The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.

In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

Solutions to the problem of a two-dimensional, laminar jet of incompressible fluid issuing into a uniform stream in the direction of the main flow are considered. Two co-ordinate-type expansions are developed. A direct expansion, when suitably transformed, predicts approximately the velocity along the plane of symmetry of the jet for all values of the abscissa, with a maximum error of 7·6% far downstream from the origin. This error is established by comparison with a second, asymptotic expansion valid only at large values of the abscissa. The two expansions are subsequently joined, permitting an approximate determination of a constant which multiplies a third-order term in the asymptotic series and which initially remained unknown even after satisfying all boundary conditions imposed on these series.

The decay of velocity excess along the plane of symmetry of the jet is accelerated by the presence of the external stream.

The steady, two dimensional, incompressible flow of a viscoelastic fluid past a wedge of 90 degrees coated with the viscoelastic material is studied theoretically using constitutive equations proposed by Oldroyd in 1958. The effect of diffusion of the coating as well as its material properties (viscosity, relaxation time, retardation time, etc.) on the frictional force is investigated.

A boundary-layer analysis is performed on the constitutive equations as well as on the momentum equations. A similarity transformation is found for the set of boundary-layer equations. Series expansion and Laplace's method are employed to obtain the solution in the asymptotic series of gamma functions.

The results obtained show that:

1. The thinner displacement thickness does not necessarily imply a large frictional force for the viscoelastic flow.

2. For a homogeneous viscoelastic flow, the frictional force increases as the degree of dilatancy of the material increases, and decreases with increasing degree of pseudo-plasticity of the material.

3. For a non-homogeneous viscoelastic flow with given material constants, depending on whether the material is pseudoplastic or dilatant and on the ratio of the material concentration of outer flow and the concentration at the body, the frictional coefficient will decrease or increase from that of the homogeneous flow with the concentration at the body as the Schmidt number increases, and will approach a limit when the Schmidt number becomes very large. This limit is the frictional coefficient of the homogeneous flow with the concentration of the outer flow.

A formal expansion method for analysis of the non-linear development of an oblique wave in a parallel flow is presented. The present approach constitutes an extension and modification of the method of Stuart and Watson. Results are obtained for plane Poiseuille flow, and for a combination of plane Poiseuille and plane Couette flow. The Poiseuille flow exhibits finite-amplitude subcritical instability, and relatively weak but finite disturbances markedly reduce the critical Reynolds number. The combined flow, which becomes stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently great, remains unstable to finite disturbances.

Temperature measurements in the test section of a hypersonic gun tunnel using a modified line-reversal two-beam optical pyrometer have been performed. Measurements were made in the shock layer at the front of an ‘end-on’ flat-faced cylinder using argon or air as the driven gas. The measured temperatures were greater than the calculated isentropic temperatures but much less than the originally predicted shock-heated values. The experimental values agreed reasonably well with the theoretical temperatures for shock-heated air corrected for boundary-layer and heat-loss effects in the gun barrel and also with previously reported experimental values found using heat-transfer methods. Preliminary studies of the vibrational relaxation of nitrogen and air in the test section have been carried out using a flat-sided ‘relaxation’ tube (a Pitot tube with glass windows). They demonstrated the feasibility of this approach.

The theory of Fay & Riddell (1958) is used to calculate stagnation temperatures from stagnation-point heat-transfer rates measured in the working section of a hypersonic gun tunnel at a Mach number of 9·8. Measurements using both thin-film gauges and calorimeters are described. The temperatures measured using this technique are found to be lower than predicted by Lemcke (1962) from measurements of shock strengths and final pressures in the gun barrel. This discrepancy is attributed to heat losses in the barrel during the initial shock compression cycle. A simple theory is developed to take into account these losses. There is good agreement between this theory and the experimental results.

Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U(z) depends on height z only. If the Richardson number R is everywhere larger than 1/4, the waves are attenuated by a factor $\exp\{-2\pi(R - \frac{1}{4})^{\frac{1}{2}}\}$ as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing.

A wind tunnel in which an arbitrary negative pressure gradient could be developed has been built for boundary-layer studies. The effects of selected pressure gradients on boundary layers grown on one of the walls of the tunnel were studied. It was possible to obtain equilibrium boundary layers of the type first suggested by Clauser: that is, layers whose non-dimensional velocity-defect distribution is invariant along the direction of flow. The velocity-defect distributions for two such boundary layers were established, corresponding to values of Clauser's dimensionless pressure-gradient parameter β of −0·35 and −0·53. These velocity profiles are compared with the profiles predicted theoretically by Mellor & Gibson (1966). The agreement between the two is very good.

The motion induced by sources and sinks distributed along the vertical side wall of a cylinder filled with fluid and rotating about a vertical axis is considered. Vertical motion, and hence vertical transport, is confined to a vertical boundary layer of thickness E1/3, where E is the Ekman number. The horizontal transport occurs through the interior of the fluid. The Ekman layers do not play any active role in the transport process. The results of a very simple experiment confirm these conclusions.

An analytical model of the interaction of a fibre tangle with an airflow is proposed. This model replaces the discrete fibres by a continuum medium with a non-linear stress-strain law. The governing equations have been examined for one-dimensional unsteady flow configurations and have been found to possess five characteristic directions.

A numerical-solution procedure, based upon the method of characteristics, has been outlined and applied to the flow within a dilation chamber. A fibre sample is located at the centre of the chamber, which is alternately pressurized and depressurized.

A field of homogeneous turbulence generated at an initial instant by a distribution of random impulsive forces is considered. The statistical properties of the forces are assumed to be such that the integral moments of the cumulants of the force system all exist. The motion generated has the property that at the initial instant \[ E(\kappa) = C\kappa^2 + o(\kappa^2), \] where E(k) is the energy spectrum function, k is the wave-number magnitude, and C is a positive number which is not in general zero. The corresponding forms of the velocity covariance spectral tensor and correlation tensor are determined. It is found that the terms in the velocity covariance Rij(r) are O(r−3) for large values of the separation magnitude r.

An argument based on the conservation of momentum is used to show that C is a dynamical invariant and that the forms of the velocity covariance at large separation and the spectral tensor at small wave number are likewise invariant. For isotropic turbulence, the Loitsianski integral diverges but the integral \[ \int_0^{\infty} r^2R(r)dr = \frac{1}{2}\pi C \] exists and is invariant.

Various methods of establishing fluid media stably stratified in either temperature or the concentration of a foreign constituent are compared. A method used to construct a salt solution with precisely predetermined stratification is described. The Euler-Lagrange equations are solved exactly for a medium whose refractive index varies linearly in one dimension; the solution shows that a schlieren system may be used. For media with non-linear one-dimensional variation of refractive index, a modified shadowgraph technique may be employed. A device for the quantitative evaluation of a one-dimensional stratification is outlined and an example of its use is given.

Laboratory and numerical experiments on non-steady convective flows in a porous medium are reported. The main objective is to note the detailed comparison found between the time-dependent solutions and the time-like development of the iterative solutions of the steady equations originally pointed out by Garabedian (1956) and others.

Two flows are chosen for study. The first is the flow which develops when a blob of hot fluid is released at the base of a porous slab. The second is the flow which develops when a portion of the base of a porous slab is suddenly heated. The former flow is very simple and ideally suited for establishing the numerical scheme. The latter flow, however, produces several unexpected features. The gross features of the time development, when the motion is strongly non-linear, show an alternation between periods of slow gradual adjustment and periods of rapid change.