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This paper studies the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude. The effects are interpreted in terms of the radiation stresses in the waves.The first example is of wave groups propagated in water of uniform mean depth. The problem is solved first by a perturbation analysis. In two special cases the second-order currents are found to be proportional simply to the square of the local wave amplitude: (a) when the lengths of the groups are large compared to the mean depth, and (b) when the groups are all of equal length. Then the surface is found to be depressed under a high group of waves and the mass-transport is relatively negative there. In case (a) the result can be simply related to the radiation stresses, which tend to expel fluid from beneath the higher waves.The second example considered is the propagation of waves of steady amplitude in water of gradually varying depth. Assuming no loss of energy by friction or reflexion, the wave amplitude must vary horizontally, to maintain the flux of energy constant; it is shown that this produces a negative tilt in the mean surface level as the depth diminishes. However, if the wave height is limited by breaking, the tilt is positive. The results are in agreement with some observations by Fairchild.Lastly, the propagation of groups of waves from deep to shallow water is studied. As the mean depth decreases, so the response of the fluid to the radiation stresses tends to increase. The long waves thus generated may be reflected as free waves, and so account for the 'surf beats’ observed by Munk and Tucker.Generalle speaking, the changes in mean sea level produced by ocean waves are comparable with those due to horizontal wind stress. It may be necessary to allow for the wave stresses in calculating wind stress coefficients.

Short gravity waves, when superposed on much longer waves of the same type, have a tendency to become both shorter and steeper at the crests of the longer waves, and correspondingly longer and lower in the troughs. In the present paper, by taking into account the non-linear interactions between the two wave trains, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated. The results differ in some essentials from previous estimates by Unna. The variation in energy of the short waves is shown to correspond to work done by the longer waves against the radiation stress of the short waves, which has previously been overlooked. The concept of the radiation stress is likely to be valuable in other problems.

This paper describes the use of a hot film flowmeter in the sea and presents experimental measurements of the ‘downstream’ component of turbulent velocity in a tidal channel. The Reynolds number of the flow is about 108 and the scale of the turbulence is so large that a ship is carried about to a considerable extent by the energy-containing eddies. Under these conditions, a velocity measuring probe attached to a ship cannot be used for reliable measurements in the energy-containing range of the spectrum. It is possible, however, to observe the intertial and dissipation ranges. Records have been made at various stages of the tide. The one-dimensional spectra are found to be proportional to $k|^{-{\frac {5}{3}}}$ for several decades in k as predicted by Kolmogoroff, and a value is given for Kolmogoroff's constant. In the dissipation range there is close agreement with both Kovasznay's theory and Heisenberg's theory. These two theories are not very different in the low wave-number end of the range and the observations do not extend to high enough wave-numbers to distinguish between them.

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.