Solution of inverse Lamb's problem with an unknown source

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

Download Full-text

Uniqueness of the Solution of Inverse Initial Value Problems for the Burgers Equation with an Unknown Source

Differential Equations ◽

10.1134/s001226612106001x ◽

2021 ◽

Vol 57 (6) ◽

pp. 701-710

Author(s):

A. V. Baev

Keyword(s):

Initial Value Problems ◽

Burgers Equation ◽

Initial Value ◽

Unknown Source ◽

Uniqueness Of The Solution

Download Full-text

The Story of Hero and Leander: A Possible Unknown Source of Celestina

A Companion to <i>Celestina</i> ◽

10.1163/9789004349322_010 ◽

2017 ◽

pp. 141-158

Keyword(s):

Unknown Source

Download Full-text

Numerical method of identification of an unknown source term in a heat equation

Mathematical Problems in Engineering ◽

10.1080/10241230212907 ◽

2002 ◽

Vol 8 (2) ◽

pp. 161-168 ◽

Cited By ~ 2

Author(s):

Afet Golayoğlu Fatullayev

Keyword(s):

Inverse Problem ◽

Heat Equation ◽

Numerical Method ◽

Minimization Problem ◽

Unknown Function ◽

Source Term ◽

Numerical Procedure ◽

Numerical Examples ◽

Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

Download Full-text