In this dissertation some characteristics of the horizontal water velocity for single, multiple, and random gravity wave trains are studied. This work consists of two parts, an analogue study and hydraulic measurements. An important aspect in this work is to suggest the horizontal water velocity asymmetry as a criterion for sand movement by water waves. In the first chapter an analogue model is developed to measure the probability distribution of an ensemble of wave trains. The work in this chapter can apply to the water velocity at a point in space, E uler coordinates, as well as the water particle velocity, Lagrange coordinates, because the velocity is described as a function of the ratio of the first two terms of the Fourier series solution. Expressions are found for the second and third moments of the velocity of an endless wave train assuming linear superposition of such waves; interaction terms are neglected. The moments and distributions are discussed for application to sand movement by waves in shallow water. The second chapter deals with some direct measurements of the second and third moments of the horizontal water velocity on the bed of a shallow water wave tank. This was done for single, multiple, and random gravity wave trains with the purpose of observing the moments as a random wave field is built up out of the single wave trains. The analogue study of chapter 1 shows that forward movement of sand by waves should diminish as the number of superimposed wave trains is increased provided the total energy of the waves is constant. This is true only for this superposition of wave trains consisting of the first two terms of the Stokes theory. Furthermore, it has been found that the distribution of four or more superimposed wave trains is nearly Gaussian. A Gaussian random wave field can, under certain conditions, be approximated by the simultaneous generation of only four wave trains. The use of less than four wave trains can lead to some special effects as far as the statistical sorting of sand particles is concerned. For the hydraulic measurements of chapter 2, a positive asymmetry for the horizontal water velocity of single wave trains in shallow water was found. On the other hand it was mostly negative for the multiple and random wave trains studied here. The measurements indicate a negative asymmetry that increases in magnitude with increasing band width for random wave trains with a constant center frequency. It also appears that the asymmetry tends toward positive for an increasing wave length and constant band width. This predicts movement of sand in a seaward direction for short, broad band wave trains and landward movement for long, narrow band wave trains. This is in agreement with the known observation that long, quiet, relatively narrow band summer wave trains build beaches while short, broad band winter wave trains wash beaches away. It is thus possible that the beachward and seaward movement of sand in shallow water due to waves can be described by means of the asymmetry of the horizontal water velocity.