(ϖμ/ϖT) P of the lower mantle

We estimate (/T) _P of the lower mantle at seismic frequencies using two distinct approaches by combining ambient laboratory measurements on lower mantle minerals with seismic data. In the first approach, an upper bound is estimated for |(/T) _P | by comparing the shear modulus () profile of PREM with laboratory room-temperature data of extrapolated to high pressures. The second approach employs a seismic tomography constraint ( lnV _S / lnV _P ) _P =1.8–2, which directly relates (/T) _P with (K _S /T) _P . An average (K _S /T) _P can be obtained by comparing the well-established room-temperature compression data for lower mantle minerals with theK _S profile of PREM along several possible adiabats. Both (K _S /T) and (/T) depend on silicon content [or (Mg+Fe)/Sil of the model. For various compositions, the two approaches predict rather distinct (/T) _P vs. (K _S /T) _P curves, which intersect at a composition similar to pyrolite with (/T) _P =–0.02 to –0.035 and (K _S /T) _P =–0.015 to –0.020 GPa/K. The pure perovskite model, on the other hand, yields grossly inconsistent results using the two approaches. We conclude that both vertical and lateral variations in seismic velocities are consistent with variation due to pressure, temperature, and phase transformations of a uniform composition. Additional physical properties of a pyrolite lower mantle are further predicted. Lateral temperature variations are predicted to be about 100–250 K, and the ratio of ( lnp/ lnV _S ) _P around 0.13 and 0.26. All of these parameters increase slightly with depth if the ratio of ( lnV _S / lnV _P ) _P remains constant throughout the lower mantle. These predicted values are in excellent agreement with geodynamic analyses, in which the ratios ( ln / lnV _S ) _P and ( / lnV _S ) _P are free parameters arbitrarily adjusted to fit the tomography and geoid data.     

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.     

Determination of the gravitational effect of extensive spherical laminae (platforms) using generalized Talwani's method

u uuuuaumau uu u m, ¶rt;u a mumau nmu a u nmu mau u nmmu. au n¶rt; nuau ¶rt;mam u u m. nua m¶rt; m u m¶rt;a aau (1960) ¶rt; uu .     

Some exact solutions for the flow of fluid through tubes with uniformly porous walls

Summary This paper considers an incompressible fluid flowing through a straight, circular tube whose walls are uniformly porous. The flow is steady and one dimensional. The loss of fluid through the wall is proportional to the mean static pressure in the tube. Several formulations of the wall shear stress are considered; these formulations were motivated by the results from Hamel's radial flow problem, boundary layer flows/and boundary layer suction profiles. For each of these formulations exact solutions for the mean axial velocity and the mean static pressure of the fluid are obtained. Sample results are plotted on graphs. For the constant wall shear stress problem, the theoretical solutions compare favorably with some experimental results.Notations A, B, D, E constant parameters - a, b constant parameters - Ai(z), Bi(z) Airy functions - Ai, Bi derivatives of Airy functions - k constant of proportionality betweenV andp - L length of pores - p,p mean static pressure - p ₀ static pressure outside the tube - p ₀ value ofp atx=0 - Q constant exponent - R inside radius of the tube - T wall shear stress - T ₀ shear parameter - t wall thickness - U free stream velocity - ,u mean axial velocity - u ₀ value ofu atx=0 - V,V mean seepage velocity through the wall - v ₀ mean seepage velocity - x,x axial distance along the tube - z transformed axial distance - z ₀ value ofz atx=0 - mean outflow angle through the wall - cos - density of the fluid - wall shear stress - dynamic viscosity of the fluid - over-bar dimensional terms - no bar nondimensional terms The National Center for Atmospheric Research is sponsored by the National Science Foundation     

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)     

Consequences of transducer inductance for seismograph calibration

Summary The magnification achieved with the standard sine-wave method using seismometers with the calibration and signal coils tightly wound on the same coil former can be erroneous at high frequencies due to the mutual inductance between both coils. An attempt was made to eliminate this influence from the calibration data. The application of theoretical equations was tested with a short-period digital seismograph.

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ma¶rt;ama auau aa nu ¶rt;uu ¶rt;uu maauu m m m a u amma ua u a au u¶rt;mumu ¶rt; ua u auau am, u u a¶rt;m n¶rt;m umu. a ¶rt;aa nnma muam auau ¶rt;a. uu mmuu au u¶rt;a nu auauu mnu¶rt;u aa u anu.

     

The new radiation chart

Summary Basing on the analysis of the most reliable data concerning the atmospheric absorption of long-wave radiation by water vapour, carbon dioxide and ozone, was obtained the integral transmission function of long-wave radiation by the atmosphere. The results are used for plotting the new radiation chart intended for calculating the thermal radiative fluxes in the atmosphere. Comparison of the results of calculations of thermal radiative fluxes has been carried out according to the new chart as well as by the preceding ones.

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Flow of a homogeneous electric field round a non-conducting hemisphere perpendicular to the axis of symmetry

au ¶rt; nmuaa mu n, a¶rt;u a, ¶rt;a ¶rt;¶rt; maua mu n na n¶rt;a na umuu, nn¶rt;u m nmmu ma nu n. a¶rt;aa a u um ¶rt;uam. a u ¶rt;m um u au. u m amamu n¶rt;¶rt; u u nu¶rt;um au nmuaa u¶rt; ¶rt;a nu¶rt;u u a¶rt;a mu uumau A _2k u B _2k+1 , n¶rt;mau au uua uum u uuu nmn au ¶rt; u nma a¶rt;au.     

The magnetic observatory of Pendeli (Athens,Greece)

Summary A new magnetic observatory, named the Magnetic Observatory of Pendeli, was established and put in operation in 1958 near Athens (Greece). This Observatory was organized by and belongs to the Greek «Institute for Geology and Subsurface Research». The geographical position of the Pendeli Observatory is given by =38° 02.8, =23°51.8 andh=495 m (above sea level). The gemagnetic coordinates of the same are =36°.2, =102°.0. The Observatory is situated near Pendeli Mt. (18 km NEE of Athens). The site of the Observatory consists of marmor underlain by mica schists, both magnetically inactive.The building of the Observatory is constructed of stone and its roof made of tiles. The magnetograph room is in the underground of the building. The magnetic and thermal conditions in the variometer room are satisfactory enough.The variometers of the Observatory forH andZ are of the magnetic balance type. ForD a fibre suspension declinometer is used. The scale values of the variometers are _H =7.2 /mm, _D =1.0/mm (7.6 /mm), _Z =11.5 /mm. The speed of recording amounts to 12 mm/h and the width of the record is 9 cm.The Pendeli Observatory has been operating since April 1958. The record is changed every day. The scale and base-line values are determined every 10 days. The room for the absolute measurements is found in the ground floor of the building. A field magnetic theodolite is being used in the absolute measurements.D is measured with two magnets in a fibre suspension declinometer.H is measured by means of the deflection oscillation method and with a QHM as well. For the measurement ofI an earthinductor is available. The values of the magnetic elements are properly corrected in order to represent the external normal field.     

On the characteristics of vlf emissions in the upper ionosphere and on the ground

auuau uu muna a f>1,5 , aumua ¶rt; a nmua uu m u a ¶rt; a mau. mu a aum nm uu, umu a mauu aa (L=2,1) n¶rt; nm uu (L=5). m mmmum mu umua uu, umu a mu mau. au uu a nm m, m a um nmam a 2000–3000 u anu u a L=2,2–5,9. au mmmu nma aamumu a u nmu uu a¶rt;am u amu aua uu a L 3,5. aa, m a mauu aa u ¶rt;a a¶rt; nuu uu a , umum uuu n¶rt; a¶rt;a a nmu. au a n¶rt;num, m am a L 3,5,¶rt; aam au uu u au mmu nma aamumu a u nmu uu, aa ¶rt;amua anau a nana. m u m amu mm au anum u ¶rt; ¶rt;a ua n¶rt;u anmau u ¶rt; — ua.     

The mean energetic level. theory

Summary One of the important atmospheric levels, the mean energetic level (MEL), which in a sense reflects the energetics of the whole atmosphere, is defined. Its fundamental properties are shown. In order to describe the MEL correctly a new vertical coordinate is introduced and discussed. The new coordinate, , is defined as the ratio of height and temperature. The MEL is shown to be a level with constant value of . Some incorrect conclusions concerning the MEL, derived in the past, have been corrected.List of symbols used c _p specific heat of air at constant pressure - c _v specific heat of air at constant volume - e base of natural logarithms - E total potential energy - f Coriolis parameter - g acceleration of gravity - i specific internal energy - I internal energy - J enthalpy - k unit vector pointing upwards - p pressure - Q diabatic heating rate - R gas constant of the air - t time - T temperature - v horizontal velocity - v ⁽³⁾ three-dimensional velocity - w vertical velocity in thez-system - z height - temperature growth rate (T/z) - Pechala's vertical coordinate (z/T) - generalized vertical velocity in the -system (d/dt) - specific potential energy - potential energy - density of the air - Ruppert function - T(1–)^–1 - ( ) _S quantity at the sea level - ( )* quantity at the MEL     

Power radiated by a VLF loop antenna in the ionospheric plasma

Summary The radiation power a VLF loop antenna with an arbitrary orientation of the loop's plane relative to the direction of the external magnetic field is calculated and its portion, transferred to the electromagnetic part of the excited spectrum, is determined.

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am umaa m uu am c nu umau nmu uma n m¶rt; a¶rt;um¶rt; n u n¶rt;a ma am mu uu, u¶rt;a ma¶rt;um am am cnma ¶rt;a au.

     

On the propagation of finite Love Waves

Summary The problem of the propagation of finite Love Waves in a heterogeneous elastic half space lying over a homogeneous elastic half space, using the quasilinear stress-strain relation due toS. Ferhst [4] is considered in detail. The variations of the parameter in the layer assumed to be of the form 1= 0e ^z, 0e ^z where is a constant andz is distance measured from the surface into the layer.     

Statistical analysis of values measured in condensating geodetic nets

Summary Procedure for verifying the agreement between parameters common to the basic and connecting trigonometric net. Procedure of determining the accuracy of the connecting net. This determination concerns not only the relativized accuracy of the points of the connecting network, but also the mutual accuracy of the points of the basic net relative to the points of the connecting net and the global accuracy of the resultant net. The procedure takes into account the accuracy of the points of the basic net which remain unchanged in computing the coordinates of new points.

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m¶rt; ¶rt; nu au u naam¶rt;mu u nu¶rt;u m. m¶rt; ¶rt; u mmu nu¶rt;u mu. a aam m mum nua mmu m nu¶rt; mu, ma au mmu m mu n mu ma nu¶rt;u mu ua mmu mu mu. m¶rt; umam mm m mu, m aa uu ¶rt;uam m mam uu.

     

The intensity of the earth's magnetic field in the past derived from the archaeomagnetic data

a n¶rt;u¶rt; amuu amua uuu anmuaum n a n¶rt;u 8500 m nm um au aam m uuu m naama. ¶rt;u u ¶rt;a ¶rt; u ¶rt; m¶rt; mumu auu nau nuu aau mama [1–3] (Puc. 1.B). nma aau auau (Puc.2) mnaua m m m¶rt;a, ¶rt; au n¶rt; amu u, m n¶rt;u 2- ma m, mmu naa auu nu¶rt; n¶rt;a 1000 m u 350 m, au a aumu, mmm 80%- mmu. am au ¶rt;a u anmu ¶rt; m¶rt; mumu (mum) naa auu u¶rt; ma u ¶rt; m naama[5] (Puc. 3). uma a mam nma aaua naaa auu nu¶rt; n¶rt;a 750 m, 200 m u a — 300 m. (Puc. 4.) nmu u ¶rt;a nma amua an¶rt;u n¶rt;u n u ma m (au) auumu m ¶rt;m m (Puc. 5). aa a m¶rt;m nu naamaum n aam ¶rt;uu mu au, anum¶rt;a m uua nm m¶rt;a.

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Dedicated to Academician Alois Zátopek on His 65th Birthday     

Palaeomagnetism of the Börzsöny Mountains (Hungary)

Summary Mean directions of magnetization (29 normal and 31 reversed) were recorded for 60 magmatic localities of middle Miocene age from the Börzsöny Mountains (Hungary). The overall mean direction of RM, irrespective of polarity, isD=0,9°;I=59,8°; withk=8,3 and ₉₅=6,8°. The coordinates of the corresponding geomagnetic north pole are =82,7°, A=193,8 with p=7,7° and m=10,2°.     

Analytical study of the energy rate balance equation for the magnetospheric storm-ring current

We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, E_y, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of E_y, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter compatible with the observed Dst peak values. To facilitate the analytical integration we assume a constant value for through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed Dst the most appropriate values of are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, _m, are found to be very sensitive to the adopted efficiency factor, , decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of and the resulting values for the decay time parameter, _r, do not vary greatly as varies. As a consequence, the relative values of between the main and the recovery phase, _m/_r, can be greater or smaller than one as varies from 0.1 to 1.     

The secular variation of the geomagnetic field in Egypt in the last 5000 years

The palaeo-intensities (F _a) of the geomagnetic field in Egypt at some ages are determined by archaeomagnetic measurements and found to be:F _a=36.2 T at 3100 B.C., F_a=46.8 T at 3000 B.C.,F _a=36.5 T at 2780 B.C., 49.0 T at 2500 B.C., 36.4 T at 2200 B.C., 57.5 T at 1990 B.C., 62.1 T atca 1400 B.C., 61.5 T at 1400 B.C., 69.9 T at 600 B.C., 59.3 T at 550 B.C., 79.9 T at 460 B.C., 73.7 T at 450 B.C., 69.7 T at 320 B.C., 56.2 T at A.D. 50, 64.9 T, at A.D. 400, 54.4 T at A.D. 300, 57.5 T at A.D. 700 and 43.0 T at A.D. 1975.The palaeo-inclinations (I _a) at some ages are found to be:I _a=24.2° at 420 B.C., 44° at A.D. 50, 60.7° at A.D. 703 and 42° at A.D. 1795.The measured values ofF _a are affected by the anisotropy of magnetic susceptibility of the samples by 13% to 20% of the expected correct value. The suitable correction of this effect is by multiplyingF by 1/((1+0.2(/90)) andF by 1/((1–0.13 (/90)), whereF andF are the resultant values ofF _a if the laboratory field is perpendicular or parallel to the wall of the sample during the Thelliers' experiments, respectively, and is the angle between the direction of natural remnant magnetization of the sample and the direction of the laboratory field.The results of this paper, together with the previous results for Egypt and the neighbourhoods, lead to the production of the secular variation curve of the geomagnetic field in Egypt for the last 5000 years. The intensity of the field shows a periodicity of about 400 years with multiples.     

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     

Transmission of Rayleigh waves over the surface of a heterogeneous medium

Summary The frequency equation of Rayleigh waves propagating over the free surface of an isotropic, perfectly elastic, heterogeneous semi-infinite medium with material properties varying as = ₀ e ^az , = ₀ e ^az , = ₀ e ^az (a>0) has been obtained. Solution of the frequency equation in closed form is obtained in two cases (i) =0, (ii) =, and the Rayleigh wave dispersion curves for phase and group velocities drawn. In both the cases the medium yields single Rayleigh modes which cannot propagate below certain cut-off frequencies. It is found that in case (i), <c<c ₀ and 0.87500 <c _g <c ₀, and in case (ii), 1.03082 <c<c ₁ and 0.90850 <c _g <c ₁, wherec andc _g denote phase nad group velocities respectively, is the constant shear wave velocity of the mediumc ₀ andc ₁ are the corresponding Rayleigh wave velocities of the homogeneous medium of the same Poisson's ratio. The motion of the surface particles is found to be retrograde elliptical as in the homogeneous case, but the ratic of the major and minor axes now becomes frequency dependent and is plotted against frequency. In both the cases (i) and (ii), the ratio starts at a lower value at the cut-off frequency and approaches the corresponding value of the homogeneous medium at high frequencies.