Rayleigh Approximation Research Articles

Numerical Computations of Elastic Scattering Cross Sections

A brief review is given of the calculation of cross section expressions for the scattering of an elastic wave in an elastic medium. With scatterers that are elastic and spherical in shape, numerical calculations are carried out for non-Rayleigh conditions of scattering. Comparisons of such cross sections are made with those obtained from the Rayleigh approximation. The deviation from Rayleigh behavior becomes pronounced, for the elastic scatterer, beyond ka≅0.5. In addition a ``peaking'' and ``fluctuating'' of the cross section occurs beyond this point.

RADAR CROSS-SECTIONS FOR WATER AND ICE SPHERES

Radar cross-sections for water and ice spheres, evaluated by using the complete Mie theory and the method of Aden, are presented for wavelengths ranging from 0.43 to 16.23 cm. The validity of the Rayleigh approximation is shown as a function of drop diameter, and the temperature dependence is shown for a wavelength of 3.2 cm. Oscillations of the normalized cross-section for water are explained in terms of multipole resonance effects.

The flow past a flat plate of finite width

An experimental study of the incompressible flow near the side edge of a finite flat plate at zero incidence is reported for the Reynolds number range 104 to 106, but comparison with data already published shows that the conclusions are quantitatively valid for Reynolds numbers up to 109. The laminar velocity field is everywhere convex and does not contain any secondary flow other than that of the normal diffusive growth of the layer, but has a logarithmic singularity at the edge where the stress is controlled by the local radius of curvature. The excess skin friction due to the edge is considerably greater than that given by calculations based on the Rayleigh approximation but agrees with a recent Pohlhausen calculation by Varley (1958). Near the edge the high stress and ease of momentum diffusion makes the flow very unstable and turbulent spots occur well upstream of the normal transition zone in the middle of the plate. The spots originate from a nearly point-like region at the edge of the plate and grow linearly in time at the same rate as ordinary turbulent spots to sweep out a narrow tongue of turbulent fluid near the edge until they merge with the normal transition zone. Within this tongue a weak secondary flow of Prandtl's second kind, driven by the anisotropic Reynolds stresses, begins to develop. In fully turbulent flow when the secondary flow is largely localized to within a few boundary-layer thicknesses of the edge the secondary flow velocities are everywhere less than 0.04 of the free-stream velocity. Nevertheless, the secondary flows from each of the side edges interact, regardless of the width of the plate, to increase the total drag coefficient by an amount 0·0004 which is independent both of the Reynolds number and the width of the plate except when it is very narrow. This simple result allows apparent discrepancies between various formulations of the drag coefficient of a finite plate to be reduced to less than ± 1% S.D. Of these formulations Schoenherr's (1932) empirical relation agrees best with the present data.

Scattering of a Plane Transverse Wave by a Spherical Obstacle in an Elastic Medium

An analysis of the scattering of transverse elastic waves by spherical obstacles is presented. The scatterer is taken to be (a) a cavity, (b) a rigid sphere, (c) a fluid-filled cavity, and (d) to consist of an elastic material with properties different from those of the surrounding material. The problems are carried as far as possible analytically without approximations and are reported as matrix equations. The solution of these equations yields the expansion coefficients that describe the waves which are scattered outward from the obstacle and which are excited within the scatterer. A general expression for the scattering cross section offered to a transverse wave has been derived. The Rayleigh approximation is then considered in detail for three of the cases.

“On the Wave Pressure Acting on Fixed Cylindrical Body” (Second Report)

Following to the preceding report, the author has solved numerically the wave pressure problem acting on parabolic waterline ships with infinite draft fixed amongst waves.Firstly, the steady drifting force has been obtained when the wave propagates at right angle to the ship's logitudinal center line. (Fig. 4, Table 3.) Secondly, the surging force has been obtained in these cases, that is, when the wave propagates parallel to and at right angle with the ship's longitudinal center line. (Fig. 8) For this calculation the wave height around the ship surface is necessary and calculated. (Fig. 5, 6) When the wave length tends to infinity we are to remind a virtual mass effect of the ship, remembering Rayleigh's approximation in such case of other wave problem.It is observed that the consideration for this connection leads us to the appropriate result of the surging force in the range of fairly large wave length.

The flow of conducting fluids in circular pipes under transverse magnetic fields

The flow rate of liquid metals is commonly measured by electromagnetic flowmeters. In these the fluid moves through a region of transverse magnetic field, inducing a potential difference between two electrodes on the walls of the pipe. The ratio of signal to flow rate is dependent on the velocity profile, and this is affected by electromagnetic forces.In this paper the ultimate steady velocity profile and its associated pressure gradient and induced potential are calculated for the case of laminar flow in a circular pipe whose walls are conducting but without contact resistance. Laminar flow is encouraged by a transverse field. When the fluid conductivity and field strength are sufficiently high, boundary layers occur with a thickness inversely proportional to normal field intensity. The induced potential difference is then 0·926 of the value corresponding to the case of uniform velocity if the walls are non-conducting.The distance the fluid must travel after entering the transverse field before the steady state is reached is next estimated by a Rayleigh approximation. The inlet velocity is taken to be uniform and effects which occur at the edge of the field are neglected. The process falls into two stages, first a boundary-layer growth and then an adjustment of the velocity away from the walls, occupying a much greater length of pipe. The entry length is shorter than it is in the case of flow in a rectangular pipe, but is still too long for appreciable distortion of the velocity profile to occur within practical flowmeters except at low flow rates. The pressure drop associated with the adjustment of the velocity profile is found to be independent of field strength, if this is high, and about one-eighth of the drop which occurs in the non-conducting case.Experiments are described in which steady-state pressure gradients and induced potential differences were measured in mercury flowing along Perspex pipes of 0·5 and 0·25 in. bore in transverse fields up to 14500 gauss. The results confirmed the steady-state theory within the limitations of experimental accuracy and the assumption in the theory of high conductivity and an intense field. The experiments also covered the entry region in many cases, and showed that the theoretical entry lengths were correct in order of magnitude but over-estimated. However, the exact entry condition was uncertain, and steady readings were difficult to obtain in the entry region.

Entry of conducting and non-conducting fluids in pipes

ABSTRACTThe problem of estimating the entry length over which the laminar motion of an electrically conducting, viscous fluid in a pipe under a transverse field adjusts to a steady state from some initial state is linearized by a Rayleigh approximation adapted for cases where there is a pressure gradient. Solutions are found which express the difference between the initial and ultimate velocity profiles as a series of orthogonal, exponentially decaying terms. Approximate entry lengths for the most persistent terms are calculated for flow between two infinite parallel planes and in rectangular tubes when magnetic effects are present. Entry lengths in circular and rectangular pipes and between infinite planes are calculated in the case where magnetic effects are negligible to permit a comparison of the results with those derived by other methods. Pressure losses at entry are considered in the simpler cases.

The transmission of Rayleigh waves across Eurasia*

Summary The problem of the propagation of Rayleigh waves in a double surface layer, although mathematically tractable, would involve very lengthy computations, scarcely justified by the rather inaccurate data available. An approximate treatment by “Rayleigh's principle” gives, for any postulated ratio of the thicknesses of the layers, an overestimate of these thicknesses which needs reduction by 10 or 15 per cent. Applied to W. Rohrbach's data on the velocities of transmission (i.e., the group velocities) of Rayleigh waves of known period, this method yields values which are consistent with those previously obtained for Eurasia from Love waves. For a single uniform granitic layer of thickness T1 resting on uniform ultrabasic material of great depth the Rayleigh-wave velocities from a track from the Kwen Lun Mountains to Göttingen give T1 = 39.4 km., without use of the Rayleigh approximation. This may be compared with the estimate of 34.6 km. obtained on slightly differing hypotheses from Love waves. If the granitic layer is underlain by a uniform layer of basic rock of thickness T2, some hypothesis must, in our present state of knowledge, be made concerning the ratio T2: T1. For Rayleigh waves the Kwen Lun earthquake gives, on the two hypotheses T2 = T1 and T2 = 2T1, the values 27 km. and 19 km. respectively for T1. Reduced by, say, 15 per cent these values are 23 km. and 16 km., which may be compared with the estimates 22.1 km. and 17.6 km. derived on the same data from Love waves. For a shock in Eastern Bengal the Rayleigh waves (both vertical and E-W components) give 30 km. and 25 km. on the two hypotheses, that is, “reduced values” of 25.5 km. and 21 km., which exceed the Love-wave estimates of 22.1 km. and 17.6 km. quoted above. However, these tracks cross the Himalayas and the excess values are not surprising. There is evidently scope for further investigations along these lines.

The fundamental frequency of vibration of rectangular wood and plywood plates

Equations are derived expressing the frequency of vibration of rectangular wood and plywood plates in terms of elastic constants, dimensions and density. For supported edges, the solution is exact and gives the complete series of overtones. For clamped edges, the fundamental frequency is estimated by the approximate Rayleigh method, but evidence is presented which indicates that the errors introduced by the use of the Rayleigh method are, in fact, negligible. A method of measuring the frequencies is described; comparison of calculated with observed frequencies shows that the former are on the average about 23% high for clamped edges and about the same amount low for supported edges. This discrepancy is attributed to lack of correspondence between the experimental edge conditions and those assumed in the theoretical treatment. Finally, an empirical method, based on the rough proportionality between elastic constants and density of wood, is suggested and justified for obtaining approximate estimates of the fundamental frequencies in terms of plate dimensions only.

Skin Effect in Solenoids

SKIN effect in long, single layer solenoids wound with solid round wire and used at very high frequencies has been treated by Sommerfeld,1 Lenz,2 and Abraham, L. Bloch, and E. Bloch.3 (The frequency is supposed so high that the Rayleigh approximation applies.)