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critical speed. Only recently can one observe a worldwide interest in the commercial exploitation of supercritical speeds for ships, especially ferries.

It is probably less well-known that the favorable effect of wave interference between twin hulls of a catamaran in shallow water, which diminishes total wave resistance, is more significant in the supercritical speed range than in the subcritical speed range (and, of course, in deep water). This phenomenon has been reported earlier by Eggers (1955, Figs.2 and 18) and Kirsch (1966, Figs.2–9). It was recently rediscovered by Chen & Sharma (1994a, Fig.5a) on the basis of a nonlinear theory and confirmed by model experiment. Even more exciting is the fact that when we tried to seek a fuller theoretical understanding of this interesting phenomenon, see Chen & Sharma (1995), we found that the wave resistance of a single-hull ship in a channel of suitable width can be made to vanish totally within the framework of a linear shallow-water wave approximation and, furthermore, also in a more accurate nonlinear theory, namely the standard KP equation. To put it simply in one sentence, the mechanism is that the bow wave after reflection from the channel sidewall hits the after-body and counteracts the stern wave so that the resultant wave in the ship wake disappears totally, if the hull geometry is adapted to ship speed and channel width according to certain rules. To our knowledge, this is a new type of ship and channel configuration marked by zero wave resistance. By analogy to electrical conductors, we have proposed the name “shallow channel superconductivity” for this phenomenon.

We would like to insert in the following three paragraphs a short review of shallow water wave theory as applied to ship hydrodynamics. The first relevant paper was by Tuck (1966) who developed a strictly linear technique of matched asymptotic expansions for a slender ship in shallow water. Lea & Feldman (1972) partly took account of nonlinearity and used an established transonic-flow numerical method for computing the transcritical motion of ships. Later on, Mei (1976) extended this work to include the dispersion effect in the near-critical speed range while still dealing with the steady problem. With due consideration to remarkable early towing tank experiment reports, e.g. Thews & Landweber (1935, 1936), Helm (1940), Kinoshita (1946), Graff (1962), and Graff, Kracht & Weinblum (1964), it may be stated that the recently revived interest in upstream solitons generated by a ship moving steadily at near-critical speeds stems from the experimental and numerical works by Huang et al. (1982) and Wu & Wu (1982). Since then various follow-up investigations have been done, e.g. Ertekin, Webster & Wehausen (1986) and Katsis & Akylas (1987) solved the 3-D free-surface pressure disturbance problem. Mei (1986), using matched asymptotic expansions, derived an inhomogeneous Korteweg-de Vries (KdV) equation for a slender ship moving at near-critical speed in shallow water and theoretically demonstrated solitons propagating upstream. Mei & Choi (1987) further developed this theory to calculate hydrodynamic forces on the ship but only crude agreement with experiments was obtained because this theory cannot predict the two-dimensional waves around a real ship and in its wake. So Choi & Mei (1989) improved their theory by using a Kadomtsev-Petviashvili (KP) equation in the far field to take account of the 2-D effect. More numerical results were reported in Choi, Bai, Kim & Cho (1990) with another finite element method.

Chen & Sharma (1992) pursued this method further with the aim of practical application. The slender body theory in the near-field was refined by taking account of local wave elevation, longitudinal disturbance-flow velocity and ship squat. Moreover, it was extended to the more general case of asymmetric motion. The KP equation in the far field was solved numerically by an efficient finite difference method, namely, a fractional step algorithm with Crank-Nicolson-like schemes in each half step. Very good agreement with towing tank experiments was achieved in wave resistance, sinkage, and trim for several ship models. Furthermore, Chen & Sharma (1994a) derived a KP equation from the Boussinesq equations by keeping a higher-order nonlinearity in the lateral direction, thus making it valid for a wider speed range. More numerical results from this modified KP equation for a ship in a wider speed range were reported in Chen & Sharma (1994b) and compared with old model experiments of Graff et al. (1964) on a Taylor Standard Series hull as well as with new tests on a Series 60 hull. It was concluded that the method based on nonlinear shallow-water wave theory holds enough precision for the practical ship problem if higher-order effects in the near field are taken into account as indicated above.

Extension of the theory to the case of asymmetric ship motion also proved useful. The corresponding computer codes enabled us to treat a ship moving parallel to the channel axis off-center and/or at a drift angle. With vertical sidewalk this configuration is mathematically equivalent to a catamaran moving in a channel of twice the width. Numerical results were reported in Chen & Sharma (1994a). The calculated wave resistance, lateral force, yaw moment, sinkage and trim agreed very well with towing tank measurements on a Series 60 model hull both in off-center and in oblique motion, except for the lateral force and yaw moment in oblique motion at higher speeds. Especially the significant wave resistance reduction in the supercritical off-center case led us on to the discovery of superconductive ship-channel geometries marked by no trailing waves and



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