By J. T. Fokkema,P. M. van den Berg
The seismic purposes of the reciprocity theorem built during this publication are partially in response to lecture notes and guides from Professor de Hoop. each pupil Professor de Hoop has taught is aware the egg-shaped determine (affectionately often called "de Hoop's egg") that performs such a major position in his theoretical description of acoustic, electromagnetic and elastodynamic wave phenomena.
On the single hand this determine represents the area for the applying of a reciprocity theorem within the research of a wavefield and nonetheless it symbolizes the facility of a constant wavefield description of this theorem.
The roots of the reciprocity theorem lie in Green's theorem for Laplace's equation and Helmholtz's extension to the wave equation. In 1894, J.W. Strutt, who later grew to become Lord Rayleigh, brought in his booklet The thought of Sound this extension lower than the identify of Helmholtz's theorem. these days it really is referred to as Rayleigh's reciprocity theorem.
Progress in seismic facts processing calls for the data of all of the theoretical features of the acoustic wave concept. The reciprocity theorem used to be selected because the crucial subject matter of this e-book because it constitutes the fundaments of the seismic wave concept. In essence, states are unusual during this theorem. those might be different, even supposing sharing a similar time-invariant area of software, and they're similar through an interplay volume. the actual number of the 2 states determines the acoustic program, in flip making it attainable to formulate the seismic scan by way of a geological method reaction to a identified resource function.
In linear method idea, it's popular that the reaction to a identified enter functionality should be written as an critical illustration the place the impulse reaction acts as a kernel and operates at the enter functionality. as a result temporal invariance of the approach, this crucial illustration is of the convolution kind. In seismics, the temporal behaviour of the process is handled in a similar way; but the spatial interplay wishes a distinct process. The reciprocity theorem handles this interplay via determining one country with the spatial impulse functionality, sometimes called the Green's functionality, whereas the opposite kingdom is hooked up with the particular resource distribution. quite often, the ensuing crucial illustration isn't a spatial convolution. in addition, the systematic use of the reciprocity theorem results in a hierarchical description of the seismic test by way of expanding complexity. additionally from a tutorial viewpoint this strategy presents a hierarchy and the scholar learns to collapse the seismic challenge into constituent partial solutions.
This e-book should still give a contribution to the certainty that the reciprocity theorem is a robust software within the research of the seismic experiment.