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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 148 proach. If we are to be certain of the accuracy of these computations, experimental validation at the higher order will be necessary, though it has never been attempted. Theoretical approach we use here for the 2.5D theory is not a fully nonlinear time domain computation but a perturbation analysis up to the second order in the frequency domain. A reason is to make the second order effect more distinct; direct comparison of the fully nonlinear computational results in time domain and the measured wave elevation might not necessarily provide a quantitative information on the higher order effect. We have to recognize that there is a demerit in the perturbation analysis that it does not predict really large nonlinear effect but moderate nonlinearity. and ζ(x, y, z, t) represent respectively the velocity potential of the unsteady flow and the unsteady Let wave elevation generated by the ship motion. Conditions of them to be satisfied at the free surface z=ζ are (7) (8) Suppose the expansions and ζ2 are of higher order than the first terms and ζ1 with respect to the and assume that the second terms magnitude δ of ship motions. It is because we simply assume that the steady flow around the ship is uniform flow U into the x direction. Then one can find the first and second order free surface conditions by substituting the expansions into the free surface conditions (7) and (8) to separate the first and the second order terms from each others. and ζ1, and (11) and (12) are the second order condition for Equations (9) and (10) are the first order condition for and ζ2. (9) (10) (11) (12) All the conditions are to be satisfied on z=0 We introduce here a crucial assumption so that we may proceed to the next stage of analysis: the magnitude of the slenderness parameters ε is independent of that of δ i.e. we retain δ 2 terms that are of the lowest order with respect to ε despite that practically ε>δ . We may assume for any flow quantity f the authoritative version for attribution. Yet we retain U∂f/∂x in the first and second order free surface conditions above because of high speed U of the ship. and ζ2 From now on our analysis is concerned with the oscillatory part of the flow i.e. we understand hereafter that represent the oscillatory part of the second order. Our analysis is in the frequency domain and everything of the first order is oscillating at the frequency ω . Consequently it is straightforward to show (13)

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 156 DISCUSSION X.-B.Chen Bureau Veritas, France I am interested in the steady part of ship waves you measured. From the second-order theory, there is a contribution of second-order to the steady part, which maybe the difference of wave pattern between that in calm water and the total steady part in the unsteady measurement. Is it true? If so, have you any information (results) on the steady part of measurement in unsteady tests? AUTHOR'S REPLY We are too interested in the second order interaction of the steady and unsteady ship waves. First of all we must know that the effect of the unsteady waves upon the steady ones will be observable but not vice versa. Reason is that the unsteady waves not under the influence of the steady waves is not realistic for the case of our concern i.e. when a ship has forward speed.. While we have not reported the result here but certainly we measured the steady ship wave without the ship motion or incident waves, and compared it with the steady wave component of the wave motion when a ship is forced to oscillate or in the incident waves. What we have found is that the steady wave is hardly affected by the unsteady wave at least for the magnitude of the oscillation we adopted in our measurement In other words the steady component of the wave motion when the ship is oscillating agrees with the steady wave at no ship motion. One exception is that the steady wave in front of very blunt bow is depressed slightly when the ship is in incident waves. the authoritative version for attribution.