The changes in amplitude of short gravity waves on steady non-uniform currents

Abstract

The common assumption that the energy of waves on a non-uniform current U is propagated with a velocity (U + c) where cg is the group-velocity, and that no further interaction takes place, is shown in this paper to be incorrect. In fact the current does additional work on the waves at a rate γijSij where γij is the symmetric rate-of-strain tensor associated with the current, and Sij is the radiation stress tensor introduced earlier (Longuet-Higgins & Stewart 1960).In the present paper we first obtain an asymptotic solution for the combined velocity potential in the simple case (1) when the non-uniform current U is in the direction of wave propagation and the horizontal variation of U is compensated by a vertical upwelling from below. The change in wave amplitude is shown to be such as would be found by inclusion of the radiation stress term.In a second example (2) the current on the x-axis is assumed to be as in (1), but the horizontal variation in U is compensated by a small horizontal inflow from the sides. It is found that in that case the wave amplitude is also affected by the horizontal advection of wave energy from the sides.From cases (1) and (2) the general law of interaction between short waves and non-uniform currents is inferred. This is then applied to a third example (3) when waves encounter a current with vertical axis of shear, at an oblique angle. The change in wave amplitude is shown to differ somewhat from the previously accepted value.The conclusion that non-linear interactions affect the amplification of the waves has some bearing on the theoretical efficiency of hydraulic and pneumatic breakwaters.

The modulation of short gravity waves by long waves or currents

“…-* 0). In this same limit (2.27) agrees with the wave energy equations derived by LonguetHiggins and Stewart [10,11]. Further details of this limit will be described in the next section.…”

“…is the basic current along the interface, and that here \a\ is the wave amplitude measured vertically, whereas in Phillips [16] the wave amplitude is measured normal to the interface. The present results thus extend those of Phillips [16] to include forced waves (Po ^ 0)-Longuet-Higgins and Stewart [11] derived analogous equations to (3.3) [10] Modulation of short gravity waves 419 and (3.5a, b) for the case when the basic flow is a steady current (c = 0), with no applied forcing (Po = 0), under the additional restriction that the free-surface slope is small (/? -• 0).…”

“…The modulation equations are the local dispersion relation of the short waves, the kinematic equation for conservation of wave crests and the wave action equation. The results incorporate and extend the earlier work of Longuet-Higgins and Stewart [10,11]. …”

“…Since the pioneering work by Longuet-Higgins and Stewart [10,11] on the modulation of short gravity waves by long waves or currents, there has been a large amount of research done, much of which has been summarised in texts such as those by Whitham [18], Le Blond and Mysak [9], and Mei [14]. However, at least in the early stages of the development of this topic, the unifying nature of the general concepts of wave kinematics and wave action conservation were not fully appreciated.…”

Effect of wave boundary layer on sea-to-air dimethylsulfide transfer velocity during typhoon passage

“…Without currents, the energy of a wave package is conserved. With currents the energy of a spectral component is no longer conserved (Longuet-Higgins and Stewart, 1961), but the wave action spectrum, N ðk; h; x; tÞuFðk; h; x; tÞ=r; is conserved (Whitham, 1965;Bretherthon and Garrett, 1968). In WWATCH, the basic equation is for the wave action spectrum.…”