J. Vaněk

Expansion of a Plane Wave into Gaussian Beams

An integral expansion which expresses a plane monochromatic wave as a superposition of Gaussian beams is found. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces.

Reflection and transmission coefficients for transition layers

SummaryFormulae are derived for the reflection and transmission coeficients of plane elastic waves for a transition layer. Haskells technique and the so-called delta matrices[5, 7] are used for this purpose. No problems are encountered in deriving the reflections and transmission coefficients from Haskells matrices[3]. However, in some cases Haskells matrices do not guarantee the accuracy required. For this reason attention is mainly devoted to deriving the reflection and transmission coefficients from the delta matrices. In deriving the transmission coefficients use is made of the fact that some3×3 subdeterminants of the delta matrices are squares of the3×3 subdeterminants of Haskells matrices.

Ray amplitudes of seismic body waves in inhomogeneous radially symmetric media

РезюмеИсхо¶rt;я uз сnецuaльноŭ annроксuмaцuu скоросmного рaзрезa, nолучены nросmые aнaлumuческuе вырaженuя ¶rt;ля лучевых uнmегрaлов,геомеmрuческого рaсхож¶rt;енuя u лучевых aмnлumу¶rt; обьёмных волн, рaсnросmрaняющuхся в сферuческu сuммеmрuчноŭ мо¶rt;елu Землu. Эma annроксuмaцuя обесnечuвaеm неnрерывносmь скоросmu u её nервоŭ u вmороŭ nроuзво¶rt;ных, не обрaзуеm слоев nонuженноŭ скоросmu u uскуссmвенных aномaлuŭ aмnлumу¶rt;ных крuвых. Пре¶rt;ложеннaя annроксuмaцuя maкже знaчumельно nовышaеm усmоŭчuвосmь aмnлumу¶rt;ных крuвых.

Partial derivatives of dispersion curves of love waves in a single-layered medium

ZusammenfassungEs werden die ersten und zweiten partiellen Ableitungen der Dispersionskurven von Love-Wellen im Einschichtenmilieu studiert. Die Berechnung der partiellen Ableitungen führt man nicht anhand numerischen Ableitens durch, sondern mit einer schnelleren und genaueren, auf dem Satz der impliziten Funktionen gegründeten Methode. Die sich aus dem Studium partieller Ableitungen für die Interpretation der beobachteten Dispersionskurven ergebenden Folgerungen werden diskutiert.

Low-frequency and high-frequency expressions for the reflection and transmission coefficients of seismic waves for transition layers

SummaryLow-frequency and high-frequency expressions are derived for the reflection and transmission coefficients of SH waves on transition layers, in which the modulus of torsion μ and density ϱ vary with depth only. Both the expressions are exact for any finite frequencies. However, the use of the former formulae is more convenient for low frequencies and, likewise, the latter formulae for high frequencies. Explicit expressions for several first terms of the expansions in terms of ω and 1/ω are given.

Theoretical dispersion curves for the seismic profile Kašperské Hory (Czechoslovakia) — Ksiaz (Poland)

РезюмеРaссчumaны mеореmuческuе ¶rt;uсnерсuонные крuвые nоверхносmных сеŭсмuческuх волн ¶rt;ля nрофuляКНС-КSР. Рaссмamрuвaемaя мо¶rt;ель земноŭ коры основaнa нa резульmamaхглубuнного сеŭсмuческого зон¶rt;uровaнuя в¶rt;оль VII меж¶rt;унaро¶rt;ного nрофuля. Эma мо¶rt;ель можеm служumь нaчaльноŭ мо¶rt;елью в бу¶rt;ущuх эксnерuменmaльных uссле¶rt;овaнuях ¶rt;uсnерсuu в¶rt;оль nрофuляКНС-КSР.

Effects of causal absorption on seismic body waves

SummaryThe effects of the causal absorption on seismic wave fields are discussed. Several computer programs for the evaluation of seismic wave fields have been modified to include the causal absorption. The programs are based mostly on matrix and ray methods. The main attention is paid to ray methods. A simple “plane wave algorithm” is described, which can be used along the ray to compute elementary seismograms in dissipative media. For a certain class of signals, commonly used in seismology, its application does not require any convolution or frequency-domain computations, and is only slightly slower than the evaluation of elementary seismograms in non-dissipative media. The derived algorithm even describes the velocity dispersion related to absorption and the decrease of the prevailing frequency of the signal well. Numerical examples are presented, both exact and approximate.

Approximate expression for the hilbert transform of a certain class of functions and their application to the ray theory of seismic waves

SummaryApproximate expressions for the Hilbert transform of the functionf(t)=exp(-ω02t2/γ2) cos(ω0t+v) are determined. This function, given a suitable choice of the three parameters ω0, γ and v, approximates a wide class of seismic signals very well. The approximate expressions for the Hilbert transform enable very simple formulae to be given for the elementary seismograms of the individual seismic body waves (in the zero approximation of the ray theory). This accelerates the computation of ray theoretical seismograms considerably.

Ray amplitudes in a three-dimensional inhomogeneous medium

SummaryA system of five ordinary differential equations of the first order to compute the geometrical spreading of the wave front of seismic body waves in a three-dimensional medium is suggested.

Elementary seismograms of seismic body waves in dissipative media

Approximate expressions for elementary seismograms of seismic body waves propagating in media with small causal absorption are derived. Special attention is devoted to modulated signals with a smooth envelope, for which especially simple formulae were obtained. The derived expressions give a good picture of all important effects of causal absorption, viz., the frequency dependent exponential decrease of amplitude, the velocity dispersion related to absorption, and the decrease of the prevailing frequency.