Internal gravity waves propagate away from a localized monochromatic disturbance inside beams, which develop around a St Andrew's Cross in two dimensions and around a double cone in three dimensions. The structure of the beams depends on three mechanisms, which couple together the different directions of propagation of the waves within the fluid. These mechanisms are associated with the size of the disturbance, the start-up of the motion and the viscosity of the fluid, respectively. The present paper considers each mechanism in isolation, for three-dimensional generation. The analysis is asymptotic and relies on far-field and large-time approximations. For each mechanism, three expressions of the waves are found: one, exact for an extended disturbance, that involves all the wavenumber vectors satisfying the dispersion relation; and two others, respectively uniform and non-uniform asymptotic expansions, that involve only the wavenumber vectors associated with group velocity vectors pointing toward the observer. For each mechanism, profiles of pressure and velocity are presented. A new time-independent characterization of the waves is introduced, in terms of the intensity or average energy flux; it is applied to the definition of the beam width. For an extended disturbance this width is a constant, the diameter of the disturbance. For an impulsive start-up the width increases linearly with the distance from the disturbance, and decreases in inverse proportion to the time elapsed since the start-up. For a viscous fluid the width increases as the one-third power of the distance. In all three cases, for a disturbance of multipolar order $2^n$, at constant strength, the power output varies in inverse proportion to the $(2n+1)$th power of the beam width.

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Limit states of internal wave beams

Experiments are described in which a horizontal circular cylinder is moved vertically and also horizontally at constant velocity normal to its axis in density stratified brine which has a constant buoyancy frequency, N. A six mirror Mach Zehnder interferometer is used to measure the density distributions within the far field wave systems. The Reynolds numbers based on the cylinder diameter, D, are between 1 and 10 and the frequency parameter, DN/U, where U is the body velocity, is of order one.

Viscous effects in some internal waves

A horizontal cylinder is traversed horizontally normal to its longitudinal axis in density stratified brine, and the lee waves which are produced over a range of model velocities are viewed with a six mirror Mach Zehnder interferometer which allows the density distribution to be evaluated. At the lowest model velocities, viscous effects limit the number of visible waves to no more than three. The viscous internal wave theory of Chang and Stevenson is extended and used to calculate lee waves over a ridge. Although the linear theory does not include the no-slip condition at the lower boundary and does not include the upstream wake, which is clearly visible in the experiments, the way in which the waves attenuate away from the model is reproduced quite well by the theory. The viscous results will be compared with the standard inviscid calculations.

Viscous effects in lee waves

In the present paper construction of the modified function of Green equation for internal gravity waves in the stratum of the stratified medium at presence of constant average flows is considered, properties of the corresponding spectral problems, the modified eigenfunctions and eigenvalues are investigated. Usage of the modified function of Green equation can give in some physically interesting events more friendly representations of the solutions for the fields of the internal gravity waves, including the wave fields disturbed by the non-local disturbing bodies.

Green modified function of the equation of the internal gravity waves in the stratum of the stratified medium with constant average flow

This issue of the Aeronautical Journal celebrates the 50th anniversary of the foundation of the Honours Degree Course in Aeronautical Engineering at the Victoria University of Manchester. The following article therefore describes the aeronautical research and teaching activities of that university up to its recent amalgamation with the University of Manchester Institute of Science and Technology (UMIST) to form the present-day University of Manchester. This juncture provides a further justification for recording the Victoria University’s achievements. Both the Victoria University and UMIST had their roots in the nineteenth century although, apart from the relatively brief period of the First World War, neither of them was particularly involved in aeronautics until after the Second World War. However, as Sections 6.0-10.0 seek to demonstrate, thereafter the Victoria University’s involvement became considerable. The preceding Sections describe the origins of the Victoria University and UMIST and, in the case of the former institution, the subsequent activities of its staff and graduates in engineering and mathematics which, although not always specifically aeronautical in content, nonetheless had a profound influence on the development of the aeronautical sciences.

The Victoria University of Manchester’s contributions to the development of aeronautics

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas & Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075003278

W. L. Chang

Papers

Internal waves in a viscous atmosphere