Tuak and Taylor 



i.e. c()(x,y,-h) satisfies the Helmholtz equation 



V^4) + k^cf) = (5.5) 



in the (x,y) plane. 



The Helmholtz equation is of course simply the ''reduced" 

 wave equation, and so applies to any scalar wave problem in two 

 dimensions, for sinusoidal time dependence. In particular, it 

 describes linear acoustics in two dinnensions (e.g. Morse [ 1948]), 

 and many results obtained in solving acoustic problems may be 

 utilized. 



For example, we may treat immediately the scattering of a 

 thin cylindrical ship, as in Michell's model of Section 3, which 

 extends from top to bottom of the water. An important difference 

 from the theory of Section 3 is that, even in the limit as the thick- 

 ness tends to zero, the thin "ship" is capable of scattering beam 

 waves. Thus, to leading order, the problem is independent of 

 thickness, and reduces to acoustic scattering by a ribbon or strip 

 of zero thickness placed broadside on to the waves, with a "hard" 

 boundary condition 



-^ = - -^ \i = constant on y = 0^, |x|<i. (5.6) 



The exact solution can be written down as a series of Mathieu 

 functions (Morse a'nd Rubenstein [ 1938]). Results for the scattering 

 cross section, the far-field polar diagram and the force on the strip 

 can be computed from this series, but only with some difficulty, 

 especially for high frequency. Alternatively, integral equation for- 

 mulations of the problem can lead to useful high and low frequency 

 asymptotic solutions (Honl, Maue and Westphal [ 196l] ) or even to 

 efficient numerical solutions (Taylor [ 1971]). Such numerical 

 results are included with the discussion of the general case in 

 Section 7. 



Once again this idealized ship is deficient from the practical 

 point of view. In particular, it allows no account to be taken of flow 

 beneath the keel of the ship. In any situation of real interest, wave 

 energy is not only scattered, diffracting around the ends of the ship, 

 but also transmitted underneath the ship if there is any reasonable 

 amount of clearance. The most interesting situation is that which 

 applies when the amounts of disturbance scattered and transmitted 

 are of the same order of magnitude; we shall see later that this is 

 true for draft /water depth ratios in the range . 5 to 0,95. 



We shall retain the approximation that the ship is thin;, and 

 hence slender, since it must have small draft. However, the 



648 



