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During the past ten years or so there has been considerable discussion in the literature regarding the author's 1963 contention that (neglecting temperature effects and spin) the Earth's liquid core cannot be stable unless the Adams-Williamson condition relating density distribution and compressibility holds there. The present paper throws light on this question by showing mathematically that a sphere of gravitating compressible liquid cannot be internally stable unless this condition is fulfilled. Physical reasons for the necessity of this condition, which implies that particles of the liquid are in neutral equilibrium, are also discussed. By internal stability is meant stability of the density distribution while the spherical shape is maintained. The question of shape stability is not treated here, since it may be assumed that the Earth's mantle is sufficiently rigid to keep the core essentially spherical. The liquid is assumed to be a perfect fluid, elastic, and in the discussion only small strains are considered from an equilibrium configuration of initial hydrostatic stress. Furthermore thermodynamic effects are neglected and there is no spin.

In a recent publication, Ricker (1977) gave integral representations for his generating, displacement, velocity, and acceleration functions pertaining to the propagation of a seismic pulse in a visco‐elastic medium. These functions are solutions of the Stokes wave equation in one dimension, and were given by Ricker as infinite integrals of oscillatory functions. Here these integrals are transformed into finite integrals of an integrand which is usually one‐signed, and these new integrals are readily evaluated by simple numerical quadrature (e.g., Simpson's rule), and are easily evaluated to high precision. Some numerical results are given and compared with Ricker’s tables.

Rational function approximations $\bar g_n (p)$ in partial form to a function $\bar f(p)$ of the Laplace transform operator p are determined to have inverses $g_n (t)$ which approximate the inverse $f(n)$ of $\bar f(p)$ in a least squares sense. Examples are presented and compared with Pade table and other rational approximations.

Results of a previous paper by the author on the summation of power and Fourier series are extended, and the summation of other types of series is now treated. Furthermore, the summation formula is now developed in such a way as to decrease the influence of the integral contained therein, and which has to be computed by numerical quadrature. In some cases this development can be done to such an extent that the integral itself can be neglected, an this gives rise to efficient convergence acceleration formulas, in which one has a very close control on the error.

In two previous papers by the author, closed expressions for the sums of Fourier, Chebyshev, and Legendre series were obtained in terms of integrals, and these are readily evaluated by numerical quadrature in cases where the coefficients of the series are moments of a function f(u) which is known, or which can readily be obtained from those moments-usually by Mellin- or Laplace-transform inversion. In many cases, however, f(u), although existing, is either unknown or not readily obtainable. This paper shows how the sums of such series may be approximated as accurately as we please in many cases, without explicit knowledge of f(u).

In previous papers the author has treated the question of convergence acceleration of alternating series, and of power series in x having positive coefficients. In the latter case it was assumed that 0 < x < 1, but even for x extremely close (but not equal) to 1, a powerful convergence acceleration was obtained. The present paper propounds an improved convergence acceleration method for alternating series, and also an efficient convergence acceleration technique which is valid for many monotonic series of importance in applied mathematics, i.e. the case x = 1 is now treated. In essence this technique obtains from the positive terms of the monotonic series an alternating sequence of convergents, which are then handled by the method for alternating series or sequences. It is assumed that the coefficients (or terms) of the series treated in this paper are moments of a function itf(u) over the interval 0 =< u =< 1, and the methods are based on properties of the shifted Legendre polynomials P^*"n(u), and on the author's earlier summation formula for power series having moment coefficients. The function itf(u) does not need to be known.

In a recent paper by the author a method for the convergence acceleration of power series was given, in which one has a very close control on the error at any stage. The main purpose of this note is to present a compact and convenient formula for the evaluation of the approximants in terms of the sequence of partiak sums of the series. A measure of the error is presented and this is compared with the corresponding measure for the generalized Euler transformation and for the original sequence of partial sums.

A function $$\bar f$$ (p) of the Laplace transform operatorp is approximated by a finite linear combination of functions $$\bar \phi $$ (p+α r ), where $$\bar \phi $$ (p) is a specific function ofp having a known analytic inverse φ(t), and is chosen in accordance with various considerations. Then parameters α r ,r=1, 2,...,n, and then corresponding coefficientsA r of the $$\bar \phi $$ (p + α r ) are determined by a least-square procedure. Then, the corresponding approximation to the inversef(t) of $$\bar f$$ (p) is given by analytic inversion of Σ =1 A r $$\bar \phi $$ (p+α r ). The method represents a generalization of a method of best rational function approximation due to the author [which corresponds to the particular choice φ(t)≡1], but is capable of yielding considerably greater accuracy for givenn.

In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

This paper deals with the question of obtaining from the sequence {s"n} of partial sums of a convergent series s a new sequence {t"n} which converges to the same limit s as s"n, but more rapidly. When the general term u"n of the series s possesses certain types of expansion involving inverse powers of n, it is shown how t"n is obtained by adding a fixed number of terms to s"n. When the series s is convergent, these terms tend to zero as n tends to infinity, but they are such as to make t"n much more rapidly convergent to s-in fact we can make the convergence rate as great as we wish. Explicit general formulas are obtained for a wide range of important series.

The paper develops a method for the summation of a series of positive terms given by Gismalla et al. [2]. Here we are concerned with the summation of power series ∑∞n = 1 μnxn where μn > 0 and 0 < x < 1, and x may be very close to 1 (but not equal to 1). Using ideas of Longman [4] we show how a bound can be obtained for the error when the process is curtailed at a particular stage. The method supposes that the coefficients in the summation are moments but, in practice, seems to work even when the coefficients are not identifiable as moments.

In previous papers the author has treated the summation of power and other series whose coefficients re moments of a function f(u), usually over the interval 0

Summary In this investigation we determine the motion of the surface of a uniform elastic half-space produced by the application of a torque-pulse at a point below the surface. The axis of the torque is taken to be parallel to the surface and its time variation is assumed to be represented by the Heaviside unit function H(t). Our results are compared with those of Pinney (1954), who treated the same problem by a different method. The principal feature of physical interest which we found is that, for ranges where the direct S wave is preceded by a diffracted SP wave, the displacement at the surface starts with an infinite amplitude at the time of arrival of the SP wave, and that this is followed by an even stronger infinity at the time of arrival of the S wave. Also, for small ranges, for which there is no SP wave, the displacement starts with a sharp pulse in the form of a Dirac delta function. None of these features was brought out by Pinney's curves. The results are shown in Figures 2, 3 and 4.

Methods are demonstrated for the numerical calculation of the inverse Laplace transform g(t) of a function g(p) given on the positive real axis of the p-plane.

However in physical problems the finiteness of the range of integration is often associated with a kind of natural boundary of f(x), such that it is impossible to extend f(x) to values of x beyond the upper limit b while preserving the general character of f(x). Analytically speaking, x = b may be a branch point of f(x). Alternatively, it may be possible to extend the range of integration to infinity as in equation (2), but the infinite integrals may not converge. As an example of the branch point difficulty we can consider the integral

In a recent paper Papadopoulos [1]1 claims to have obtained solutions to the problem of sound propagation from an impulsive point source in a semi-infinite elastic solid which differ in certain respects from solutions previously obtained by Pekeris [2], [3], and by Pekeris and Lifson [4]. The results of Papadopoulos are based on a singular integral source representation developed by him [5]. By tests based on simple special cases we show that, in a number of cases where Papadopoulos claims a result different from that of Pekeris, the results of Papadopoulos are unacceptable whereas the results of Pekeris are acceptable. Reasons for these errors in Papadopoulos's work are suggested. A number of other errors in the paper of Papadopoulos are also noted. 1 Numbers in brackets refer to references at rear.

A rational approximation technique is applied to the interpretation of the operational solution of an important problem arising in the design of an electrical delay line with continuously distributed parameters. The electrical system is essentially a resistor-capacitor ladder network smoothed out into a space-wise continuous distribution.Only the interpretation of the operational solution is considered in this paper. A description of the electrical problem and its operational solution will be given in a separate paper by D. Hazony. In the rational approximation technique great importance is attached to the ease with which the Pade table can be recursively computed by a method developed by the author.

The equations given in part 1 of this paper are further developed so as to facilitate computation. Numerical results for the surface deformation and the perturbation in the superficial gravity field of the earth are presented for the Gutenberg earth model, which is assumed to be subjected to surface mass loading. A certain difficulty in computing the deformation of the earth close to a concentrated load is experienced, but a method for resolving this difficulty is outlined. Values of Love numbers hn, kn, ln and the associated load-deformation coefficients hn′, kn′, ln′ are presented to three places of decimals for values of n up to 25, and approximate values of hn′, kn′, ln′ are given for n = 26 through 40.

In order to test the applicability of ray theory even to extremely large ranges, we have applied it to the problem of determining the exact shape of the pressure pulse received at a range r equal to 460 times the depth of the layer H, in the case of an explosion in a layered liquid. The time variation of the pressure at the source was assumed to be given by a Heaviside unit function. Comparison is made with a previous solution of this problem which was obtained, for the identical conditions, by the use of the normal mode theory. The exact ray-theory solution exhibits the well-known characteristic features of a ground wave followed by a dispersive water wave, but the pattern of the received pressure pulse is more ruffled than in the normal mode solution, in which the higher modes, as well as branch-line integrals, were neglected. The applicability of ray theory to long-range propagation is made feasible by virtue of the mutual cancellation at long ranges of all but a group of the last-arriving rays.

The method of approximate Laplace transform inversion, by analytically inverting suitable rational approximations to a function f¯(p) of the operator p is applied to the problem of propagation from a quadratic pulse point source in a uniform liquid sphere. Apart from this main problem, the relation between the efficiency of the method and the sharpness of the pulse is illustrated in a general way by considering three simple examples (in descending order of sharpness), namely the approximate inversion of the Laplace transforms of a delayed Heaviside unit function, a simple function having a gradient discontinuity, and a quadratic pulse. In order to suggest how the technique can be extended to more difficult theoretical seismic problems, the method is applied to a simple function f¯(p) having branch points on the imaginary axis in the p-plane.

A method is given for the determination of the surface deformation of the earth and the perturbation in its superficial gravity field caused by an idealized unit point-mass load placed on the surface. By use of this Green's function the response of an accepted earth model to any superficial mass layer such as the variable ocean tide can readily be calculated. The theory is presented in this paper. Numerical results will be given in a later publication.

Summary A method is developed for the solution of problems of pulse propagation in elastic half-spaces. It is based on Laplace transform inversion by means of rational approximations applied to the factor muItiplying the Bessel function in the integral representing the operational solution. The discontinuities (at arrival times of rays) are generated automatically by analytic inversion of the rational approximations, followed by the analytic evaluation of some Bessel function integrals. transform In a previous paper (Longman 1966) approximate inversion of the Laplace ai J 0 by means of suitable rational approximations to J(p) was applied to the solution of the problem of pulse propagation from a point source in a fluid sphere. The method, based on the work of Luke (1962,1964), was found to be superior to a previous method suggested by the author (Longman 1964) which was based on the numerical values of f(p) at points on the real axis in the p-plane. However, difficulties associated with discontinuities in the solution at the arrival times of rays from too sharp a pulse were still encountered. In the present paper discontinuities in the solution appear automatically in our approximate solutions, being derived from certain discontinuous integrals involving Bessel functions. In other words, we do not attempt to approximate the discontinuous solution by a sequence of continuous functions (a process which would lead to poor convergence), but the discontinuities are inherently present in the approximation functions themselves. In addition to the discontinuities appropriate to the solution, our approximate solutions may exhibit spurious singularities which are in no way related to the physical problem. Thc spurious nature of these singularities can be recognized from the fact that they become progressively thinner and more numerous as we increase the order of the approximation. They can effectively be eliminated by applying the method appropriately in a number of different ways to a given problem, in such a way that the spurious singularities are outside various time intervals. Thus we can build up an approximate solution over the whole time range of interest without spurious singularities. The essential feature in the present paper is that for an operational solution of the form

Results of world-wide earth-tide observations made by the Institute of Geophysics during the IGY are presented. The observations were made at 13 stations distributed around the world with two LaCoste and Romberg tidal gravity meters, each observation lasting 30 to 40 days. The data were analyzed by a method consisting of (a) a power spectrum analysis to determine the general noise background in the spectrum and (b) Fourier scanning of the major tidal lines to determine the amplitudes and phases of the gravity variation at each site as compared with the variation computed for a perfectly rigid earth. A Chebyshev window was used in the Fourier scanning to reduce contamination of the estimates by neighboring lines. Knowledge of the noise levels permits error limits to be assigned to the determination of the amplitude and phase relationships. The results obtained at the various stations differ significantly in relation to the estimated error limits and may be summarized as follows in comparison with theoretical results computed for various earth models. The amplitude ratio of the static tide on a homogeneous incompressible fluid earth to that on a rigid earth would be 1.25. For a semidiurnal tide on model earths of the Bullard I, II, Bullen B, and Gutenberg types, this ratio would have the values 1.161, 1.150, 1.162, and about 1.156, respectively. Our maximum observed value for the largest semidiurnal tide, M2, is 1.249 (Bermuda) and the minimum is 1.153 (New Delhi). The mean for the eight oceanic stations is 1.192, that for the four continental stations is 1.169, and the mean for all stations is 1.184. Thus our continental value is about 1 per cent greater than those computed for the Bullen B or Gutenberg models, but our mean oceanic value is 3 per cent greater. The causes of these variations require further investigation. It is believed that both the oceanic tides and regional geologic differences contribute to the variations. The O1 tide, whose period differs significantly from the solar day, also has a mean amplitude ratio of 1.18, but the P1 and K1 combination (whose period is 24 hours) gives the significantly different ratio 1.14 and is possibly influenced by the daily temperature cycle. The observed average phase lag of the M2 tide is compatible with the astronomical evidence concerning the changing rotation rate of the earth. However, the phases exhibit rather large local variations, the causes of which need further examination.

(Received 1970 January 6) Summary Methods are presented for the numerical solution of theoretical seismic problems. The methods are based on approximate Laplace transform inversion techniques developed by the author, special attention being given to location and magnitude of ray arrivals.

A summary of formulas with which the tidal accelerations due to the moon and the sun can be computed at any given time for any point on the earth's surface, without reference to tables, is presented in this paper. These formulas are convenient for computer use.

The theoretical analysis of problems in geophysics, and particularly in seismology, leads in many cases to solutions expressed in terms of integrals that cannot be evaluated by conventional numerical methods. The usual difficulties are connected with the oscillatory nature of the integrand and with the presence of one or more poles in the path of integration. In the simpler problems these difficulties can sometimes be circumvented by appropriate transformations in the complex plane, though this often involves lengthy analysis. A method is presented for the direct and accurate evaluation of such solutions with a minimal use of algebraic manipulations. By way of illustration the method is applied to Lamb's problem with an oscillatory source.

Summary Computations of Love numbers and load deformation coefficients for a Gutenberg model Earth are described. The computations present some special features which are absent in the calculation of the periods of free oscillation of the Earth. This is due to the fact that the absence of inertia terms in the static problem involves a certain indeterminacy in the particle displacement in the liquid core. It is found that a necessary condition for the existence of a solution of the problem of the static deformation of the Earth under surface mass loads or externally applied gravity fields is that the Adams-Williamson condition holds in the core. This is without any assumption as to chemical homogeneity, but is merely a consequence of the assumption of zero rigidity in the core. Another special feature is the computation of the load deformation coefficient h′0(of order zero), which has a significance even though the ordinary Love number h0 does not exist. h′0 has not previously been evaluated by other authors, nor has its existence been noted. Three special dimensionless constants play a significant role in the calculations. Results of the computation are compared with results due to Takeuchi et al, and due to Kaula.

Summary In previous papers the first author has demonstrated the application of rational approximations to Laplace transform inversion in theoretical seismic problems. One of the difficulties in the use of rational approximations is the computation of the roots (usually complex) of the denominator in order to effect the inversion. Calculation of the roots of a highorder polynomial can be very time-consuming even when a large computer is available. The present paper demonstrates a method for performing the inversion without calculating these roots.

A method is presented whereby earth-tide gravity records can conveniently be interpolated. It is based on the theoretical rigid-earth gravity tide go(t) and its time derivative. The possibility is envisaged that earth-tide gravity records for any station may be generated in their entirety by this method for any station for which a pair of constants has been determined.