An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ 1 (or equivalently when the Reynolds number R 1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ 1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ 4 0 the proposed solution becomes the inviscid one at each point in the flow field.For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate.Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1 %, with the (constant) inviscid values provided λs\a is less than about 10 −$ . This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs\a is about 10 −$ . For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1 % when λs\a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs\a, their similarity solution applies.In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero.Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

We consider the internal gravity waves that are produced in an inviscid Boussinesq fluid, whose Brunt-Va$ isa$ la$ frequency N is constant, by the small rectilinear vibrations of a horizontal elliptic cylinder whose major axis is inclined at an arbitrary angle to the horizontal. When the angular frequency ω is greater than N, no waves are produced and the governing elliptic equation is solved using conformal transformations. Analytic continuation in ω to values less than N, when waves are produced, is then used to determine the solution. It exhibits the surprising feature that, apart from certain phase differences, the form of the velocity distributions in each of the beams of waves that occur is the same for all values of the thickness ratio of the ellipse, the inclination of its major axis to the horizontal and the plane in which the vibrations are occurring. The Fourier decomposition of the velocity distribution is found and is used in a sequel, Part 2, to investigate the effects of viscous dissipation.In an important paper Makarov et al. (1990) have given an approximate solution for a vibrating circular cylinder in a viscous fluid. We show that the limit of this solution as the viscosity tends to zero is not the exact inviscid solution discussed herein. Further comparison of their work and ours will be made in Part 2.

A train of surface gravity waves of wavelength λ in a channel of depth H is incident on a horizontal plate of length l that is submerged to a depth c. Under the assumption that both λ and l are large compared with H, the method of matched asymptotic expansions is used to show that, to first order, the reflexion coefficient R and the transmission coefficient T are given by \[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \] and \[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \] where \begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ is the angular frequency and g the acceleration due to gravity.

The paper describes an investigation of the internal waves that are produced in a stratified fluid having constant Brunt—Väisälä frequency by a cylinder which executes small vibrations at a lower frequency. Explicit solutions are found for slender cylinders having arbitrary cross-sections. When the cross-sectional area of the cylinder varies with time it is found necessary in calculating the surface pressures and power output to take account of terms in the governing equations that are significant only at distances from the cylinder comparable to or larger than the scale height of the density variations. For this case a simple expression for the power output is obtained in terms of the rate of change of the cross-sectional area of the cylinder.When the vibrating cylinder is rigid its cross-sectional area is independent of time and then the expression for the power output is very similar to von Kármán's expression for the drag of a body of revolution in supersonic flow.In both the above cases it is found that one quarter of the power is radiated in each of the four directions that are inclined at a particular angle to the horizontal.

The paper describes a simple but general method for solving 'steady-state’ problems involving internal gravity waves in a stably stratified liquid. Under the assumption that the motion is two-dimensional and that the Brunt-Väisälä frequency is constant, the method is used to re-derive in a very simple way the solutions to problems where the boundary of the liquid is either a wedge or a circular cylinder. The method is then used to investigate the effect that a model of the continental shelf has on an incident train of internal gravity waves. The method involves analytic continuation in the frequency of the disturbance, and may well prove to be effective for other types of wave problem.