Flutter effect and emission in the region of anomalous and normal Doppler effects

Membrane flutter induced by radiation of surface gravity waves on a uniform flow

“…which is nothing else but the shallow water dispersion relation derived by Nemtsov (1985). In order to get the dispersion relation (3.21), one must take the pressure difference (2.7), make it non-dimensional and use in the expression (2.17) which then reads as…”

“…In figure 4 we show that the exact neutral stability boundaries obtained from equating the discriminant of the fourth-order polynomial (3.30) in σ to zero and their approximation corresponding to β = 0 and κ = κ 0 shows four straight lines intersecting at five points, including the origin. The upper horizontal line corresponds to the fast surface gravity wave with σ = M 0 + √ (tanh κ 0 )/κ 0 ≈ 3, whereas the lower horizontal line to the slow surface gravity wave (Nemtsov 1985) with σ = M 0 − √ (tanh κ 0 )/κ 0 ≈ 1. The two inclined lines correspond to the forward and backward elastic waves in the membrane with σ = ±M w .…”

“…In Abramovich, Mareev & Nemtsov (1986) the ADE for such an oscillator was demonstrated due to radiation of surface gravity waves in a layer of an incompressible fluid. Nemtsov (1985) was the first who considered flutter of an elastic membrane resting at the bottom of a uniform horizontal flow of an inviscid and incompressible fluid as an anomalous Doppler effect due to emission of long surface gravity waves. In the shallow water approximation, he investigated both the case of a membrane that spreads infinitely far in both horizontal directions and the case when the length of the membrane in the direction of the flow (or the chord length) is finite whereas the span in the perpendicular direction is infinite.…”

“…Nevertheless, the case of flow of arbitrary depth has not been studied in Nemtsov (1985), and no numerical computation supporting the asymptotic results has been performed. Another issue that has not been addressed in Nemtsov (1985) is the relation of stability domains for the membrane of finite length to those for the membrane of infinite length. Vedeneev (2004) studied flutter of an elastic plate of finite and infinite length at the bottom of a uniform horizontal flow of a compressible gas occupying the upper semi-space.…”

“…(2.6) For the sake of completeness, we present also the analogous expression for the pressure difference for the case when there is a vacuum below the membrane (Nemtsov 1985) ΔP…”