In physics and mathematics, the spacetime triangle diagram (STTD) technique, also known as the Smirnov method of incomplete separation of variables, is the direct space-time domain method for electromagnetic and scalar wave motion.
Basic stages (Electromagnetics) The system of Maxwell's equations is reduced to a second-order PDE for the field components, or potentials, or their derivatives. The spatial variables are separated using convenient expansions into series and/or integral transforms—except one that remains bounded with the time variable, resulting in a PDE of hyperbolic type. The resulting hyperbolic PDE and the simultaneously transformed initial conditions compose a problem, which is solved using the Riemann–Volterra integral formula. This yields the generic solution expressed via a double integral over a triangle domain in the bounded-coordinate—time space. Then this domain is replaced by a more complicated but smaller one, in which the integrant is essentially nonzero, found using a strictly formalized procedure involving specific spacetime triangle diagrams (see, e.g., Refs.). In the majority of cases the obtained solutions, being multiplied by known functions of the previously separated variables, result in the expressions of a clear physical meaning (nonsteady-state modes). In many cases, however, more explicit solutions can be found summing up the expansions or doing the inverse integral transform.
STTD versus Green's function technique The STTD technique belongs to the second among the two principal ansätze for theoretical treatment of waves — the frequency domain and the direct spacetime domain. The most well-established method for the inhomogeneous (source-related) descriptive equations of wave motion is one based on the Green's function technique. For the circumstances described in Section 6.4 and Chapter 14 of Jackson's Classical Electrodynamics, it can be reduced to calculation of the wave field via retarded potentials (in particular, the Liénard–Wiechert potentials). Despite certain similarity between Green's and Riemann–Volterra methods (in some literature the Riemann function is called the Riemann–Green function ), their application to the problems of wave motion results in distinct situations:
The definitions of both Green's function and corresponding Green's solution are not unique as they leave room for addition of arbitrary solution of the homogeneous equation; in some circumstances the particular choice of Green's function and the final solution are defined by boundary condition(s) or plausibility and physical admissibility of the constructed wavefunctions. The Riemann function is a solution of the homogeneous equation that additionally must take a certain value at the characteristics and thus is defined in a unique way. In contrast to Green's method that provides a particular solution of the inhomogeneous equation, the Riemann–Volterra method is related to the corresponding problem, comprising the PDE and initial conditions,
and it was the Riemann–Volterra representation that Smirnov used in his Course of Higher Mathematics to prove the uniqueness of the solution to the above problem (see, item 143).
In the general case, Green's formula implies integration over the entire domain of variation of coordinates and time, while integration in the Riemann–Volterra solution is carried out within a limited triangle region, assuring the boundness of the solution support. Causality of the (unique) Riemann–Volterra solution is provided automatically, without need to recur to additional considerations, such as the retarded nature of the argument, wave propagation in certain direction, specific choice of the integration path, etc. (Usually the descriptive equations, such as the classical scalar wave equation, possess the T-symmetry. It is the time-asymmetric initial conditions that define the arrow of time through the limitation of the integration domain in the Riemann formula to t > 0 {\displaystyle t>0} , see more in and a particular example given below.) Green's function can be readily derived from the Liénard–Wiechert potential of a moving point source, but concrete calculation of the wavefunction, inevitably involving the analysis of the retarded argument, may develop in a rather complicated task unless some special techniques, like the parametric method, are invoked. The Riemann-Volterra approach presents the same or even more serious difficulties, especially when one deals with the bounded-support sources: here the actual limits of integration must be defined from the system of inequalities involving the space-time variables and parameters of the source term. However, this definition can be strictly formalized using the spacetime triangle diagrams. Playing the same role as the Feynman diagrams in particle physics, STTDs provide a strict and illustrative procedure for definition of areas with the same analytic representation of the integration domain in the 2D space spanned by the non-separated spatial variable and time.
Drawbacks of the method The method can only be applied to problems possessing known Riemann function. Application of the method and analysis of the results obtained require more profound knowledge of the special functions of mathematical physics (e.g., operating with the generalized functions, Mathieu functions of different kinds and Lommel's functions of two variables) than Green's function method. In some cases the final integrals require special consideration in the domains of rapid oscillation of the Riemann function.
Most important concretizations
General considerations Several efficient methods for scalarizing electromagnetic problems in the orthogonal coordinates x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} were discussed by Borisov in Ref. The most important conditions of their applicability are h 3 = 1 {\displaystyle h_{3}=1} and
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