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Lighthill, in his elegant and classic theory of jet noise, showed that the far-field acoustic pressure of noise generated by turbulence is proportional to the integral over the jet volume of the second time derivative of the Lighthill stress tensor, the integrand being evaluated at a retarded time. The purpose of this paper is to generalize the above results to include the effects of mean flow (velocity and temperature) surrounding the source of sound. It is shown quite generally that the integrand is now a certain functional of the Lighthill stress tensor evaluated at a retarded time. More important, however, at low and high frequencies this functional assumes an extremely simple form, so that the acoustic field can once more be given by integrals of the time derivatives of the Lighthill tensor. Both the self- and the shear-noise contributions to the pressure are evaluated.

This paper investigates the propulsive performance of the lunate tails of aquatic animals achieving high propulsive efficiency (the hydromechanical efficiency being defined as the ratio of the work done by the mean forward thrust to the mean rate at which work is done by the tail movements on the surrounding fluid). Small amplitude heaving and pitching motions of a finite flat-plate wing of general planform with a rounded leading edge and a sharp trailing edge are considered. This is a generalization of Chopra's (1974) work on model rectangular tails. This motion characterizes vertical oscillations of the horizontal tail flukes of some cetacean mammals. The same oscillations, turned through a right angle to become horizontal motions of side-slip and yaw, characterize the caudal fins of certain fast-swimming fishes; viz. wahoo, tunny, wavyback skipjack, etc., from the Percomorphi and whale shark, porbeagle, etc., from the Selachii. Davies’ (1963, 1976) method of finding the loading distribution on the wing and generalized force coefficients, through approximate solution of an integral equation relating the loading and the upwash (lifting-surface theory), is used to find the total thrust and the rate of working of the tail, which in turn specify the hydromechanical swimming performance of the animals. The physical parameters concerned are the tail aspect ratio ((span)2/planform area), the reduced frequency (angular frequency x typical length/forward speed), the feathering parameter (the ratio of the tail slope to the slope of the path of the pitching axis), the position of the pitching axis, and the curved shapes of the leading and trailing edges. The variation of the thrust and the propulsive efficiency with these parameters has been discussed to indicate the optimum shape of the tail. It is found that, compared with a rectangular tail, a curved leading edge as in lunate tails gives a reduced thrust contribution from the leading-edge suction for the same total thrust; however, a sweep angle of the leading edge exceeding about 30° leads to a marked reduction of efficiency. Another implication of the present analysis is that no negative work is involved in the actual oscillation of the tail.

The present results are used to obtain an estimate of the drag coefficient for the motion of the animals, based on observed data and the computed thrust. The results show some evidence of differences between the CD's for cetacean mammals and scombroid fish respectively. Some discussion of this difference is also given.

Natural modes of water oscillation inside harbours are known to occur with periods of the order of minutes. It seems likely that these oscillations are excited by water fluctuations of similar period outside the harbour and an often quoted cause of such fluctuations is the phenomenon of surf beats. These are thought to be long waves which are reflected back out to sea when a primary wave system breaks upon a beach. In this paper it is shown theoretically that the natural oscillations of a harbour can be excited directly, without breaking of the primary wave system, by set-down beneath wave groups, which is a long-period disturbance travelling towards the shore line at the group velocity. This theory is in agreement with model experimental results which show that, when the group period is close to a natural period of the harbour, resonance will occur with the set-down behaving as if it were a real long wave.

A double scale technique is used to determine the asymptotic behaviour of a rolled-up vortex sheet. The technique relies on a process of averaging out the saw-tooth-like behaviour of the flow variables, which generates a continuous solution having the structure of a vortex filament. The fine-scale behaviour of the flow is described and includes concentrated vorticity on the sheet. Application to the conical vortex sheet allows the solution of Mangler & Weber (1967) to be rederived. A further application, to Kaden's problem, is worked out and the results are in complete agreement with Moore's asymptotic formulae for the shape of the spiral.

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α1 and α2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin2ϕ α1,2 or strong if ϕ2 = O(α1,2). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O(α1α2), then is determined from these basic solutions.

Strong interactions are intrinsically nonlinear. It is shown that these interactions are phase-conserving (the sum of the phases of the incoming waves is equal to the sum of the phases of the outgoing waves) if |α2-α1 > (2ϕ)2 but not if |α2-α1| (2ϕ)2 (e.g. the reflexion problem, for which the interacting waves are images and α2 = α1). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech2 profiles yield singular outgoing waves with - csch2 profiles, if \[ \psi_{-}< |\psi| < \psi_{+},\quad{\rm where}\quad\psi_{\pm}={\textstyle\frac{1}{2}}\left|(3\alpha_2)^{\frac{1}{2}}\pm(3\alpha_1)^{\frac{1}{2}}\right|. \]

Regular interactions appear to be impossible within this singular regime, and its end points, |ϕ| = ϕ±, are associated with resonant interactions.

Resonant (phase-locked) interactions among three obliquely oriented solitary waves are studied. It is shown that such interactions are associated with the parametric end points of the singular regime for interactions between two solitary waves. The latter include regular reflexion at a rigid wall, which is impossible for ϕi < (3α)½ (ϕ = angle of incidence, α = amplitude/depth [Lt ] 1), and it is shown that the observed phenomenon of ‘Mach reflexion’ can be described as a resonant interaction in this regime. The run-up at the wall is calculated as a function of ϕi/(3α)½ and is found to have a maximum value of 4αd for ϕi = (3α)½. This same resonant interaction also describes diffraction of a solitary wave at a corner of internal angle π − ψi, −(3α)½, and suggests that a solitary wave cannot turn through an angle in excess of (3α)½ at a convex corner without separating or otherwise losing its identity.

Self-diffusion coefficients were determined experimentally for lateral dispersion of spherical and disk-like particles in linear shear flow of a slurry at very low Reynolds number. Using a concentric-cylinder Couette apparatus, recurrent observations were made of the lateral position of a particular radioactively labelled particle. The self-diffusion coefficient D was calculated by means of random-walk theory, using the ergodic hypothesis. Owing to great experimental difficulties, the calculated values of D are not of high accuracy, but are correct to within a factor of two. In the range 0 < ϕ < 0·2, D/a2ω increases from zero linearly with ϕ up to D/a2ω ≈ 0.02 (where ϕ = volumetric concentration of particles, a = particle radius, ω = mean shear rate of suspending fluid). In the range 0.2 < ϕ < 0.5, the trend of D/a2ω is not clear because of experimental scatter, but in this range D/a2ω ≈ 0·025 to within a factor of two. Within the experimental accuracy, spheres and disks have the same value of D/a2ω.

Self-diffusion coefficients were determined experimentally for lateral dispersion of spherical and disk-like particles in linear shear flow of a slurry at very low Reynolds number. Using a concentric-cylinder Couette apparatus, recurrent observations were made of the lateral position of a particular radioactively labelled particle. The self-diffusion coefficient D was calculated by means of random-walk theory, using the ergodic hypothesis. Owing to great experimental difficulties, the calculated values of D are not of high accuracy, but are correct to within a factor of two. In the range 0 < ϕ < 0·2, D/a2ω increases from zero linearly with ϕ up to D/a2ω ≅ 0·02 (where ϕ = volumetric concentration of particles, a = particle radius, ω = mean shear rate of suspending fluid). In the range 0·2 < 0·5, the trend of D/a2ω is not clear because of experimental scatter, but in this range D/a2ω ≅ 0·025 to within a factor of two. Within the experimental accuracy, spheres and disks have the same value of D/a2ω.