General theory of elastodynamic source fields

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Abstract
The most important and interesting source of elastic radiation in geophysics is an earthquake, or tectonic source, because the radiation field from such an energy source provides information on the largely unknown stress field within the earth. The actual mechanisms or processes of material failure undoubtedly can be described parametrically by the radiation field in terms of rupture velocity, rupture geometry, and the initial and residual stress within the region of failure. Accurate estimates of stress and the parameters of failure are therefore of particular significance in any description of the physical state of the material and would not be unrelated to the larger‐scale dynamical processes taking place within the earth. A number of methods and theories are presently used in estimating some of these parameters. The present study is intended to extend the dynamical theory of tectonic sources in order to provide a more complete description of earthquakes in terms of these basic parameters of rupture, including prestress. No assumptions are made concerning the nature of equivalent forces at the source or of their time dependence. The theory predicts the spatial and temporal form of the radiation field in terms of the initial prestress field and the basic rupture parameters. These predictions follow from the recognition that an earthquake is a relaxation source and that such a phenomenon is described analytically as an initial‐value problem. Consequently, such a source satisfies the conservation of energy and linear and angular momentum conditions required for a spontaneous source. The radiation field is produced by the continuous reduction of stored potential strain energy in the elastic medium surrounding a growing rupture zone, where it is assumed that the rupture, or at least a part of the total rupture zone, has a well‐defined boundary at a given time to which boundary conditions are applicable. The compatibility of this geometrically sharp, time‐varying boundary condition with probable failure processes in the earth is examined and judged to be good. Analytical expressions for the radiation field from an arbitrary source of elastic radiation are given, and within the framework of this formulation the properties of a spontaneous tectonic source are contrasted with ‘applied force’ sources and their special properties, as well as with some of the field observations of earthquake radiation fields. These considerations demonstrate the need for a more general and complete description of tectonic sources in order to explain all the observations and, more fundamentally, to deduce more precisely the nature of the physical processes of failure in the earth. It is concluded that a relaxation theory will provide the flexibility required to describe the characteristics of the observed radiation field and will also provide estimates of rupture parameters bearing on the processes of failure and the state of the material.A complete development of the dynamical relaxation theory for tectonic sources, including considerations of the total energy release and the final equilibrium field, constitutes the main result of this study, providing explicit expressions for the dynamic and static displacement and energy fields. It is shown that the radiation field will generally have a frequency‐dependent shape, dependent to first order on the ratio of rupture length to radiation wavelength. A relatively simple example is considered, and the radiation pattern and displacement and energy spectra computed. The radiation pattern is contrasted with the pattern from a more complicated rupture geometry and serves to demonstrate the pattern‐shape dependence on frequency, rupture geometry, and prestress. The energy and displacement spectra are also found to have maxima and minima, the number being dependent on the rupture geometry and their location and spacing being dependent on rupture length and velocity.

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