This chapter deals with low periodic long waves and oscillations such as tides and seiches, taking into account inertia and resistance. The aim of this chapter is to provide insight into the dynamics of these waves, in particular with respect to the effects of resistance, which were mainly ignored so far. Using linear approximations, valid for low waves, explicit yet simple analytical solutions are obtained in the form of complex exponential functions, representing damped harmonic (sinusoidal) waves or oscillations. Solutions will be derived for uniform as well as non-uniform channels, which are used subsequently to analyze periodic wave motion in channels that are closed at one end or at both ends. The chapter concludes with a discussion of nonlinear influences, such as the variation of the cross-sectional parameters with the free-surface elevation, or the quadratic nature of the resistance.

Introduction

A typical situation occurs when an initial state of rest is affected by continual periodic disturbances at a boundary, which propagate into the domain considered. Given enough time, a periodic state of motion is established in the entire domain, without memory of the initial situation. In such cases the motion is unsteady within each wave cycle, but the cycles themselves do not vary in time, as is approximately the case for the variation of amplitudes and phases in seiches and tidal waves. The principal features of such waves can reasonably well be represented with a linear model, assuming a harmonic variation of the state variables involved.

To the authors’ knowledge, analytical modelling of the one-dimensional propagation of long harmonic waves was pioneered by Parsons (1918), for the prediction of the tidal flows in the then future Cape Cod Canal. Parsons’ linearization of the quadratic friction was incorrect (see Box 6.1), and the system considered by him was simple, but his work was nevertheless a commendable achievement. Yet it does not seem to have found a recognizable follow-up. In fact, it is not even mentioned in the review of tidal computations with the harmonic method by Ippen and Harleman (1966).