Peter G. Malischewsky

Surface Waves and Discontinuities

1. Introduction. General background. General remarks about wave-guides with discontinuities. 2. Mathematical Aspects. The boundary conditions and their seismological interpretation. The differential operators in closed domains. The differential operators in open domains. 3. The Undisturbed Wave-Guide. The mode conception. The closed wave-guide. The open wave-guide. 4. The Disturbed Wave-Guide. General remarks. The bracket formalism (interrelations with quantum theory). The closed wave-guide with a vertical discontinuity or an impedance jump, respectively. The open wave-guide with vertical discontinuity and an impedance jump. Lateral inhomogeneities and seismic surface waves. 5. The Wiener-Hopf Technique. General remarks. Application of the Wiener-Hopf technique to an impedance jump in an open wave-guide. Application of the Wiener-Hopf technique to an impedance jump in a closed wave-guide. 6. The Three-Dimensional Problem. General remarks. Some special cases. Some remarks concerning Rayleigh waves. 7. Appendix. Calculation of the integrals I 1 , I 2 , I 3 . Perturbation method for complex roots. 8. References. 9. Index.

Comment to “A new formula for the velocity of Rayleigh waves” by D. Nkemzi [Wave Motion 26 (1997) 199–205]

Abstract A new formula for the phase velocity of Rayleigh waves for all Poisson values by Nkemzi [Wave Motion 26 (1997) 199–205] is discussed and commented. By applying Cardan’s formula and using the advantages of mathematica the author succeeds in extracting a formula which is also valid in the whole range of Poisson ratios and is, contrary to Nkemzi, as well correct as probably the simplest representation of the relevant real root of Rayleigh’s equation. Additionally, a similar formula is presented for the complex roots. In this connection, for the first time an analytic expression of the critical Poisson ratio, which limits the range of complex roots, is derived.

Comparison of approximated solutions for the phase velocity of Rayleigh waves (Comment on ‘Characterization of surface damage via surface acoustic waves’)

An approximation for the SAW velocity (Rayleigh waves) in dependence on Poissons ratio, which was misprinted in a former paper (P D Warren et al 1996 Nanotechnology 7 295–301), is corrected. At the same time, the connection of this approximation with the recent history of exact formulae for the Rayleigh-wave velocity is established.

Mode series expansions at vertical boundaries in elastic waveguides

Abstract This paper investigates the mathematical foundations of modal expansion series of the Love and Rayleigh type at vertical boundaries in elastic waveguides. A functional analytic treatment in a product Hilbert space of displacement–stress vector functions is given. For Love modes, an expansion theorem is achieved. The Love mode series are shown to be Fourier-like expansion series, and this provides a clear insight into their convergence behaviour. For the Rayleigh modes, a formulation of the eigenvalue problem is introduced and a completeness proof is obtained. It is shown how the orthogonality relation, which is known from seismology, can be represented in the product Hilbert space. This also leads to the normability of Rayleigh modes due to the orthogonality relation.

Improved Approximations of the Rayleigh Wave Velocity

In this article we have derived some approximations for the Rayleigh wave velocity in isotropic elastic solids which are much more accurate than the ones of the same form, previously proposed. In particular: (1) A second (third)-order polynomial approximation has been found whose maximum percentage error is 29 (19) times smaller than that of the approximate polynomial of the second (third) order proposed recently by Nesvijski [Nesvijski, E. G., J. Thermoplas. Compos. Mat. 14 (2001), 356—364]. (2) Especially, a fourth-order polynomial approximation has been obtained, the maximum percentage error of which is 8461 (1134) times smaller than that of Nesvijskis second (third)-order polynomial approximation. (3) For Brekhovskikh—Godins approximation [Brekhovskikh, L. M., Godin, O. A. 1990, Acoustics of Layered Media: Plane and Quasi-Plane Waves. Springer-Verlag, Berlin], we have created an improved approximation whose maximum percentage error decreases 313 times. (4) For Sinclairs approximation [Malischewsky, P. G., Nanotechnology 16 (2005), 995—996], we have established improved approximations which are 4 times, 6.9 times and 88 times better than it in the sense of maximum percentage error. In order to find these approximations the method of least squares is employed and the obtained approximations are the best ones in the space L2[0, 0.5] with respect to its corresponding subsets.

Surface waves in media having lateral inhomogeneities

SummaryOn the basis of Alsops method (1966), approximated reflection and transmission coefficients are obtained for surface waves obliquely incident at a vertical discontinuity. Rayleigh and Love waves are arranged in such a way that they form a homogeneous eigenfunction system and their mutual conversion into each other is compatible with this theory. The structure of the eigenvalue equation is responsible for the mathematical complications in many surface wave problems. On the basis of a computer routine, first result concerning the angular dependence of the reflection and transmission coefficients including mode conversion are presented. The problem of phase jump is discussed. A comparison of theoretical and experimental results shows the applicability of Alsops method for 90°-corners, too.

A Review and Some New Issues on the Theory of the H/V Technique for Ambient Vibrations

In spite of the Horizontal-to-Vertical Spectral Ratio (HVSR or H/V) technique obtained by the ambient vibrations is a very popular tool, a full theoretical explanation of it has been not reached yet. A short excursus is here presented on the theoretical models explaining the H/V spectral ratio that have been development in last decades. It leads to the present two main research lines: one aims at describing the H/V curve by taking in account the whole ambient-vibration wavefield, and another just studies the Rayleigh ellipticity. For the first theoretical branch, a comparison between the most recent two models of the ambient-vibration wavefield is presented, which are the Distributed Surface Sources (DSS) one and the Diffuse Field Approach (DFA). A mention is done of the current developments of these models and of the use of the DSS for comparing the H/V spectral ratio definitions present in literature. For the second research branch, some insights about the connection between the so-called osculation points of the Rayleigh dispersion curves and the behaviour of the H/V curve are discussed.

An Improved Formula of Fundamental Resonance Frequency of a Layered Half-Space Model Used in H/V Ratio Technique

The resonance frequency of the transmission response in layered half-space model is important in the study of site effect because it is the frequency where the shake-ability of the ground is enhanced significantly. In practice, it is often determined by the H/V ratio technique in which the peak frequency of recorded H/V spectral ratio is interpreted as the resonance frequency. Despite of its importance, there has not been any formula of the resonance frequency of the layered half-space structure. In this paper, a simple approximate formula of the fundamental resonance frequency is presented after an exact formula in explicit form of the response function of vertically SH incident wave is obtained. The formula is in similar form with the one used in H/V ratio technique but it reflects several major effects of the model to the resonance frequency such as the arrangement of layers, the impedance contrast between layers and the half-space. Therefore, it could be considered as an improved formula used in H/V ratio technique. The formula also reflects the consistency between two approaches of the H/V ratio technique based on SH body waves or Rayleigh surface waves on the peak frequency under high impedance contrast condition. This formula is in explicit form and, therefore, may be used in the direct and inverse problem efficiently. A numerical illustration of the improved formula for an actual layered half-space model already investigated by H/V ratio technique is presented to demonstrate its new features and its improvement to the currently used formula.

Approximate Formula of Peak Frequency of H/V Ratio Curve in Multilayered Model and Its Use in H/V Ratio Technique

The main peak frequency of the Horizontal-to-Vertical (H/V) ratio curve is the key factor used in the H/V ratio technique since the resonance frequency of the transmission response of the site is estimated from this frequency. However, there has not been explicit formula of the main peak frequency of the H/V ratio curve in multilayered models. In the present study, an approximate explicit equation of the peak frequency of H/V ratio is derived for the multilayered models of high impedance contrast between the half-space and surface layers. This approximate equation is then generalized for model of an functionally graded material (FGM) layer over half-space. Then, the approximate equation is used to obtain an explicit approximate formula of the main peak frequency of H/V ratio curve. The principle formula of H/V ratio technique is used along with the obtained approximate formula of the main peak frequency to formulate a new average formula of the shear-wave velocity of a composite layer composed of an arbitrary number of horizontal, homogeneous layers. The new average formula is shown to be more suitable in the use of H/V ratio technique than the currently used ones in the sense that it takes into account the effect of the mass density and the position of sublayers. Finally, some numerical calculation to illustrate the application of the peak formula and the new average formula of shear-wave velocity is presented.

H/V spectral ratio analysis and Rayleigh modelization in Eastern Thuringia, Germany

We use records from the East Thuringian Seismic Network (OTSN, Ostthuringer Seismisches Netzwerk) to characterize the site response for each station, and to analyze the scope and limits of the Rayleigh modeling for H/V spectral ratio. The stations considered in this work can be classified by their seismic response as hard rock sites or as sites with some site effect. From these results we propose velocity models based on Rayleigh modeling (theoretical Rayleigh wave ellipticity). Our results show that for locations affected by site effects the H/V spectral ratio can be modeled by the theoretical ellipticity of layered velocity models. For hard rock sites the spectral ratio is rather flat and the modeling with the theoretical ellipticity was not very clear. This may be explained by the fact that for hard rock sites the conditions for a clear fundamental frequency associated with S-wave resonance, and therefore with Rayleigh wave ellipticity, are not fulfilled.