Complex resonances in the water-wave problem for a floating structure

Abstract

Citation: MCIVER, P., 2005. Complex resonances in the water-wave problem for a floating structure. Journal of Fluid Mechanics, 536, Additional Information:• This work is concerned with the linearized theory of water waves applied to the motion of a floating structure that restricts in some way the motion of a portion of the free surface (an example of such a structure is a floating torus). When a structure of this type is held fixed in incident monochromatic waves, or forced to move time harmonically with a prescribed velocity, the amplitude of the fluid motion will have local maxima at certain frequencies of the forcing. These resonances correspond to poles of the scattering and radiation potentials when extended to the complex frequency domain. It is shown in this work that, in general, the positions of these poles in the scattering and radiation potentials will not coincide with the positions of the poles that appear in the velocity potential for the coupled problem obtained when the structure is free to move. The poles of the potential for the coupled problem are associated with the solution for the structural velocities of the equation of motion.When physical quantities such as the amplitude of the fluid motion are examined as a function of (real) frequency, there will in general be a shift in the resonant frequencies in going from the radiation and scattering problems to the coupled problem. The magnitude of this shift depends on the geometry of the structure and how it is moored.

Experimental observation of a near-motion-trapped mode: free motion in heave with negligible radiation

“…Note that an equivalent statement is that a complex resonance in this mode occurs on the real axis of the complex frequency plane (McIver 2005). The 8 cylinder structure discussed in Wolgamot et al (2015) formed of a ring of 8 truncated cylinders was a near-motion-trapping structure in the sense that the draft of the cylinders could be adjusted to align the frequency at which condition 2.2 was satisfied to the frequency of a mode with extremely low (but non-zero) damping.…”

“…Motion-trapping structures which enclosed part of the free surface within axisymmetric shapes of complicated vertical cross-section (McIver & McIver 2007), and later simple rectangular crosssection (Porter & Evans 2008) were found. While motion-trapping structures cannot be excited by incident regular waves at the trapping frequency (as shown by McIver (2005) and investigated experimentally by Kyozuka & Yoshida (1981)) they can be excited given appropriate initial conditions. According to linear inviscid theory, a motion-trapping Figure 1.…”

Trapped modes in the water-wave problem for a freely floating structure

“…Recently, it has been established that the trapped modes supported by a fixed structure, as described above, cannot be excited when that structure is allowed to float freely (and hence respond to the hydrodynamic forces on it), with or without incident waves (McIver 2005). For motion in a single mode this follows immediately from the frequency-domain equation of motion which shows that the pole in the radiation potential at the trapped-mode frequency is annulled by a corresponding zero in the velocity.…”

The motion of a freely floating cylinder in the presence of a wall and the approximation of resonances

“…Newman (1977); Linton & Evans (1992); Yeung & Seah (2007)) and it has been found that, as frequency increases, the resonance peaks in the added mass and damping coefficients become higher and narrower. It should be noted that, in general, the locations of the resonances of complexforce coefficients q w jk (ω) do not coincide with those of the displacements (McIver 2005), and a consequence of this is that highly detailed computation of the hydrodynamic coefficients around a resonance may be unnecessary when performing computations of the motion of a freely-floating structure. Rather, attention should be paid to the resonances in the displacement which occur at nearby, but distinct, complex frequencies (Lewandowski 2008), although it should be noted that the differences in the locations can be quite small.…”

“…Newman (1977) investigated the forces on a floating slender torus and connected peaks in the hydrodynamic forces to near standing waves within the toroidal ring; in addition he calculated the responses of the torus due to incident waves. In both Fredriksen et al (2015) and Newman (1977) it is reported that peaks in the displacements of a freelyfloating structure are shifted relative to those in the standard hydrodynamic coefficients, a phenomenon studied in detail by McIver (2005).…”