Spacetime triangle diagram technique
In physics and mathematics, the spacetime triangle diagram 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
- 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 . 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.
- 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. 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 most well-established method for the inhomogeneous 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.
Despite certain similarity between Green's and Riemann–Volterra methods, 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 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,
- 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 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.
- 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,
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 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 were discussed by Borisov in Ref.The most important conditions of their applicability are and
, where are the
metric coefficients. 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 meeting the required boundary conditions at the surface of the guiding system.
Cartesian and cylindrical coordinates
In the Cartesian and general-type cylindrical coordinatesseparation of the spatial variables result in the initial value problem for a hyperbolic PDE known as the 1D Klein–Gordon equation
Here is the time variable expressed in units of length using some characteristic velocity,
is a constant originated from the separation of variables, and represents a part of the source
term in the initial wave equation that remains after application of the variable-separation procedures.
The above problem possesses known Riemann function
where is the Bessel function of the first kind of order zero.
Passing to the canonical variables one gets the simplest STTD diagram reflecting straightforward application of the Riemann–Volterra method, with the fundamental integration domain represented by spacetime triangle MPQ.
Rotation of the STTD 45° counter clockwise yields more common form of the STTD in
the conventional spacetime.
For the homogeneous initial conditions the solution of the problem is given by the Riemann formula
Evolution of the wave process can be traced using a fixed observation point successively increasing the triangle height or, alternatively, taking "momentary picture" of the wavefunction by shifting the spacetime triangle along the axis.
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.
Spherical coordinates
In the spherical coordinate system — which in view of the General considerations must be represented in the sequence, assuring — one can scalarize problems for the transverse electric or transverse magnetic waves using the Borgnis functions, Debye potentials or Hertz vectors. Subsequent separation of the angular variables via expansion of the initial wavefunction and the source
where is the associated Legendre polynomial of degree and order
, results in the initial value problem for the hyperbolic
Euler–Poisson–Darboux equation
known to have the Riemann function
where is the Legendre polynomial of degree.
Equivalence of the STTD (Riemann) and Green's function solutions
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, 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 while the retarded Green's function to the Klein–Gordon equation is a Fourier transform of the imaginary exponential term reducible to
Extending integration with respect to to the complex domain, using the residue theorem
one gets
Using formula 3.876-1 of Gradshteyn and Ryzhik,
the last Green's function representation reduces to the expression
in which 1/2 is the scaling factor of the Riemann formula and the Riemann function, while the Heaviside step function reduces, for, the area of integration to the fundamental triangle MPQ, making the Green's function solution equal to that provided by the STTD technique.