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.
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.
[4] For the circumstances described in Section 6.4 and Chapter 14 of Jackson's Classical Electrodynamics,[4] 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 [5]), their application to the problems of wave motion results in distinct situations: [7] [8] 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,[8] item 143).
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.
Several efficient methods for scalarizing electromagnetic problems in the orthogonal coordinates
are the metric (Lamé) coefficients (so that the squared length element is
Remarkably, this condition is met for the majority of practically important coordinate systems, including the Cartesian, general-type cylindrical and spherical ones.
For the problems of wave motion is free space, the basic method of separating spatial variables is the application of integral transforms, while for the problems of wave generation and propagation in the guiding systems the variables are usually separated using expansions in terms of the basic functions (modes) meeting the required boundary conditions at the surface of the guiding system.
separation of the spatial variables result in the initial value problem for a hyperbolic PDE known as the 1D Klein–Gordon equation (KGE) Here
is the time variable expressed in units of length using some characteristic velocity (e.g., speed of light or sound),
represents a part of the source term in the initial wave equation that remains after application of the variable-separation procedures (a series coefficient or a result of an integral transform).
one gets the simplest STTD diagram reflecting straightforward application of the Riemann–Volterra method,[7][8] with the fundamental integration domain represented by spacetime triangle MPQ (in dark grey).
For the homogeneous initial conditions the (unique[8]) solution of the problem is given by the Riemann formula Evolution of the wave process can be traced using a fixed observation point (
) or, alternatively, taking "momentary picture" of the wavefunction
More useful and sophisticated STTDs correspond to pulsed sources whose support is limited in spacetime.
Each limitation produce specific modifications in the STTD, resulting to smaller and more complicated integration domains in which the integrand is essentially non-zero.
Examples of most common modifications and their combined actions are illustrated below.
In the spherical coordinate system — which in view of the General considerations must be represented in the sequence
— one can scalarize problems for the transverse electric (TE) or transverse magnetic (TM) waves using the Borgnis functions, Debye potentials or Hertz vectors.
, results in the initial value problem for the hyperbolic Euler–Poisson–Darboux equation[3][10] known to have the Riemann function where
The STTD technique represents an alternative to the classical Green's function method.
Due to uniqueness of the solution to the initial value problem in question,[8] in the particular case of zero initial conditions the Riemann solution provided by the STTD technique must coincide with the convolution of the causal Green's function and the source term.
The two methods provide apparently different descriptions of the wavefunction: e.g., the Riemann function to the Klein–Gordon problem is a Bessel function (which must be integrated, together with the source term, over the restricted area represented by the fundamental triangle MPQ) while the retarded Green's function to the Klein–Gordon equation is a Fourier transform of the imaginary exponential term (to be integrated over the entire plane
to the complex domain, using the residue theorem (with the poles
to satisfy the causality conditions) one gets Using formula 3.876-1 of Gradshteyn and Ryzhik,[15] the last Green's function representation reduces to the expression[16] in which 1/2 is the scaling factor of the Riemann formula and
, the area of integration to the fundamental triangle MPQ, making the Green's function solution equal to that provided by the STTD technique.