Harmonic Decay Research Articles

A numerical study of the divergence of spherical harmonic series of the gravity and height anomalies at the earth's surface

The problem of the divergence of the geopotential spherical harmonic series at the earth's surface is investigated from a numerical, rather than a theoretical, approach. A representative model of the earth's potential is devised on the basis of a density layer, which, in the spherical approximation, generates a gravity field whose harmonic constituents decay according to an accepted degree variance model. This field, expanded to degree 300, and a topographic surface specified to a corresponding resolution of 67 km are used to compute the differences between truncated inner and outer series of the gravity and height anomalies at the surface of the earth model. Up to degree 300, these differences attain RMS values from 0.33 μgal to 86 μgal for the gravity anomaly and from 0.32 μm to 410 μm for the height anomaly, in areas ranging respectively from near the equator to the vicinity of the pole. In addition to these values, there is an expected truncation effect, caused by the neglect of higher degree components of the inner series, of about 30 mgal and 36 cm, respectively. The field is then subjected to a Gaussian filter which effectively cuts off information at degree 300 (at the 5% level). The RMS error to degree 300 is thereby reduced by factors of 10 to 20, with a concomitant reduction in the truncation effect to about 0.3 mgal and 0.7 cm.

Asymptotic decay of periodic spherical waves in dissipative media

In studies of plane periodic waves of finite amplitude in dissipative media, it has been found that if nα1 < αn, the nth harmonic pressure component pn, decays asymptotically as exp(−nα1x), not as exp(−αnx), where αn is the small signal absorption coefficient for the nth harmonic component. Since generally nα1 < αn, this means that the nonlinearly generated nth harmonic decays more slowly than would be expected from linear theory. Moreover, save for the fundamental, no distance is ever reached at which the linear theory decay is established. In this paper the spherical wave problem is addressed. The results are based on an asymptotic solution of Burgers' equation. It is found that for both weak and strong waves the harmonics decay as exp(−nα1r)/rn. The apparent absorption is the same as in the plane wave case. The apparent geometrical spreading, however, is r−n, not r−1. The asymptotic decay is still, however, slower than would be expected on the basis of linear theory. [Work supported by NASA, AFOSR, and ONR.]

Periodic solutions of a nonlinear diffusion equation used in groundwater flow theory: Examination using a Hele-Shaw model

A Hele-Shaw experiment which models groundwater flow is described. The experiment confirms the Philip prediction that in steady periodic nonlinear diffusion the root mean square of the oscillation, rather than the mean is constant. For the weakly nonlinear case examined, however, it is found that the logarithm of the amplitude of the oscillation still diminishes in proportion to xT − 1 2 , so from a practical viewpoint the hydrological problem could be treated as a linear one provided the displaced mean were taken into account. The anomalous appearance and subsequent decay of harmonics in the initially sinusoidal wave are discussed.

Resonance capture in certain nearly Hamiltonian systems

The difFerentia1 equation fi = F(t, 8, e), in which 0 is an angular variable and F is small and periodic with period 2rr in t and 0, represents a material body spinning about a fixed axis subject to a small, periodic, not necessarily conservative torque. Mathematically the equation is a paradigm case of non. . linear oscillations. The equation 19 = 0, of which it is a perturbation, is formally linear but exhibits the characteristic behavior of a nonlinear integrable Hamiltonian system because Q is an angular variable, The solutions e=wt+e,, where w and 0, are initial conditions, are geometrically periodic with period 297/w, and dependence of the period upon the amplitude or initial conditions, as seen in the unperturbed Duffing equation 1+ cw: + /5’S = 0, is the chief property of nonlinear oscillations. When 0 == 0 is perturbed to a nonintegrable Hamiltonian system, the results of Poincare-Birkhoe and Kolmogorov-Arnol’d-Moser apply and a complicated global picture results, which however has the simple consequence that the angular velocity w = 8 is not subject to very much “drift” along any orbit. This is due to the presence of invariant curves surrounding the cylindrical phase space of the variables (0, w) near circles w = constant for which w is “badly irrational..” At the other extreme, when the perturbation contains substantial friction, all orbits may tend to a single limit cycle on the cylinder, with cu fluctuating around a single final value. The cases studied in this paper lie between these two extremes, when the perturbation consists of a small Hamiltonian part and a still smaller friction. The problem is then to determine the possible “final values” of w (in a sense to be made precise later), and show how their number increases as the friction goes to zero. The delicacy of the problem is seen in the fact that the result depends upon the higher harmonics of 0 present in F, and that no matter how rapidly the coefficients of these harmonics decay, their presence will be felt when the friction is small enough. The application motivating this work is to planetary spin resonance, in

A theoretical study of helicopter rotor noise

A comprehensive theoretical study of the problem of helicopter rotor noise radiation is presented. The theory includes blade slap, rotation noise and vortex noise effects. Peak spectral levels over the “vortex noise” region are shown to be due to the higher harmonics of the rotational noise. An exact theoretical expression for the noise radiation has been used as the basis for the development of a comphrehensive computer program to calculate helicopter noise at any field point, including all effects of fluctuating airloads and all possible rigid and flexible blade motions. Under very reasonable approximations an analytic expression has been found for the sound field far from the helicopter. Computations based on this expression have also been made. The results show that it is the very high harmonics of the loading which contribute to the important harmonics of the sound field. For instance, calculation of the tenth harmonic of a four-bladed rotor requries a knowledge of loading harmonics up to the sixtieth. Details of such loadings are not available from theory or experiment. Rotor aerodynamic loadings have therefore been reviewed in detail and empirical harmonic decay laws derived. Loading phases appear to be best described as random, and this introduces simplification in the theory, together with the necessity for definition of a correlation length. Results of a parameter study show trends in agreement with experiment, both for overall levels and for spectrum shape. Sound at the higher harmonics is basically proportional to thrust times disc loading times tip velocity squared. For the lower harmonics the dependence on tip velocity is to the 2B power where B is the number of blades. The effect of forward speed is to increase the sound radiated forward and decrease that radiated aft, causing a difference of as much as 20 dB for the second and third harmonics at a forward Mach number of 0·25. The overall sound directionality pattern is found to have a minimum slightly above the plane of the disc and a broad maximum about 20° below. The effects of both the near field and blade motion effects are found to be small. The theory generally shows fair agreement with experiment for overall levels and good agreement for trends, and should therefore be of direct use for design trade-off studies to minimize noise in future helicopters.

Measurement of the Distortion of Finite Ultrasonic Waves in Liquids by a Pulse Method

An rf pulse system is described for investigating progressive distortion of finite amplitude waves propagating in liquids. The system employs pulses short enough to avoid reflections but long enough to approach continuous wave conditions. A 25-Mc quartz crystal backed by fused quartz serves as a nonresonant receiver for the first few harmonics of the 2.58-Mc working frequency. Harmonic content is determined by Fourier analysis of waveform photographs, or by use of a heterodyne analyzer having bandwidth adequate to pass the pulse envelope. For an initially sinusoidal wave of 3.0 or 6.0 atm pressure amplitude in water, curves are shown tracing the decay of the fundamental and the growth and decay of harmonics through the fourth, to a propagation distance of 40 cm. Results are in good agreement with published theory within the region of convergence of the latter.

The Generation of Harmonics by Small Circular Orifices

A spectral analysis was made of the sound field emanating from a small orifice set flush in the top of a table which acted as an infinite baffle. The orifice was driven at a single frequency at various intensities by an electromagnetic horn exciter, coupling through a small cavity below the baffle. At high intensities all harmonics up to the fortieth have been detected. Starting at a distance of one radius from the center of the orifice, the fundamental component of the sound pressure is observed to decrease at a rate of six decibels per distance doubled, indicating that the orifice acts as a spherical source whose radius is that of the orifice. The higher harmonic components decay at the same rate as the fundamental indicating that, for the levels considered, essentially no integrative interactions occur between the harmonics while progagating away from the orifice. Thus it appears that harmonic generation occurs locally at the orifice. The successive odd harmonics are invariably stronger than the immediately preceding even harmonics. Even at moderate levels, the third harmonic may lie less than five decibels below the fundamental component. A theory based upon the non-linear wave equation in spherical coordinates has been developed which accounts for the first two harmonics semiquantitatively. No account is taken of turbulence or rotational terms in the theory.