The effect of giving an infinite plate, immersed in a compressible fluid, an impulsive velocity U in its own plane is examined on the assumption that M = U/a0 ≫ 1 where a0 is the velocity of sound in the undisturbed fluid. To begin with the flow field is very complex but very soon after the motion has begun a shock wave moves at right angles to the plate gradually weakening and eventually degenerating into a sound wave. Attention is concentrated on the flow field when there is a distinct strong shock wave, it being assumed, and justified a posteriori, that the flow behind it may be divided into an inviscid region and a boundary layer region. The equation of the shock wave and a complete description of the flow behind it is found when log M, where ν0 is the coefficient of kinematic viscosity of the undisturbed air. An approximate method is also given to enable the solution to be joined up to Van Dyke's (10) which is valid when .

In this paper there is considered the irrotational motion of an infinite fluid when the normal velocity across a plane is specified as a stationary random function of position in the plane, and a solution is obtained in terms of the specified boundary conditions. It is shown that the mean square velocity normal to the plane is equal to the sum of the mean squares of the velocities in the other two orthogonal directions. The asymptotic variations with distance normal to the plane are found for functions representing the important properties of the motion, and, in particular, the energy of the fluctuations is shown to be inversely proportional to the fourth power of the distance from the plane. The conditions postulated are shown to correspond closely to the motion outside a free turbulent boundary, and good agreement is found between the predictions of the theory and the available experimental results.

The homomorphism of the title is a homomorphism of πr(Sn) into πr+n−1(Sn) given by

where ι generates πn(Sn). Various results on the P-homomorphism were proved in (4); in particular it was shown that, if ηn generates πn+1(Sn), n ≥ 2, then