A theory is proposed to describe the generation of sound by turbulence at high Mach numbers. The problem is formulated most conveniently in terms of the fluctuating pressure, and a convected wave equation (2.8) is derived to describe the generation and propagation of the pressure fluctuations.

The supersonic turbulent shear zone is examined in detail. It is found that, at supersonic speeds, sound is radiated as eddy Mach waves, and as the Mach number increase, this mechanism of generation becomes increasingly dominant. Attention is concentrated on the properties of the pressure fluctuations just outside the shear zone where the interactions among the weak shock waves have had little effect. An asymptotic solution for large M is derived by a Green's function technique, and it is found that radiation with given frequency n and weve-number K can be associated with a coresponding critical layer within the shear zone.

It is found that $(\overline {p - p_0})^2 $ increases approximately as $\rm {M}^{\frac {3} {2}}$ for M [Gt ] 1 contrasting with the M8 variation found by Lightill for M [Lt ] 1. The acoustic efficiency thus varies as $\rm {M}^-{\frac {3} {2}}$ for M [Gt ] 1, and as M5 for M [Lt ] 1, indicating a maximum acoustic efficiency for Mach numbers near one. The directional distribution of the radiation is discussed and the direction of maximum intensity is shown to move towards the perpendicular to the shear zone as M increases. The predictions of the theory are supported qualitatively by the few available experimental observations.

An inviscid fluid is bounded above by a horizontal plane and below by finite-amplitude ridges. The fluid is rotating and stratified. A uniform transport is forced across the ridges at small dynamical Rossby number, although the boundary conditions are such that motion cannot remain geostrophic. The most significant parameter is found to be a thermal Rossby number based upon the vertical density difference and both vertical and horizontal length scales, but independent of the transport. Conditions determining whether or not effects of the bottom topography will penetrate vertically throughout the fluid are discussed. Some numbers characteristic of flow in the deep ocean are presented.

Three classes of exact solutions for steady two-dimensional flows of a stratified fluid are found. The flows which correspond to these solutions have arbitrary amplitude (however defined). Two of the three classes of solutions have close bearings on the lee-wave problem in meteorology. It is also shown that the amplitudes of the lee-wave components (if there is more than one component) depend not on the details of the shape of the barrier, but only on certain simple integral properties of the function for the singularity distribution generating the barrier.

A steady slightly non-uniform flow with a free surface is subject to a concentrated surface pressure which gives rise to a pattern of surface waves. (For gravity waves on deep water this is the well-known Kelvin ship-wave pattern.) The motion is assumed inviscid, and the waves are assumed small. A theory is developed for the wave pattern, based on the following assumptions:

1. The stream velocity component normal to a wave crest is equal to the phase velocity based on the local wavelength;

2. the separation between consecutive crests is equal to the local wave-length. These assumptions are expressed in mathematical form, and the existence of a set of characteristic curves (associated with the group velocity) is deduced from them. These characteristics are not identical with the crests. Let the additional assumption be made that

3. the characteristics all pass through the point disturbance; the characteristics are then completely defined and may be constructed by a step-by-step process starting at the point disturbance. The same construction gives the direction of the wave crests at all points. The wave crests can then be deduced.

Assumptions of the same type as (1) and (2) have long been familiar in various applications of ray tracing. For uniform flows the present theory gives the same pattern as the method of stationary phase.

The frequency dependence of the velocity and attenuation of sound waves in a gas which undergoes vibrational relaxation have been investigated theoretically. At low audible frequencies the attenuations due to viscosity, thermal conduction and relaxationin the gas, add linearly, while the velocity is the relaxation velocity diminished by the Helmholtz-Kirchhoff factor. The relations have been confirmed experimentally, and the free gas velocities of sound at zero frequency, one atmosphere pressure and 30 °C, found for carbon dioxide, air and oxygen, are 270·57 ± 0·04 m sec−1, 349·18 ± 0·02 m sec−1 and 331·33 ± 0·03 m sec−1, respectively. The corresponding specific heats are Cp/R = 4·537 ± 0·008 for carbon dioxide and Cp/R = 3·547 ± 0·003 for oxygen.

Earlier work on the dynamics of a perfectly conductive gas in situations where all variables depend on one space co-ordinate only is extended to the case where the magnetic field has a component in the direction of variation. The theory is developed for an arbitrary gas in equilibrium, subject only to certain reasonable restrictions.

The first main section studies the variation of the transverse field component in slow and fast simple waves and the tendency of compressive waves in which the transverse field does not change sign to steepen into shocks.

The next section develops a symmetrical treatment of the Rayleigh, Fanno and other processes of ordinary steady one-dimensional gasdynamics, generalized to allow for electromagnetic effects. The slow, fast and Alfvén wave speeds are critical. A particular case of this analysis is a generalization of ordinary gasdynamics which allows for the effect of transverse forces such as occur in turbo-machinery.

The final section is an exhaustive study of shocks in the presence of a field component normal to the shock front. From the generalized Rayleigh line it is established that there are up to six different types of shocks, all compressive and distinguishable by the relative magnitudes of the upstream and downstream normal velocities in comparison with the local slow, fast and Alfvén wave speeds.

Some aspects of shock structure are discussed briefly.

A Theory is developed, based on the very limited available experimental evidence, to predict the distortion and disintegration of a water drop when it is exposed to a stream of air with continuously increasing relative velocity. The theory is applied to water drops situated in the path of a solid sphere moving through the air.

Coriolis terms are introduced into the equations governing the motion of a finite tidal wave. Various types of solution are found, all of which travel without change of shape and some which are periodic with sharp crests and broad troughs. The classical result that such waves cannot be propagated without change of shape is therefore untrue in these circumstances.

The kinematic approach to group velocity given in Lighthill & Whitham (1955) for one-dimensional waves is extended to cover the general three-dimensional case. The ideas have particular bearing on the theory developed by Ursell (1960) for treating steady wave patterns on non-uniform steady fluid flows.

Although this note was written in ignorance of the fact, all the main ideas presented here are implicit in §§66 and 67 of the book by Landau & Lifshitz (1959). However, these ideas do not seem to be well known to fluid dynamicists, and it was suggested to the author by the editor that a useful purpose would still be served by publishing this note as an expository article amplifying the paragraphs in Landau & Lifshitz. It also serves the original purpose of providing a supplement to Ursell's paper.

It is supposed that a heated liquid is rising very slowly through a semi-infinite porous medium towards the permeable horizontal surface, where it mixes with a layer of cool overlying fluid. In the steady state a thermal boundary layer of exponential form exists in the medium. It is shown that the layer is stable provided that the Rayleigh number for the system does not exceed a critical positive value, and that the wave-number of the critical neutral disturbance is finite. The stability properties of the layer are explained qualitatively from physical considerations.

Experimental data are presented for the growth of vapour bubbles in various superheated liquids, such as carbon tetrachloride, benzene, ethyl alcohol, and methyl alcohol. These data are compared with the theoretical results obtained by Plesset & Zwick (1953) who derived these results by taking into account the heat diffusion across the bubble boundary. The agreement in all cases between experiment and theory is found to be good.

The growth of vapour bubbles in slightly superheated water is also presented in the form of experimental data for bubbles just beginning to grow from a point of equilibrium which is presumed to be dynamically unstable. The radii corresponding to the points of equilbrium are of the same order of magnitude as those predicted by theoretical considerations.

The structure of the steady magnetohydrodynamic switch-on shock wave is investigated for several orderings of the four diffusivities involved in the problem. The various orderings are approximated to by allowing one or more of the appropriate diffusivities to approach zero, and approximate solutions that are uniformly valid to order unity are sought. In general, singular perturbation problems are encountered, the number occurring (from zero to a maximum of three) depending upon the ordering of the diffusivities and the magnitude of the downstream velocity normal to the shock relative to certain critical velocities downstream of the shock. Where necessary, the approximate solutions are rendered uniformly valid to first order by the insertion of boundary layers, for which the approximate equations are determined to first order. For most of the cases considered, the limiting forms of the integral curves are determined and they are sketched in appropriate three-dimensional phase spaces.

The theoretical study presented in this paper was inspired by the recent report (Krämer 1960) of experiments showing that considerable reductions in the drag of an underwater solid body were achieved by covering it with a skin of flexible material; apparently this effect was due to the boundary layer being stabilized in the presence of the skin, so that transition to a turbulent condition of flow was prevented or at least delayed. The stability problem for flow past a flexible boundary is here formulated in a general way which allows a full exploration of the possibility of a stabilizing effect without the need to assign specific properties to the flexible medium; the collective properties of possible boundaries are represented by a ‘response coefficient’ α (a sort of ‘effective compliance’) measuring the deflexion of the surface under a travelling sinusoidal distribution of pressure.

A remarkably simple analytical connexion is established between the present general problem and the corresponding stability problem for the boundary layer on a rigid plane wall, and hence many details of the existing theory of hydrodynamic stability are immediately useful. However, the presence of the flexible boundary admits possible modes of instability additional to those which already exist when the boundary is rigid, and clearly every mode must be considered with regard to practical measures for stabilization—that is to say, it might be useless to inhibit one mode by a device which lets in another. What is believed to be an essentially complete interpretation of the over-all possibilities is deduced on recognizing three more or less distinct forms of instability. The first comprises waves resembling the unstable waves which can arise in the presence of a rigid boundary, but now being modified by the effects of flexibility. These waves tend to be stabilized when the boundary has a compliant response to them, which means the respective wave velocity has to be less than the velocity of free surface waves on the boundary; but it is found that the effect of internal friction in the flexible medium is actually destabilizing. The second form of instability is essentially a resonance effect and comprises waves travelling at very nearly the velocity of free surface waves. These waves can only be excited when the latter velocity falls below the free-stream velocity; they are scarcely affected by the viscosity of the fluid since the ‘wall friction layer’ is largely cancelled, so that damping due to the medium itself becomes the only stabilizing factor. The third form is akin to Kelvin–Helmholtz instability.

This interpretation of the theoretical results seems to point to the essential factors in the operation of a flexible skin as a stabilizing device, and accordingly in the concluding secttion of the paper two alternative sets of criteria are proposed each of which would provide a logical basis for designing such a device. The principle of the first alternative explains the success of Gamer's invention, but the second appears equally promising and the relative advantages of the two can really be proved only by further experiment.

This paper is concerned with the non-linear interactions between pairs of intersecting gravity wave trains of arbitrary wavelength and direction on the surface of water whose depth is large compared with any of the wavelengths involved. An equation is set up to describe the time history of the Fourier components of the surface displacement in which are retained terms whose magnitude is of order (slope)2 relative to the linear (first-order) terms. The second-order terms give rise to Fourier components with wave-numbers and frequencies formed by the sums and differences of those of the primary components, and the amplitudes of these secondary components is always bounded in time and small in magnitude. The phase velocity of the secondary components is always different from the phase velocity of a free infinitesimal wave of the same wave-number. However, the third-order terms can give rise to tertiary components whose phase velocity is equal to the phase velocity of a free infinitesimal wave of the same wave-number, and when this condition is satisfied the amplitude of the tertiary component grows linearly with time in a resonant manner, and there is a continuing flux of potential energy from one wave-number to another. The time scale of the growth of the tertiary component is of order of the (−2)-power of the geometric mean of the primary wave slopes times the period of the tertiary wave. The Stokes permanent wave appears as a special case, in which the tertiary wave-number is the same as that of the primary, but its phase is advanced by ½π. The energy transfer to the tertiary component in this case is usually interpreted as an increase in the phase velocity of the wave.

The dynamical interactions in water of finite depth are considered briefly, and it is shown that the amplitude of the secondary components becomes large (though bounded in time) as the water depth becomes smaller than the wave-length of the longest primary wave.

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.

It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.

Two mechanisms by which a viscous fluid can be deposited on a plane surface are described. Measurement of the thickness of the deposit are compared with calculated values. It is found that the two agree within rather wide limits of experimental error provided the effect of surface tension can be neglected, and the conditions under which this is legitimate are discussed.

A transformation has been found which reduces steady non-homentropic flows of a compressible fluid to homentropic flows, provided diffusive and gravity effect are negligible. With this transformation, the equation governing steady two-dimensional flows of a compressible fluid with variable entropy in a gravitational field is derived, which is then applied to the study of atmospheric waves in the lee of mountains. The corresponding equation governing swirling axisymmetric flows is also given.

It is shown that for the steady isoenergetic rotational flow of an ideal gas, both the specific enthalpy and the speed of sound can be expressed as functions of the velocity. As a result, it is possible to formulate the equations of motion so that the velocity is the only dependent variable. For a gas whose enthalpy and sound speed are functionally related, the results are a generalization of those for a perfect gas. If the enthalpy and sound speed are independent variables, the new formulation leads to a single vector equation whose solution completely determines the flow.

In Part 1 by Stuart (1960), a study was made of the growth of an unstable infinitesimal disturbance, or the decay of a finite disturbance through a stable infinitesimal disturbance to zero, in plane Poiseuille flow, and that paper gave the most important terms in a solution of the equations of motion. The greater part of the present paper is concerned with a re-formulation of this problem which readily yields the complete solution. By the same method a solution for Couette flow is obtained. This solution is only a formal one for the present because the conditions imposed in deriving the solution may not be valid for Couette flow; this flow is believed to be stable to infinitesimal disturbances of the type considered.