Abstract We use eikonal theory to investigate the “induced diffusion” interaction in the ocean between small-scale internal waves and a much larger scale internal wave field. Surprising results are found. The eikonal description follows transport in both position and wavenumber space. The approach consists of modeling the small-scale portion as a superposition of wave packets, each of which moves through the large-scale flow according to the laws of particle mechanics: trajectories are determined by the solution of Hamilton's ordinary differential equations of motion, with a Hamiltonian given by a dispersion relation H( k, x ) = ω = σ( k ) + v 0 · k . The first term on the right-hand side is the dispersion relation in the absence of the large-scale flow, v0, and the second term is a Doppler shift that describes the interaction between the different scales. We give a careful derivation of the eikonal equations for a fluid, starting with a Hamiltonian description of the fluid flow (Section 2). A discussion is given of the important difference in meaning between the total energy of the wave packet, ωA, and the intrinsic energy, σA (A is the action). We also clarify the existence of a Stokes drift for internal waves (Section 3). Numerical experiments were performed which consist of following the motion of the center of a wave packet as it propagates through a Garrett-Munk field of internal waves (Section 4). The initial conditions of the packet were chosen to lie within the induced diffusion kinematic regime. A total of 50 trajectories were obtained from which average properties were calculated. The results of our numerical experiments show that horizontal transport is unimportant, whereas transport through vertical wavenumbers is very significant. We find a mean motion of kv to large values with the same sign as at t = 0, and fluctuations about the mean. The individual excursion in kv(t) are of large magnitude, and we argue that the traditional idea of diffusion in kv−t space, implying a random walk, is inappropriate. Thirtyfour out of the 50 trajectories in our sample reached a vertical wavelength cut-off placed at 5 m, doing so in essentially the same way as trajectories which approach a critical layer in a time and horizontal position independent shear flow. On the basis of our numerical results we provide a simple model which describes much of the transport that occurs. We argue that diffusion occurs in σ − z space, and a simple “mean first passage” calculation allows us to derive an expression for the probability density of critical layers.