# A note on the cylindrical wave-maker problem in a liquid with an inertial surface

Abstract: This note is concerned with the problem of forced motion due to a vertical circular cylindrical wave-maker immersed in a liquid with an inertial surface composed of uniformly distributed floating particles. The techniques of Laplace transform in time and Weber transform in the radial co-ordinate are used to obtain the velocity potential and hence the inertial surface depression. For the special case of a time-harmonic wave-maker, the potential function is analysed for its steady-state development. It is shown that if the inertial surface is “too heavy”, the disturbance due to the wave-maker remains confined within a short distance only.

Topics: Inertial wave (63%), Velocity potential (54%), Inertial frame of reference (53%), Laplace transform (52%)

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Abstract: The problem of generation of waves in a liquid of uniform finite depth with an inertial surface composed of a thin but uniform distribution of disconnected floating particles, due to forced axisymmetric motion prescribed on the surface of an immersed vertical cylindrical wave-maker of circular cross section under the influence of surface tension at the inertial surface, is discussed. The techniques of Laplace transform in time and the modified Weber transform involving Bessel functions in the radial coordinate have been employed to obtain the velocity potential. The steady-state development to the potential function as well as the inertial surface depression due to time-harmonic forced oscillations of the wave-maker are deduced. It is found that the presence of surface tension at the inertial surface ensures the propagation of time-harmonic progressive waves of any angular frequency.

3 citations

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Abstract: The velocity potential due to a submerged circular ring source of time-dependent strength outside an immersed vertical co-axial circular cylinder is obtained for a liquid of finite depth bounded by an inertial surface composed of uniformly distributed noninteracting floating particles. The modified Weber transform formula as used in [1] in the radial coordinate has been employed to obtain the transformed velocity potential after using Laplace transform in time. For time-harmonic source strength, the steady-state development shows that the progressive waves generated by the ring source in the presence of the cylinder cannot propagate if the inertial surface is too heavy.

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Abstract: In the present note the classical vertical cylindrical wave‐maker problem in water of finite depth under the influence of surface tension is reinvestigated by employing a suitable Weber transform in the radial coordinate.

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01 Jan 1946-

1,623 citations

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01 Jan 1962-

1,294 citations

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265 citations

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01 Sep 1971-

Abstract: The classical wave-maker problem to determine the forced two-dimensional wave motion with outgoing surface waves at infinity generated by a harmonically oscillating vertical plane wave-maker immersed in water was solved long ago by Sir Thomas Havelock. In this paper we reinvestigate the problem, making allowance for the presence of surface tension which was excluded before, and obtain a solution of the boundary-value problem for the velocity potential which is made unique by prescribing the free surface slope at the wave-maker. The cases of both infinite and finite constant depth are treated, and it is essential to employ a method which is new to this problem since the theory of Havelock cannot be extended in the latter case of finite depth. The solution of the corresponding problem concerning the axisymmetric wave motion due to a vertical cylindrical wave-maker is deduced in conclusion.

59 citations