases. Here we quote only the final result for the hydrodynamic part 

 of the pressure distribution over the body surface, namely 



P = 





Z-n-J 



00 



b'(e) de 



00 



r2 



F < 1 



(3.9) 



p^ b'(x), F> 1, 



ZirVF - 1 



the bar denoting a Cauchy principal value. Note that the pressure 

 given by (3.9) is a function of x only. The z-dependence has been 

 neglected as part of the shallow-water approximation and there is no 

 y-dependence because of the thin-ship approximation. The complete 

 pressure distribution is obtained by adding to (3.9) the hydrostatic 

 pressure. 



The only possible force on this cylindrical body is in the x- 

 direction, and there is no net moment. Michell found by integration 



636 



