Propagation of surface waves on a nonhomogenous spherically aeolotropic shell

Summary This paper studies the propagation of Surface Waves on a spherically aeolotropic shell surrounded by vacuum. The elastic constantsc _ij and density of the material of the shell are assumed to be of the form _ij r ^l and _o r ^m respectively, where _{ij o} are constants andl, m are any integers.     

Propagation of surface waves on the surface of a non-homogeneous aeolotropic cylindrical shell

Summary The object of the present paper is to investigate the propagation of surface waves on a non-homogeneous aeolotropic cylindrical shell surrounded by vacuum. The elastic constantsc _ij (i, j=1,2...) and density of the material of the shell are assumed to be of the form and respectively, where _ij, ₀ are constants andk ₁,k ₂ are any integers.     

La risoluzione dell'equazione di Laplace nel campo esterno a uno sferoide

Riassunto L'Autore dimostra che, nel sistema di coordinate polari , , , si possono determinare un numeros di funzioni della sola variabile :Q ₁,Q ₃, ....Q _2s–1 tali che la sommatoria delleQ _2i–1/^2i–1 rappresenti il potenzialeV di un geoide di rotazione. La condizione di armonicità determina ciascunaQ (che si riduce a un polinomio nelle potenze di sen ) a meno di una costante arbitraria; si dispone pertanto dis costanti che servono per soddisfare la natura dellaV sulla superficie del geoide. Come esempio l'Autore ha determinato la gravità sul geoide sferico, confermando i risultati delSomigliana, e su uno sferoide generico dove ha ritrovato la relazione diClairaut.

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Summary The Author proofs that, in the system of polar coordinates , , , it is possible to determine a numbers of functions only of the variable :Q ₁,Q ₃ ....Q _2s–1 in such a way as to make the summatory of theQ _2i–1/^2i–1 represent the potential function of a rotational geoid. The condition of harmonicity determines, saving an arbitrary constant, each of theQ which is reduced to a polynom developed by the sin powers; therefore one disposes of a number of constants to make use for satisfing theV on the geoid. To illustrate his theory the Author determines the gravity on the spherical geoid, thus confirmingSomigliana's formulas and on a spheroidal on which he pointed outClairaut's relations.

     

Physical parameters governing the formation of Pele's hair and tears

The physical parameters that affect the formation of Pele's hair and Pele's tears during lava fountaining are discussed. Experiments on ink jets produced from a nozzle under different Weber number (We) and Reynolds number (Re) show the following results: if (Re) is relatively large compared with (We), an ink droplet is produced. However, if (Re) is relatively small and (We) is large, the spurting ink becomes thread-like. I define the Pele number (Pe) as (We)/(Re), which is expressed as v/₀, where v is the spuring velocity from an erupting vent, and are viscosity and interfacial tension of the erupting magma, and and ₀ are density of air and magma. The experimental results from ink jets suggest that Pele's hair will be produced for larger (Pe), while Pele's tears are very likely produced for relatively small (Pe). I conclude that Pele's hair could be produced when the spurting velocity of erupting magmas is high, and Pele's tears when it is relatively low. As an additional point of interest, the similarity of SEM photographs of the characteristic shape of Pele's hair to those of the failed products of commercial glass fibre are shown.     

Reservoir lifetime and heat recovery factor in geothermal aquifers used for urban heating

Simple models are discussed to evaluate reservoir lifetime and heat recovery factor in geothermal aquifers used for urban heating. By comparing various single well and doublet production schemes, it is shown that reinjection of heat depleted water greatly enhances heat recovery and reservoir lifetime, and can be optimized for maximum heat production. It is concluded that geothermal aquifer production should be unitized, as is already done in oil and gas reservoirs.Nomenclature a distance between doublets in multi-doublet patterns, meters - A area of aquifer at base temperature, m² drainage area of individual doublets in multidoublet patterns, m² - D distance between doublet wells, meters - h aquifer thickness, meters - H water head, meters - Q production rate, m³/sec. - r _e aquifer radius, meters - r _w well radius, meters - R _g heat recovery factor, fraction - S water level drawdown, meters - t producing time, sec. - T aquifer transmissivity, m²/sec. - v stream-channel water velocity, m/sec. - actual temperature change, °C - theoretical temperature change, °C - water temperature, °C - heat conductivity, W/m/°C - _r rock heat conductivity, W/m/°C - _aC_a aquifer heat capacity, J/m³/°C - _aC_r rock heat capacity, J/m³/°C - _WC_W water heat capacity, J/m³/°C - aquifer porosity, fraction     

Formation of cracks due to moving source over a thick crust

Summary In this paper the problem of a point source of stress moving over the surface of a thick aelotropic plate resting of a rigid foundation has been considered. Following the method ofAleksandrov & Vorovich (1960) the stress componentsZ _x andZ _z have been expanded in series of ascending powers of 1/h when the source velocity is less than (c ₄₄/)^1/2. When the velocity exceeds (c ₄₄/)^1/2 it has been shown that two cracks are produced in different directions and their successive reflections at the upper and lower surface are also obtained.     

Su unaVexata quaestio di gerarchia nei dispositivi dei sondaggi elettrici

Riassunto L'A. mostra corne dal metodo più semplice di misura di rcsistività d'un suolo del polo singolo, si possano dedurrc tutti gli altri (Wenner, Schlumberger, ecc). CiÒ dà la possibilità di risalire immediatamente dai grafici _s di un dispositivo a quelli di un altro. L'A., dopo rettificato recenti valutazioni del metodo dei due punti, rimuove la pretesa da altri formulata di porre gerarchie nei dispositivi di misure _s .

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Summary The Author shows how departing from simplest method of the ground resistivity measurement with the single pole can be deduced all another methods (Wenner, Schlumberger, etc.). That gives the practical possibility to pass immediately the _s graphics of one arrangement to these of other types of arrangement. After having rectified recently mades evaluations about the two points method, the Author removes the requirements made from other side to graduate the _s measuring arrangements.

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Zusammenfassung Der Verfasser zeigt wie man von der einfachsten Methode der Widerstandsmessung eines Bodens mit Einzelpol alle anderen Methoden (Wenner, Schlumberger, usw.) ableiten kann. Dadurch hat man die praktische Möglichkeit, von den _s Diagrammen einer Anordnung sofort auf jene anderer Anordnungen überzugehen. Nach Richtigstellung von kürzlich geäusserten Ansichten über die Zweipunktmethode, beseitigt der Verfasser die von Anderen gestellten Ansprüche, die _s Messanordnungen vermeintlichen Vorzügen nach staffeln zu wollen.

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Relazion e presentata il 5 Aprile 1956 alla Quarta Assemblea Generale dcllaSocietà Italiana di Geofisica e Meteorologia (Genova: 5–8 Aprile 1956).     

Torsional vibration of non-homogeneous isotropic hollow thick cylinder

Summary Free vibration of a thick hollow cylinder of non-homogeneous elastic material having rigidity and density varying according to the laws: (i) =₀(₀+₀ z) and Q=Q₀(Q₀+₀ z) for finite and infinite length of the cylinder and (ii) ₁/r = and Q = ₁/r for finite length only are considered in this paper. Frequency equations are deduced and respective numerical solutions are obtained.     

Planar charged-particle trajectories in multipole magnetic fields

This paper provides a complete generalization of the classic result that the radius of curvature () of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particles equatorial distance () from the dipole (i.e. ³). Comparable results are derived for the radii of curvature of all possible planar chargedparticle trajectories in an individual static magnetic multipole of arbitrary order m and degree n. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (m = 0), or zonal, magnetic multipole, provided n is odd: the radius of curvature varies directly as ⁿ⁼². This result reduces to the classic one in the case of a zonal magnetic dipole (n = 1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 < m < n) magnetic multipole. These meridional planes are defined by the 2m roots of the equation cos[m()–_n^m)] = 0, where _n^m = (1/m) arctan (h_n^m/g_n^m); g_n^m and h_n^m denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (n – m) is odd. The polar axis ( = O,) of a tesseral magnetic multipole is a magnetic neutral line if m > I. A further 2_m(n – m) neutral lines exist at the intersections of the 2_m meridional planes with the (n – m) cones defined by the (n – m) roots of the equation P_n^m(cos ) = 0 in the range 0 < 9 < , where P_n^m(cos ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ⁿ⁼² and inversely proportional to cos[m(–)]. Since this last expression vanishes at the 2m equatorial neutral ines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to rⁿ⁺², where r denotes radial distance from the multiple, and inversely proportional to P_n^m(cos )/sin . Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral ine (m > 1) or any one of the 2m(n – m) neutral ines located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are pressented for low-degree (n 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the non-adiabatic scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.Also Visiting Reader in Physics, University of Sussex, Falmer, Brighton, BN1 9QH, UK     

A comparison of two bivariate extreme value distributions

There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, namely Gumbel mixed (GM) model and Gumbel logistic (GL) model. These two models have completely different structures and their dependence ranges are different. The product-moment correlation coefficient for the former is [0,2/3] and the latter is [0,1]. It is natural to ask which one is more appropriate for representing the joint probabilistic behavior of two correlated Gumbel-distributed variables. This study compares these two models by numerical experiments. The comparison is based on that: (i) if the two distribution models are identical, then the joint probability and the joint return period computed by the GM model should be the same as those by the GL model; and (ii) if a selected distribution is the true distribution from which sample data are drawn, then the probabilities computed by the theoretical model should provide a good fit to empirical ones. Comparison results indicate that in the range of correlation coefficient [0,2/3], both models provide identical joint probabilities and joint return periods, and both indicate a good fit to empirical probabilities; while for (2/3,1), only the Gumbel logistic model can be used.     

Water uptake and equilibrium sizes of aerosol particles at high relative humidities: Their dependence on the composition of the water-soluble material

Equilibrium water uptake and the sizes of atmospheric aerosol particles have for the first time been determined for high relative humidities, i.e., for humidities above 95 percent, as a function of the particles chemical composition. For that purpose a new treatment of the osmotic coefficient has been developed and experimentally confirmed. It is shown that the equilibrium water uptake and the equilibrium sizes of atmospheric aerosol particles at large relative humidities are significantly dependent on their chemical composition.List of symbols A proportionality factor - a _w activity of water in a solution - c _{p v} specific heat of water vapour at constant pressure - c _w specific heat of liquid water - f relative humidity - l _w specific heat of evaporation of water - M _i molar mass of solute speciesi - M _s mean molar mass of all the solute species in a solution - M _w molar mass of water - m ₀ mass of an aerosol particle in dry state - m _i mass of solute speciesi - m _s mass of solute - m _w mass of water taken up by an aerosol particle in equilibrium state - m total molality=number of mols of solute species in 1000 g of water - m _i molality of solute speciesi - m _k total molality of a pure electrolytek - O(m ²) remaining terms being of the second and of higher powers ofm - p ⁺ standard pressure - p total pressure of the gas phase - p pressure within a droplet - p ₁,p ₂,p ₃ coefficients in the expansion of M - p _1i, p_2i, p_3i specific parameters of ioni - p _s saturation vapour pressure - p _w water vapour pressure - R _w individual gas constant of water - r radius of a droplet - r ₀ equivalent volume radius of an aerosol particle in dry state - T temperature - T ₀ standard temperature - T ₁ temperature of the pure water drop in the osmometer - v _w specific volume of pure water - z _i valence of ioni - _i relativenumber concentration of ioni in a solution - correction term due to the adsorption of ions at liquid-solid interfaces - activity coefficient of solute speciesi in a solution, related to molalities - I bridge current - T temperature difference between solution and pure water drop in the osmometer - exponential mass increase coefficient - _w specific chemical potential of water vapour - _w specific chemical potential of water - ⁰ _w specific chemical potential of pure water vapour - ⁰ _w specific chemical potential of pure water - ₀ density of an aerosol particle in dry state - _w density of pure water - surface tension of a droplet - ₀ surface tension of pure water, i.e., at infinite dilution of the solute - osmotic coefficient - _k osmotic coefficient of a solution of a pure electrolytek - _k osmotic coefficient of a solution of a mixed solute - _M fugacity coefficient of water vapour - ^s _{i=1 i} z ² _i This work is part of a Ph.D. thesis carried out at the Meteorological Institute of the Johannes Gutenberg-Universität, Mainz.     

A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres

A numerical study has been made of the heat transfer through a fluid layer with recirculating flow. The outer fluid surface was assumed to be spherical, while the inner surface consisted of a sphere concentrically or eccentrically located with respect to the outer spherical surface. The recirculating flow was assumed to be driven by a gas flow creating stress on the fluid's outer surface so that creeping (low Reynolds number) flow developed in its interior. The present study solves the Stokes equation of motion and the convective diffusion equation in bispherical coordinates and presents the streamline and isotherm patterns.Nomenclature a _i inner sphere radius - a _d outer sphere radius - A ₁ defined by equation (5) - A ₂ defined by equation (6) - B ₁ defined by equation (7) - B ₂ defined by equation (8) - c dimensional factor for bispherical coordinates - C constant in equation (4) - d narrowest distance between the two eccentric spheres - E ² operator defined by equation (1) in spherical coordinates and by equation (21) in bispherical coordinates - G modified vorticity, defined in equation (22) - G ^* non-dimensional modified vorticity, defined in equation (28) - h metric coefficient of bispherical coordinate system, defined in equation (18) - k _w thermal conductivity of water - K ₁ defined by equation (9) - K ₂ defined by equation (10) - N _Re Reynolds number=2a _dU/gn - N _Pe,h Peclet number=2a _dU/ - n integer counter - q heat flux - r radius - r ^* non-dimensional radius=r/a _d - S surface area - t time - t ^* non-dimensional time=t/a _d ² - T temperature - T _o temperature at inner sphere surface - T _a temperature at outer sphere surface - T ^* non-dimensional temperature;=(T–T _o)/(T_a–T_o) - u velocity - u _r radial velocity in spherical coordinates - u angular velocity in spherical coordinates - u radial velocity in bispherical coordinates - u angular velocity in bispherical coordinates - U free stream velocity - u _r ^* =u _r/U - u ^* =u /U - u ^* =u /U - u ^* =u /U Greek symbols a ₁ small displacement - vorticity, defined in equation (17) - ^* non-dimensional vorticity, defined in equation (27) - radial bispherical coordinates - _o bispherical coordinate of inner sphere - _a bispherical coordinate of outer sphere - angular coordinate in spherical coordinates - thermal diffusivity - _w thermal diffusivity of water - kinematic viscosity - angular bispherical coordinate - spherical coordinate - streamfunction - non-dimensional streamfunction for spherical coordinates, = /(U a _d ² ) - ^* non-dimensional streamfunction for bispherical coordinates, defined in equation (26)     

Determination of characteristics of steam reservoirs by Radon-222 measurements in geothermal fluids

The authors conducted a Rn²²² survey in wells of the Larderello geothermal field (Italy) and observed considerable variations in concentrations. Simple models show that flow-rate plays an important part in the Rn²²² content of each well, as it directly affects the fluid transit time in the reservoirs. Rn²²² has been sampled from two wells of the Serrazzano area during flow-rate drawdown tests. The apparent volume of the steam reservoir of each of these two wells has been estimated from the Rn²²² concentration versus flow-rate curves.List of symbols Q Flow-rate (kg h^–1) - Decay constant of Rn²²² (=7.553×10^–3 h^–1) - Porosity of the reservoir (volume of fluid/volume of rock) - ₁ Density of the fluid in the reservoir (kg m^–3) - ₂ Density of the rock in the reservoir (kg m^–3) - M Stationary mass of fluid filling the reservoir (kg). - E Emanating power of the rock in the reservoir (nCi kg _rock ^–1 h^–1). - P Production rate of Rn²²² in the reservoir: number of atoms of Rn²²² (divided by 1.764×10⁷) transferred by the rock to the mass unit of fluid per unit time (nCi kg _fluid ^–1 h^–1). - N Specific concentration of Rn²²² in the fluid (nCi kg^–1) - Characteristic time of the steam reservoir at maximum flow-rate (=M/Q)     

Humidity effects on gravitational settling and Brownian diffusion of atmospheric aerosol particles

The dependency on relative humidity of the settling velocity of aerosol particles in stagnant air and of the diffusion coefficient due to Brownian motion of aerosol particles was computed for six aerosol types and different particles sizes in dry state. The computations are based (1) on mean bulk densities of dry aerosol particles obtained from measurements or from the knowledge of the chemical composition of the particles, (2) on micro-balance measurements of the water uptake per unit mass of dry aerosol substance versus water activity at thermodynamic equilibrium, and (3) on measurements of the equilibrium water activity of aqueous sea salt solutions. The results show a significant dependence of the settling velocity and Brownian diffusion of aerosol particles on relative humidity and on the particle's chemical composition.Nomenclature A surface parameter of a particle - B surface parameter of a particle - c _L velocity of sound in moist air - C 1+Kn[A+Qexp(–B/Kn]=slip correction - D diffusion coefficient of a particle - D ₁ D(=1)=diffusion coefficient of a spherical particle - f P _w /P _we (T,P)=relative humidity (f=0 dry air,f=1 saturated air) - g acceleration due to gravity - g |g| - k 1.3804×10^–16 erg/°K=Boltzmann constant - Kn _L /r=Knudsen number of a particle - Kn _{0 0L} /r ₀=Knudsen number of a dry particle - m 4r ³/3=mass of a particle - m _L 4r ³ _L /3=mass of the moist air displaced by a particle - M mobility of a particle - M ₀ molar mass of dry air - M _w molar mass of water - Ma |u–u _L |/c _L =Mach number of the particles motion relative to the ambient air - n particle number per unit volume of air - P P ₀+P _w =pressure of the moist air - P ₀ partial pressure of the dry air - P _w partial pressure of the water vapour - P _we P _we (T,P)=equilibrium partial water vapour pressure over a plane surface of water saturated with air - Q surface parameter of a particle - r equivalent radius of a particle (radius of a sphere with the particles volume) - r ₀ equivalent radius of a particle in dry state - R 1+0.13Re ^0.85=inertia correction - R ₀ specific gas constant of dry air - R _w specific gas constant of water - Re 2r _L u–u _L / _L =Reynolds number of the particles motion relative to the ambient air - t time - T absolute temperature - u velocity of a particle - u (amount of the) settling velocity of a particle in stagnant air - u ₁ u(=1)=(amount of the) settling velocity of a spherical particle in stagnant air - u _L velocity of the ambient moist air (far enough from the particle where the flow pattern remains undistorted) - W drag coefficient of a particles equivalent sphere - empirical parameter in equation (3.1) - dynamic viscosity of a particles liquid cover - _L dynamic viscosity of moist air - _0L dynamic viscosity of dry air (at the same pressure and temperature like the moist air) - celsius temperature - dynamic shape factor of a particle (=1 for a sphere) - ₀ dynamic shape factor of a dry particle - _L mean free path of the molecules in moist air - _0L mean free path of the molecules in dry air (at the same pressure and temperature like the moist air) - _Po mean free path of the molecules in dry air at the pressureP ₀ of the dry air and the temperature given - factor of solid to liquid change-over (=1 for a solid particle) - mean bulk density of a particle - _L density of the moist air - _0L density of the dry air at the same pressure and temperature like the moist air - ₀ mean bulk density of a dry particle - ₀ mean diameter of the molecules of dry air - _w diameter of water molecules - relaxation time of a particle - gradient operation - 3.141593     

Rotatory vibration of a thick spherical shell of isotropic non-homogeneous elastic material

Summary Rotatory vibrations of a thick spherical shell of isotropic non-homogeneous material with rigidity and density given by (i) = ₀ r ^-2 withQ =Q ₀ r ^-2 e ^2mr and (ii) = ₀ r ^m with =Q ₀ r ⁿ have been discussed and the frequency equation is derived with numerical enumeration of frequency in each case.     

Molecular refraction in the Earth's mantle

The Drude law (molecular refraction) for the temperature radiation in a monoatomic model of the Earth's mantle is derived. The considerations are based on the Lorentz electron theory of solids. The characteristic frequency (or eigenfrequency) of independent electron oscillators (in energy units, ) is identified with the band gapE _G of a solid. The only assumption is that solid material related to the Earth's mantle has the mean atomic weight A21 g/mole, and its energy gap (E _G) is about 9 eV. In this case the value of molecular refraction (in cm³/g) is (n ²–1)/=0.5160.52, where andn are the density and the refractive index at wavelength _D=0.5893 m (sodium light), respectively. The average molecular refraction of important silicate and oxide minerals with A21, obtained byAnderson andSchreiber (1965) from laboratory data, is , where denotes the mean arithmetic value calculated from three principal refractive indices of crystal. For the rock-forming minerals with 19A<24 g/mole the new relation was found byAnderson (1975).     

A new dissipation model based on memory mechanism

Summary The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials.List of symbols unidimensional stress - unidimensional strain - Q ^–1 specific dissipation function - c(t) creep compliance - m(t) relaxation modulus - c ₀ instantaneous compliance - m equilibrium modulus - (t) creep function - relaxation function - () spectral distribution of retardation times - spectral distribution of relaxation times - c ^*() complex compliance - m ^*() complex modulus - tang loss-tangent     

Structure and evolution of the terrestrial planets

Geo-scientific planetary research of the last 25 years has revealed the global structure and evolution of the terrestrial planets Moon, Mercury, Venus and Mars. The evolution of the terrestrial bodies involves a differentiation into heavy metallic cores, Fe-and Mg-rich silicate mantles and light Ca, Al-rich silicate crusts early in the history of the solar system. Magnetic measurements yield a weak dipole field for Mercury, a very weak field (and local anomalies) for the Moon and no measurable field for Venus and mars. Seismic studies of the Moon show a crust-mantle boundary at an average depth of 60 km for the front side, P- and S-wave velocities around 8 respectively 4.5 km s^–1 in the mantle and a considerable S-wave attenuation below a depth of 1000 km. Satellite gravity permits the study of lateral density variations in the lithosphere. Additional contributions come from photogeology, orbital particle, x-and -ray measurements, radar and petrology.The cratered surfaces of the smaller bodies Moon and Mercury have been mainly shaped by meteorite impacts followed by a period of volcanic flows into the impact basins until about 3×10⁹ yr before present. Mars in addition shows a more developed surface. Its northern half is dominated by subsidence and younger volcanic flows. It even shows a graben system (rift) in the equatorial region. Large channels and relics of permafrost attest the role of water for the erosional history. Venus, the most developed body except Earth, shows many indications of volcanism, grabens (rifts) and at least at northern latitudes collisional belts, i.e. mountain ranges, suggesting a limited plate tectonic process with a possible shallow subduction.List of Symbols and Abbreviations a=R _e mean equatorial radius (km) - A(r, t) heat production by radioactive elements (W m^–3) - A, B equatorial moments of inertia - b polar radius (km) - complex amplitude of bathymetry in the wave number (K) domain (m) - C polar moment of inertia - C _Fe moment of inertia of metallic core - C _Si moment of inertia of silicate mantle - C _p heat capacity at constant pressure (JK^–1 mole^–) - C _nm,J _nm,S _nm harmonic coefficients of degreen and orderm - C/(MR _e ² ) factor of moment of inertia - d distance (km) - d nondimensional radius of disc load of elastic bending model - D diameter of crater (km) - D flexural rigidity (dyn cm) - E Young modulus (dyn cm^–2) - E maximum strain energy - E energy loss during time interval t - f frequency (Hz) - f flattening - F magnetic field strength (Oe) (1 Oe=79.58A m^–1) - g acceleration or gravity (cms^–2) or (mGal) (1mGal=10^–3cms^–2) - mean acceleration - g _e equatorial surface gravity - complex amplitude of gravity anomaly in the wave number (K) domain - g free air gravity anomaly (FAA) - g Bouguer gravity anomaly - g _t gravity attraction of the topography - G gravitational constant,G=6.67×10^–11 m³kg^–1s^–2 - GM planetocentric gravitational constant - h relation of centrifugal acceleration (² R _e ) to surface acceleration (g _e ) at the equator - J magnetic flux density (magnetic field) (T) (1T=10⁹ nT=10⁹ =10⁴G (Gauss)) - J ₂ oblateness - J _nm seeC _nm - k ₍₀₎ (zero) pressure bulk modulus (Pa) (Pascal, 1 Pa=1 Nm^–2) - K wave number (km^–1) - K ^* thermal conductivity (Jm^–1s^–1K^–1) - L thickness of elastic lithosphere (km) - M mas of planet (kg) - M _Fe mass of metallic core - M _Si mass of silicate mantle - M(r) fractional mass of planet with fractional radiusr - m magnetic dipole moment (Am²) (1Am²=10³Gcm³) - m _b body wave magnitude - N crater frequency (km^–2) - N(D) cumulative number of cumulative frequency of craters with diameters D - P pressure (Pa) (1Pa=1Nm^–2=10^–5 bar) - P _z vertical (lithostatic) stress, see also _z (Pa) - P _n ^m (cos) Legendre polynomial - q surface load (dyn cm^–2) - Q seismic quality factor, 2E/E - Q _s ,Q _p seismic quality factor derived from seismic S-and P-waves - R=R ₀ mean radius of the planet (km) (2a+b)/3 - R _e =a mean equatorial radius of the planet - r distance from the center of the planet (fractional radius) - r _Fe radius of metallic core - S _nm seeC _nm - t time and age in a (years), d (days), h (hours), min (minutes), s (seconds) - T mean crustal thickness from Airy isostatic gravity models (km) - T temperature (°C or K) (0°C=273.15K) - T _m solidus temperature - T sideral period of rotation in d (days), h (hours), min (minutes), s (seconds), =2/T - U external potential field of gravity of a planet - V volume of planet - V _p ,V _s compressional (P), shear (S) wave velocity, respectively (kms^–1) - w deflection of lithosphere from elastic bending models (km) - z, Z depth (km) - z (K) admittance function (mGal m^–1) - thermal expansion (°C^–1) - viscosity (poise) (1 poise=1gcm^–1s^–1) - co-latitude (90°-) - longitude - Poisson ratio - density (g cm^–3) - mean density - ₀ zero pressure density - _m , _Si average density of silicate mantle (fluid interior) - average density of metallic core - _t , _top density of the topography - density difference between crustal and mantle material - electrical conductivity (^–1 m^–1) - _r , radial and azimuthal surface stress of axisymmetric load (Pa) - _z vertical (lithostatic) stress (seeP _z ) - _II second invariant of stress deviation tensor - latitude - angular velocity of a planet (=2/T) - ages in years (a), generally 0 years is present - B.P. before present - FAA Free Air Gravity Anomaly (see g - HFT High Frequency Teleseismic event - LTP Lunar Transient Phenomenon - LOS Line-Of-Sight - NRM Natural Remanent Magnetization Contribution No. 309, Institut für Geophysik der Universität, Kiel, F.R.G.     

On the propagation of wave from a cylindrical cavity

Summary This paper discusses the disturbance produced in an infinite layer of non-homogeneous elastic material characterised by =0 ⁿ and =0ⁿ(n>0) where and are the density and shear modulus respectively of the material, due to periodic torsional force applied on the wall of a cylindrical hole in the layer. The variation of the displacement component with the radius vector is shown graphically and compared with the corresponding homogeneous case.     

Frequency dependence of crustalQ β in stable and tectonically active regions

Fundamental-mode Rayleigh wave attenuation data for stable and tectonically active regions of North America, South America, and India are inverted to obtain several frequency-independent and frequency-dependentQ models. Because of trade-offs between the effect of depth distribution and frequency-dependence ofQ on surface wave attenuation there are many diverse models which will satisfy the fundamental-mode data. Higher-mode data, such as 1-Hz Lg can, however, constrain the range of possible models, at least in the upper crust. By using synthetic Lg seismograms to compute expected Lg attenuation coefficients for various models we obtained frequency-dependentQ models for three stable and three tectonically active regions, after making assumptions concerning the nature of the variation ofQ with frequency.In stable regions, ifQ varies as , where is a constant, models in which =0.5, 0.5, and 0.75 satisfy fundamental-mode Rayleigh and 1-Hz Lg data for eastern North America, eastern South America, and the Indian Shield, respectively. IfQ is assumed to be independent of frequency (=0.0) for periods of 3 s and greater, and is allowed to increase from 0.0 at 3 s to a maximum value at 1 s, then that maximum value for is about 0.7, 0.6, and 0.9, respectively, for eastern North America, eastern South America, and the Indian Shield. TheQ models obtained under each of the above-mentioned two assumptions differ substantially from one another for each region, a result which indicates the importance of obtaining high-quality higher-mode attenuation data over a broad range of periods.Tectonically active regions require a much lower degree of frequency dependence to explain both observed fundamental-mode and observed Lg data. Optimum values of for western North America and western South America are 0.0 if is constant (Q is independent of frequency), but uncertainty in the Lg attenuation data allows to be as high as about 0.3 for western North America and 0.2 for western South America. In the Himalaya, the optimum value of is about 0.2, but it could range between 0.0 and 0.5. Frequency-independent models (=0.0) for these regions yield minimumQ values in the upper mantle of about 40, 70, and 40 for western North America, western South America, and the Himalaya, respectively.In order to be compatible with the frequency dependence ofQ observed in body-wave studies,Q in stable regions must be frequency-dependent to much greater depths than those which can be studied using the surface wave data available for this study, andQ in tectonically active regions must become frequency-dependent at upper mantle or lower crustal depths.On leave from the Department of Geophysics, Yunnan University, Kunming Yunnan, People's Republic of China