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.     

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.     

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.     

The effect of the number of charges carried by the argon ion on the darkening of ilford Q2 emulsion as ion detectors in mass-spectrometer

Summary The darkening (S) of Illford Q₂ photographic plates as ion detectors in mass spectrometer has been investigated. The dependence of the darkening (S) on the ion density (n=ions/mm²) i.e.S=S(n)E for constant energy (E)=z U ranging from 4U20 Kv of the impinging⁴⁰A⁺¹-,⁴⁰A⁺²- and⁴⁰A⁺³-ions whenS does not exceed the value 0.15 and the second relationn=n(z U) _S for darkening 0.05S0.15 constructed from the above relationS=S(n) _E has been determined. The darkening was found to increase with increasing ion-density which inturn decreases with the ionenergy. For⁴⁰A⁺¹-,⁴⁰A⁺²-, and⁴⁰A⁺³-ion of equal energy and ion-density the darkening effect was independent of the number of the charges carried by the argon ion.     

Physical properties of the Earth's core

Calculations of the compression and temperature gradient of the core are facilitated by the use of the thermodynamic Grüneisen ratio, =3Ks/C _P . A pressure-dependent factor in is found to have the same numerical value for the core as for laboratory iron, justifying the use of a constant value for (1.6) in core calculations. The density of the outer core is satisfied by the assumption that it contains about 15% of light elements, particularly sulphur, whereas the inner core is probably ironnickel with very little lighter component. The presence of sulphur in the outer core reduces its liquidus at least 600° below pure iron, so that the adiabatic gradient does not intersect the liquidus, as Higgins and Kennedy have shown would occur in a pure iron core. The inner core is probably close to its melting point, 4700 K, and the adiabatic temperature gradient of the outer is calculated with this as a fixed point, giving 3380 K at the core-mantle boundary. The estimated electrical resistivity of the outer core, 3×10^–6 m, corresponds to a thermal conductivity of 28 W·m^–1·deg^–1, which, with the adiabatic core gradient gives a minimum of 3.9×10¹² W of heat conduction to the mantle. The only plausible source of this much heat is the radioactive decay of potassium in the core. As pointed out by Goles, Lewis, and Hall and Murthy, the presence of potassium becomes geochemically probable once sulphur is admitted as a core constituent. Thus it appears that the recognition of sulphur in the core resolves the two major difficulties which we have faced in attempting to understand the core.List of Symbols a equilibrium atomic spacing at zero pressure, also a constant - A surface area of core - b a constant - c a constant - C _V ,C _P specific heat at constant volume, constant pressure - D dimension of core (or core eddy) - E(r) atomic interaction energy - E energy due to atomic displacement from equilibrium - lattice energy of material - f ₁,f ₂ structure-dependent constants - F(P) pressure dependent factor in Grüneisen's ratio - g gravitational acceleration; also a constant (Equation (13)) - H latent heat of solidification - I integral (Equation (23)) - k Boltzmann's constant - K incompressibility (bulk modulus) - K _T ,K _S isothermal, adiabatic incompressibilities - N number of atoms in a volume of material - P pressure - dQ/dt core to mantle heat flux - r atomic spacing - r _e equilibrium value ofr under pressure - R _m magnetic Reynolds number - T temperature - T _c critical temperature - T _R reduced temperature (Equation (39)) - U specific internal energy of a material - v velocity of internal core motion - V volume - 3 volume expansion coefficient - compressibility - thermodynamic Grüneisen ratio (Equation(2)) - magnetic diffusivity - thermal conductivity - _e electronic contribution to - ₀ permeability of free space - density - _e electrical resistivity - R reduced conductivity,eM/e     

Finite strain theories and comparisons with seismological data

All the finite strain equations that we are aware of that are worth considering in connection with the interior of the Earth are given, with the assumptions on which they are based and corresponding relationships for incompressibility and its pressure derivatives in terms of density. In several cases, equations which have been presented as new or independent are shown to be particular examples of more general equations that are already familiar. Relationships for deriving finite strain equations from atomic potential functions or vice versa are given and, in particular it is pointed out that the Birch-Murnaghan formulation implies a sum of power law potentials with even powers. All the equations that survive simple plausibility tests are fitted to the lower mantle and outer core data for the PEM earth model. For this purpose the model data are extrapolated to zero temperature, using the Mie-Grüneisen equation to subtract the thermal pressure (at fixed density) and the pressure derivative of this equation to substract the thermal component of incompressibility. Fitting of finite strain equations to such zero temperature data is less ambiguous than fitting raw earth model data and leads immediately to estimates of the low temperature zero pressure parameters of earth materials. On this basis, using the best fitting equations and constraining core temperature to give an extrapolated incompressibilityK ₀=1.6×10¹¹Pa, compatible with a plausible iron alloy, the following numerical data are obtained: Core-mantle boundary temperature 3770 K Zero pressure, zero temperature densities: lower mantle 4190 kg m^–3 outer core (solidified) 7500 kg m^–3 Zero pressure, zero temperature incompressibility of the lower mantle 2.36×10¹¹PaHowever, an inconsistency is apparent betweenP() andK() data, indicating that, even in the PEM model, in which the lower mantle is represented by a single set of parameters, it is not perfectly homogeneous with respect to composition and phase.     

Über das Flimmern und Schweben des Zielbildes

Zusammenfassung In zwei Beobachtungsserien von 24 Stunden ist jede zweite Stunde die Schwankung des Zielbildes beobachtet und gleichzeitig die Temperaturdifferenz zwischen den Höhen 0.5 und 2.5 m gemessen worden. Aus diesen Beobachtungen gehen folgende Resultate hervor. Kurz nach Sonnenaufgang ist das Zielbild unbeweglich. Bald beginnt es zu flimmern, zuerst schwach und langsam (einmal in der Sek.) später stärker und schneller (5 mal in der Sek.). Das Flimmern erreicht sein Maximum etwas nach Mittag, vermindert sich dann und wird Null kurz vor Sonnenuntergang. Von hier an beginnt das Zielbild sehr langsam (in Perioden von Minuten) zu schweben. Der Betrag des Flimmerns ist proportional der 1.68-Potenz der Zielweite. Sie hängt von dem Temperaturgradienten ab, und z. B. bei der Zielweite 75 m beträgtF ₇₅=0.66–1.42 mm, wo die Temperaturdifferenz zwischen den Höhen 0.5 und 2.5 m bezeichnet. Der Betrag des Schwebens ist proportional der 2.05-Potenz der Zielweite. Sie hängt von dem Temperaturgradienten ab, und z. B. bei der Zielweite 125 m beträgtS ₁₂₅=1.3+4.02 mm, wo die obenangegebene Bedeutung hat. Zwischen dem ZielungsfehlerZ und dem FlimmernF besteht die BeziehungZ=0.11++0.029F mm. Das Schweben wirkt sehr stark auf die Zielungsgenauigkeit ein, obgleich beim Vorkommen des Schwebens das Zielbild nach dem Augenmass unbeweglich erscheint.

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Summary The oscillation of the sight point picture in telescope and simultaneously the difference of the air temperatures between the hights of 0.5 and 2.5 meters have been observed every two hours through two clear days and nights. Hereby the following results have been obtained. Short after the sunrise the sight point picture is quite immovable. Soon after it beginns to vibrate, at first slightly and slowly (one time in second) and then more violently and quickly (five times in second). The vibration attains its maximum short after the noon, then it decreases and becomes zero not long before the sunset. From now till the sunrise the sight point picture is swaying very slowly (in periods of several minutes). The vibration is proportional to the 1.68-power of the sight length. It correlates the vertical temperature gradient, and the vibration is e. g. for the sight of 75 m:F ₇₅=0.66–1.42 mm, where means the temperature increase from 0.5 to 2.5 m (here negative). The swaying is proportional to the 2.05-power of the sight length. It correlates the gradient, and the swaying is e. g. for the sight of 125 m:S ₁₂₅=1.3+4.02 mm, where has the same meaning as above. Between the sight errorZ and the vibrationF a relation:Z=0.11+0.029F mm is to be seen. Thus the accuracy of the sighting is not very sensitive to the vibration.

     

A relationship between phonon conductivity and seismic parameter Φ for silicate minerals

Summary The relationship between the phonon conductivity at room temperature (K _N ) and the seismic parameter () for silicate minerals is suggested. The considerations are based on the Debye model of thermal energy transport phenomena in solids and on the seismic equation of state for silicates and oxides given byAnderson (1967). The semiempirical relationship is the formK _N = 0.43^0.82 where is in km²/s² andK _N in mcal/cm s K, and the empirical relationship isK _N =(0.528±0.006) –(8.18±2.11). The laboratory data on thermal and elastic properties for several silicates were taken fromHorai andSimmons (1970).     

On the propagation of wave from a cylindrical cavity,II

Summary We have discussed in the paper the disturbance produced in an infinite layer of nonhomogeneous elastic material characterised by = ₀ Z and = ₀ Z 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 depth of the layer is shown graphically and compared with the corresponding homogeneous case.     

The role of wavegroup velocity in the blocking problem

Considering the blocking problem as a baroclinic instability problem in a dispersive wave system with diabatic heating effects, it is of great interest to investigate the role of wavegroup velocityv _gr in blocking processes, becausev _gr controls the energy transfer in the wave field. Using a Newtonian Cooling —type of forcing with a phase differencek to the main field and taking the linearized version of a two-level model, the phase speedc _r, the group velocityv _gr and the growth ratekc _i have been obtained as analytical functions of the mean zonal windU, the thermal windU _T, the coefficient of diabatic heating x, the phase differencek and the wavelengthL. Now the hypothesis is introduced, that a blocking should be expected, ifv _gr has a maximum value in the vicinity ofL _o, for whichc _r vanishes and thee-folding timet=1/kc _i (kc _i>0) is smaller than 6 days (see condition (20) in the text). One finds, that for special parameter combinations (U _T, U, ), where 15 m/secU _T25m/sec,U=10m/sec, 0.8·10^–51.5·10^–5 [sec^–1], certain valuesL _o with an appropriate phase differencek exist, which satisfy the conditions mentioned above (for values see Table 2). TherebyL _o varies within the range 8500 km <L _o<11000 km corresponding to the preferred planetary blocking wavenumber 2 in middle latitudes 50°<<70° N.     

Die Seismizität der Türkei

Summary Using the fromulae given byGutenberg andRichter, the writer has computed the magnitude and energy of 1804 earthquakes which occurred in Turkey during the period 1850–1960. For drawing the Isenerget, the formula =log₁₀ S has been used in accordance with the definitions given byToperczer andTrapp, whereS=e _i/F·p represents the energy in erg/m² h corresponding to the surface element of 0.5° Lat. x 0.5° Long. Also the relationship between the seismicity and the tectonics of Turkey has been studied by drawing the maps of the epicenters, the focus-depths and the frequences of the earthquakes with various intensities.     

Weiteres zur unmittelbaren Berechnung der durchschnittlichen Geschwindigkeit in der Reflexionsseismik

Zusammenfassung Die Anordnung der Schusspunkte I, II ...,A, B ... und der Geophone 1, 2, ...a, b ... nach Figur 1 ermöglicht — horizontalen Reflektor vorausgesetzt — bei demselben Reflexionspunkt eine gute Bestimmung der durchschnittlichen Geschwindigkeiit und des Reflektors durch Ausgleichung mit den Gleichungen (3)-(6). Die angegebenen Gleichungen können auch bei geneigtem Reflektor verwendet werden, da sogar bei 11° Neigung des Reflektors der dadurch verursachte Fehler inv undN unterhalb 0.5% bleibt. Der Reflexionspunkt wandert in diesem Falle allerdings (vgl. Figur 2) mit dem Betrag r im Sinne der Gleichung (17a) weiter.Kennt man die Neigung der Schnittgeraden in der Reflexionsebene, so kann man mit Hilfe der Gleichungen (12) und (14) die genaueren Werte vonv undN ermitteln.Zur Bestimmung des Neigungswinkels wird man vorteilhaft die Anordnung nach Figur 3 treffen, wo bei einem Schuss in I/A die Geophone in 1, 2 ...,a, b ... angeordnet sind. Aus den Messergebnissen können wir durch Ausgleichung nach den Gleichungen (23) und (24) bestimmen.Im Anhang werden Zahlenbeispiele mit praktischen Folgerungen angegeben.

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Summary The system of shot points I, II, ...A, B ... and geophones 1, 2 ...a, b according to Figure 1 assures — assuming horizontal reflector — by identical reflection point an advantageous determination of the average velocity and of the reflector by adjustment with equations (3)-(6). These equations can also be used if the reflector dips, as the error caused even by a dip of 11° of the reflector inv andN does not exceed 0.5 percent. The reflection point moves, however, simultaneously (see Figure 2) with the quantily r according to equations (17a).If the dip of the intersection line in the reflection plane is known, the more precise values ofv andN can be computed with the aid of equations (12) and (14).To determine the dip , the system of Figure 3 is most convenient, where the geophons 1, 2 ...a, b ... are attached to a shot inI/A.In the appendix some numerical solutions are given and practical consequences drawn.

     

Propagation of love waves in a non-homogeneous stratum of finite depth sandwitched between two semi-infinite isotropic media

Summary The aim of this paper is to study a problem in which the intermediate layer is non-homogeneous, the rigidity varying exponentially with depth i.e. ₂=₂ v ₀ ² e ^2pz , the density being constant, velocity varies also exponentially with depth according to the law =v ₀ e ^pz . The variability ofKH with the change of phase velocity is shown graphically.     

Coseismic slip in the 1964 Prince William Sound earthquake: A new geodetic inversion

The 1964 Prince William Sound earthquake (March 28, 1964;M _w =9.2) caused crustal deformation over an area of approximately 140,000 km² in south central Alaska. In this study geodetic and geologic measurements of this surface deformation were inverted for the slip distribution on the 1964 rupture surface. Previous seismologic, geologic, and geodetic studies of this region were used to constrain the geometry of the fault surface. In the Kodiak Island region, 28 rectangular planes (50 by 50 km each) oriented 218°N, with a dip varying from 8^o nearest the Aleutian trench to 9^o below Kodiak Island, define the rupture surface. In the Prince William Sound region 39 planes with variable dimensions (40 by 50 km near the trench, 64 by 50 km inland) and orientation (218°N in the west and 270°N in the east) were used to approximate the complex faulting. Prior information was introduced to constrain offshore dip-slip values, the strike-slip component, and slip variation between adjacent planes. Our results suggest a variable dip-slip component with local slip maximums occurring near Montague Island (up to 30 m), further to the east near Kayak Island (up to 14 m), and trenchward of the northeast segment of Kodiak Island (up to 17m). A single fault plane dipping 30°NW, corresponding to the Patton Bay fault, with a slip value of 8 m modeled the localized but large uplift on Montague Island. The moment calculated on the basis of our geodetically derived slip model of 5.0×10²⁹ dyne cm is 30% less than the seismic moment of 7.5×10²⁹ dyne cm calculated from long-period surface waves (Kanamori, 1970) but is close to the seismic moment of 5.9×10²⁹ dyne cm obtained byKikuchi andFukao (1987).     

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.     

Depression in the low-latitude horizontal intensity due to quiet-time ring current

Summary The external field due to plasma within the magnetosphere has been computed as a function ofA _p, which is a measure of solar wind velocity, for very quiet to slightly disturbed conditions using mean daily horizontal intensity from 1932 to 1968 at Alibag. The intensity, corrected for secular change and reduced to a common epoch, showed initially a small increase withA _p followed by a steady depression with further increase in the index. ForA _p7.5, which is representative of conditions over the 33-hour interval during which data relating to low-energy protons were acquired and used byHoffman andBracken [4]²) to compute current distributions, the decrease, computed here from surface data, is 6 . This is in goodagreement with the southward directed field of the quiet-time proton belt 9±5 obtained byHoffman andBracken.     

Multifractal measures,especially for the geophysicist

This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach inMandelbrot (1974). The generalization from fractalsets to multifractalmeasures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function (), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity is called Hölder exponent. In terms of the alternative functionf() used in the approach of Frisch-Parisi and of Halseyet al., one has ()=f()–E for measures supported by the Euclidean space of dimensionE. Whenf()0,f() is a fractal dimension. However, one may havef()<0, in which case is called latent. One may even have <0, in which case is called virtual. These anomalies' implications are explored, and experiments are suggested. Of central concern in this paper is the study of low-dimensional cuts through high-dimensional multifractals. This introduces a quantityD _q, which is shown forq>1 to be a critical dimension for the cuts. An enhanced multifractal diagram is drawn, includingf(), a function called (q) andD _q.This text incorporatesand supersedes Mandelbrot (1988). A more detailed treatment, in preparation, will incorporateMandelbrot (1989).     

Separation of equation of motion of love waves in curvilinearly anisotropic inhomogeneous medium

Summary The general problem of Love wave propagation, in a medium with cylindrical anisotropy of hexagonal type, is formulated. The method of seperation of variables is applied to examine the possibility of obtaining formal solutions for different types of inhomogeneities present in the medium. It is found that when the elastic parameters (C ₄₄ and ) are functions of bothv and the equation of motion is not separable. The use of the technique is illustrated, by considering radial inhomogeneity in an anisotropic cylindrical crustal layer, for obtaining the characteristic frequency equation of Love waves in such a medium.     

Instantaneous photochemical rates in the global stratosphere

Starting with the average actual distribution of ozone (Dütsch [15]) and temperature in the stratosphere, we have calculated the solar intensity as a function of wavelength and the instantaneous rates (molecules cm^–3 sec^–1) for each Chapman reaction and for each of several reactions of the oxides of nitrogen. The calculation is similar to that ofBrewer andWilson [5]. These reaction rates were calculated independently in each volume element in spherical polar coordinates defined by R=1 km from zero to 50, =5° latitude, and ø=15° longitude (thus including day and night conditions). Calculations were made for two times: summer-winter (January 15) and spring-fall (March 22). As input data we take observed solar intensities (Ackerman [1]) and observed, critically evaluated. constants for elementary chemical and photochemical reactions; no adjustable parameters are employed. (These are not photochemical equilibrium calculations.) According to the Chapman model, the instantaneous, integrated, world-wide rate of formation of ozone from sunlight is about five times faster than the rate of ozone destruction, and locally (lower tropical stratosphere) the rate of ozone formation exceeds the rate of destruction by a factors as great as 1000. The global rates of increase of ozone are more than 50 times faster thanBrewer andWilson's [5] estimate of the average annual transfer rate of ozone to the troposphere. The rate constants of the Chapman reactions are believed to be well-enough known that it is highly improbable that these discrepancies are, due to erroneous rate constants. It is concluded that something else besides neutral oxygen species is very important in stratospheric ozone photochemistry. The inclusion of a uniform concentration of the oxides of nitrogen (NO_x as, NO and NO₂) averaging 6.6×10^–9 mole fraction gives a balance between global ozone formation and destruction rates. The inclusion of a uniform mole fraction of NO_x at 28×10^–9 also gives a global balance. These calculations support the hypethesis (Crutzen [10],Johnston [24]) that the oxides of nitrogen are the most important factor in the global, natural ozone balance. Several authors have recently evaluated the natural source strength of NO_x in the stratosphere; the projected fleets of supersonic transports would constitute an artificial source of NO_x about equal to the natural value, thus promising more or less to double an active natural stratospheric ingredient.     

Waves in an elastic plate with an irregular boundary

Summary Green's function corresponding to the displacements for anSH-line source inside a wedge shaped medium finite or infinite in which the density and the elastic parameter are not constant throughout but some class of function ofr, the radial distance (i.e. and satisfy some differential equation) has been obtained. The paths of propagation of the cylindrical waves, their reflections on different boundaries and their continuous refractions within the medium has been clearly established. The scattered field from the different parts of the body have been pointed out with prescribed distribution of and in this inhomogeneous wedge. It is shown that for some distribution of and the rays are curved and the reflections at the two plane boundaries occur in such a way that no ray will go to infinity by reflection, rather they are coming back towards the apex after suitable number of reflections with variable intensity. Also there are some distributions of and in the same class as mentioned earlier, such that the behaviour of the waves is similar to that in a homogeneous wedge, i.e. these waves will go to infinity by a suitable number of reflections and ultimately die out. The first case is quite unlike the usual homogeneous medium with non parallel boundaries and so care must be taken in computing the field within the nonhomogeneous medium within the non-parallel boundaries.