Linearized Potential Flow Theory for ACVs in a Seaway 



It is clear from the above that the situation in respect of the 

 ACV leads to a very complicated non-linear problem. The position 

 of the free surface is not known a priori and the velocity potential 

 has therefore to be determined in a fluctuating and unknown domain. 

 The boundary conditions are of a mixed type. These relating to the 

 air cushion are stipulated in terms of pressure and those relating to 

 the side hulls in terms of velocity. Appropriate boundary conditions 

 at infinity may be imposed from physical considerations in order to 

 obtain a unique solution of the problem, but it is doubtful whether 

 this general non-linear problem will be mathematically tractable. It 

 will be observed that the motions of the ACV are not given in advance 

 (except in the case of forced oscillations of a pre -determined kind), 

 but are unknown functions of the time to be determined as part of the 

 solution. The motions vary the pressure distribution on the IFS and 

 therefore dictate the appropriate elevation slope of the water surface 

 which together with the pressure distribution determine the forces 

 and moments on the ACV due to the air cushion. Similarly, the mo- 

 tions vary the wetted surface of the side hulls and the pressure of 

 the water particles both of which determine the forces and moments 

 due to the side hulls. This highly non-linear problem has therefore 

 to be linearized in a suitable manner if a practical solution is to be 

 achieved and this we shall endeavour to do in the next section. 



III. THE LINEARIZED PROBLEM 



As the general non-linear problem has been shown to be highly 

 complicated, we shall not attempt the solution of the single boundary 

 value problem, but consider instead a sequence of linear boundary- 

 value problems which result when all the relevant physical variables 

 relating to the motions of the ACV and of the water are expanded in 

 terms of say, four small perturbation parameters, 8 , /3 , a and 



e , describing respectively the orders of magnitude of the width of 

 the side hulls, the cushion pressure, the amplitude of the unsteady 

 motions and the slope of the incident waves in the regular seaway. 

 The reasons for the choice of these parameters are explained in the 

 Introduction. 



The solution of the sequence of linear boundary -value pro- 

 blems is relatively simple and all relevant quantities determining 

 the motion are obtained in the form of a power series in the pertur- 

 bation parameters. An approximate solution of the general non-linear 

 problem can thus be obtained up to any desired degree of accuracy. 



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