The development during the last two decades of analytical techniques for the solution of problems relating to the optimisation of rocket trajectories is outlined and the present position in this field of research is summarised. It is shown that the determination of optimal trajectories in a general gravitational field can be expressed as a Mayer problem from the calculus of variations. The known solution to such a problem is stated and applied, first to the special case of the launching of an artificial satellite into a circular orbit with minimum expenditure of propellant and, secondly, to the general astronautical problem of the economical transfer of a rocket between two terminals in a gravitational field. The special cases when the field is uniform and when it obeys an inverse square law of attraction to a point are then considered, and the paper concludes with some remarks concerning areas in which further investigations are necessary.

The concept of the manoeuvre margin is discussed, and definitions are suggested which are applicable to all types of configurations. Calculations are presented for the manoeuvre characteristics of simplified flexible models representing conventional and slender integrated aircraft. Linear aerodynamics only are assumed.

The overall effective efficiency of a jet-propelled flight may be defined as the net useful work that is done in accelerating the vehicle, raising it to height and combating its drag, divided by the calorific value of the propellants used. The analysis presented here shows that for a single-stage vehicle this overall efficiency is equal to the integral of the fraction

with respect to each element of propellants used. In this expression u'is the effective jet velocity (allowing for jet deflection, this is √(1 + k2) times the actual exhaust velocity), υ the vehicle velocity, q the fuel/air ratio and H the calorific value of the fuel in dynamic units. The mean lift/drag ratio of the vehicle is taken as 1/k, so that the last three terms of the numerator represent the work lost in dragging, raising and accelerating each element of propellant to the range R, height h and flight velocity υ at which it is burnt. The fraction may therefore be considered as the efficiency with which that element of propellant is used with respect to the whole flight plan.

For advanced air-breathing projects—by-pass engines up to aircraft speeds around Mach 2 and ramjets between Mach 4 and 9—the analysis shows that the sum of the υ-dependent terms in the efficiency factor is practically constant. The potential energy term is insignificant in the normally accepted flight corridor. Thus the overall efficiency with which propellants are consumed in covering a range R1 at flight speed υ js closely approximated to by the fraction

and the useful work done by the mass mm of propellants is

The vehicle mass (including engine, payload, and so on) that can be dragged over the range R1 by unit mass of propellants can then be calculated.

Although such simplified expressions cannot substitute for the final detailed analysis of a projected flight plan, it is hoped that they will prove more useful in preliminary analysis than the figures based entirely on cruise performance which have often been quoted in the past.

A large number of far-field jet noise measurements obtained over a considerable period of time are presented, correlated in such a manner that they may be used to provide a complete noise assessment for any jet of which the performance parameters are known. The data are presented in two sections, one dealing with the problem of a static source near ground level, the other dealing with the case of a source in flight.

Much of the data was derived from Avon engine tests, but a wide range of other jet noise sources has also been encompassed. Comparisons are drawn which show good correlation to exist between the overall sound pressure levels produced by jets of all sizes. It appears also that even the spectra quoted, which are based only on Avon measurements, are valid over a fairly wide range of jet sizes and are not peculiar to the Avon engine.

The problem considered is the buckling of a rectangular plate under uniaxial compression. The ends may be either both clamped, both simply-supported or a mixture of the two. The sides may be elastically restrained against both deflection and rotation with any stiffnesses whatsoever. It is shown that the curve of buckling stress coefficient versus side ratio can be deduced in a simple manner from that of a plate with the same end conditions but with both sides simply-supported, provided only that the buckling stress coefficient and wavelength for an infinite strip with the same side conditions are known. Some correlations between the curves for the three types of end condition are discussed. It is also shown that if, for some given side ratio, the buckling mode is known, then it is always possible to deduce the rate of change of buckling stress coefficient with side ratio at that point. The argument is based upon an assumption which is shown to give very accurate results in a wide range of cases.

The subject of three-dimensional flow in axial compressors and turbomachines has been extensively studied since 1945. This paper gives a means of calculating the approximate three-dimensional flow in an axial compressor by giving expressions for the slope of the velocity profile as a function of the axial co-ordinate. These expressions bring out the rôle of the ratio of stage inlet annulus height to stage length in the three-dimensional flow in these machines. The effect of the three-dimensional flow on the stage temperature rise at mean radius is discussed by introducing a work done factor.

A study has been made of the effects of inlet coolant injection upon axial compressor performance, using the results of tests on turboshaft engines. It is shown that evaporation of the coolant changes the stage work distribution as well as the ideal compression work and that these effects may be estimated by elementary thermodynamic methods. Simplified prediction procedures are suggested and compared with experimental results.

An investigation has been made into the errors involved in using certain lumped parameter methods for the solution of beam frequencies. It is found that existing methods are not consistent for all boundary conditions. A new dynamic stiffness matrix has been formulated, which gives consistently good results for even a few elements. The error in the solution is always inversely proportional to the fourth power of the number of elements used.

The paper presents a solution for the buckling under shear of a rectangular plate which is clamped along one edge and simply-supported along the other edges. The authors have also re-examined the case of one pair of opposite edges clamped and the other pair simply-supported.

The vorticity shed behind a finite wing has been studied and a condition found for steady circulation. This condition is then combined with three-dimensional boundary layer theory to find the lift of a finite wing. An example shows that on a thin ellipsoid of aspect ratio 6·37 the lift as found by the present method is 2 per cent lower than the lift obtained using a sectional boundary layer approach.

A description is given of a supersonic pressure-tube wind tunnel which has been constructed at A.R.D.E. This is a blow-down tunnel which uses as a reservoir a long tube filled with gas under pressure. A quasi-steady supersonic flow is achieved by expanding in a convergent-divergent nozzle the subsonic flow behind rarefaction waves which propagate down the tube when a diaphragm at the nozzle exit is burst. The theory of the operation of the tunnel is given and calculations are made of the boundary-layer growth along the tube. Pressure-time records were obtained in the tube, and a high speed camera was used to obtain pictures of the flow round a model. Measurements also included a pitot-tube traverse of the nozzle exit, and the Mach number distribution was determined from the ratio of the pitot to the stagnation pressure. Tests showed that, as predicted, a constant stagnation pressure was obtained ahead of the nozzle, and it is considered that a tunnel of this type would be a cheap and simple way of obtaining an intermittent tunnel with adequate running time for many types of test, and capable of operating at a Reynolds number of more than 107 per inch at a Mach number of about 3·5.

There appear in the literature some discrepancies in the various mathematical expressions for the upwash constituting the wall interference on a small lifting wing placed symmetrically in a rectangular tunnel with closed side-walls and open floor and roof. The representation of the interference field as that of a doubly infinite array of image doublets is shown to lead to divergence. An independent calculation by relaxation confirms the correctness of an analytical solution by expansion in Fourier series. The discrepancies are found to arise from incomplete flow conditions at the open boundaries, and it is shown in this paper how the various theories are reconciled.

After a retrospective discussion of Glauert's theorem concerning the wall interference on a small lifting aerofoil in a wind tunnel with mixed boundaries, a simple proof of the theorem is given for a tunnel having one rigid and one free boundary. It is shown that the theorem does not necessarily hold when there are two rigid walls and two free boundaries.

It has been shown in another paper that the relation between the compressive buckling stress coefficient and side ratio for a rectangular plate with the sides elastically restrained against deflection and rotation can be obtained from the corresponding curve for a plate with the same end conditions but with both sides simply-supported, provided only that the buckling stress coefficient and wavelength of the buckle are known for an infinite strip with the given side conditions. This paper presents these numerical values for infinite strips with a wide range of edge stiffnesses.

The analysis of Freeman is extended to the hypersonic flow of an inviscid, vibrationally relaxing gas past a bluff body. Expressions for the shock shape, streamline shapes and stand-off distance are derived; these expressions have been evaluated for a sphere for various values of an appropriate non-equilibrium parameter Λ.

The stress concentration factors for equilateral triangular holes, with varying degrees of curvature at the corners, and with varying amounts of compact reinforcement on the boundary, have been computed for each of three loading cases. Poisson's ratio was assumed to be 1/3.

An understanding of radial flow between confined boundaries is of practical importance in the design of radial diffusers and air bearings. This study presents a combined experimental and theoretical analysis of radial flow, without swirl, between parallel discs using air at incompressible speeds.

Emphasis is placed on the pressure distribution sufficiently far downstream of the channel inlet for the entry conditions to be unimportant. However, a study is also made of the main features of the flow near the inlet, particularly within the annular separation bubble.

It is shown, for both turbulent and laminar flow, that a similarity solution is possible only in special cases where certain terms in the equations of motion can be neglected. Approximate solutions are obtained for the turbulent and the laminar radial pressure distributions using an integral momentum method. Both theories agree well with experiment. The critical Reynolds number for reverse transition is found to be approximately the same as that for flow in twodimensional channels and circular pipes. With the flow separating at the channel inlet, it is established that both a suitably chosen, minimum pressure coefficient of the separation bubble and the reattachment distance are functions only of the channel width for a given inlet pipe diameter and are independent of Reynolds number and the diameter of the discs.

The stability of the phugoid motion of an aircraft in the presence of wind shear is investigated. The effect of the wind shear on the phugoid frequency increases with increasing aircraft speed and can be stabilising or destabilising, depending on the aircraft orientation relative to the wind shear. The destabilising effect of wind shear is alleviated by the stabilising effect of the density gradient of the atmosphere. At the most critical combination of speed and altitude a strong wind shear may lead to divergence, with a time to double amplitude of the order of 10-15 seconds.

A numerical study of the aircraft motion with controls fixed when descending through a wind profile similar to that in a jet stream indicates that the increase in the aircraft indicated speed can be of the same magnitude as, or larger than, the maximum wind velocity increment in the jet stream core. However, as the time to reach the excess speed is of the order of one minute, the actual behaviour of the aircraft strongly depends on the pilot's action and thus is not fully predictable by theoretical analysis.

A new method of analysing the response of linear systems to sinusoidal excitation, which avoids the assumptions of present methods, is developed in this paper. It enables one to obtain the eigenvalues and righthand eigenvectors of a system with viscous damping from the response of the system. If the inertia matrix of the system is known, the damping and stiffness matrices can also be obtained. No assumptions are made about the form of these matrices, and the method is not restricted to systems with only small amounts of damping.

A solution is presented for the stresses in a half plane compactly reinforced along its straight edge and pierced by a reinforced circular hole in the vicinity of the edge. Numerical results are given for two cases of uniform applied tractions at infinity. In the special case of both boundaries unreinforced, the results obtained are in agreement with an existing solution by Mindlin.