Singular Perturb at ion Problems in Ship Eydrodynamios 



theory in this way was rejected by many workers on the ground that 

 the resulting theory would be valid only for very short incident waves, 

 whereas the most important ship motions are known to occur when 

 the waves have wave lengths comparable to ship length. 



The choice was this: 1) Follow the reasonable usual assump- 

 tions of slender-body theory and obtain a rather useless theory . 

 2) Accept the formal assumption that frequency Is high and obtain 

 a much more interesting theory -- which turns out to be very similar 

 to the intuitive but quite successful "strip theory" of ship motions. 

 In what follows, I make the second choice. 



The reasons for the success of this choice have become clear 

 in the last few years. In one of the most Important practical prob- 

 lems, namely, the prediction of heave and pitch motions in head 

 seas, we can truly say that we are dealing with a high-frequency 

 phenonmenon. Because of the Doppler shift in apparent wave frequency, 

 fairly long waves are encountered at rather high frequencies; the 

 waves are long enough to cause large excitation forces, and the 

 frequencies are high enough to cause resonance effects. At zero 

 speed, on the other hand, incident waves with frequency near the 

 resonance frequencies of a ship are likely to be much shorter in 

 length than the ship, and so their net excitation effect is much re- 

 duced through interference. For typical ships on the ocean, most of 

 the heave and pitch motion at zero speed is caused by waves with 

 length comparable to ship length, and so the frequencies of such 

 motion are well below the resonance frequencies. Thus, at zero 

 speed, prediction of ship motions can be treated largely on a quasi- 

 static basis; the system response is "spring-controlled" rather than 

 "mass controlled," 



The problem Is very much like the simple spring-mass 

 problem discussed in Section 1,2. If the mass of a spring-mass 

 system is very small, we can ignore inertia effects at low frequency. 

 Thus, if the system is described by the differential equation: 



mV + ky = Fe'*^*, 

 the exact and approximate solutions , given by: 



Newman and Tuck showed, for example, that the lowest-order 

 perturbation potential resulting from ship oscillations satisfies a 

 rigid- wall free- surface condition, even with forward speed in- 

 cluded. Maruo [ 1967] has the same result for the forced-oscilla- 

 tion problem. Newman and Tuck performed calculations with a 

 second- order theory for the zero- speed case and found practically 

 no change in their predictions due to second-order effects. They 

 did not make such calculations in the forward- speed problem. 



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