General considerations

Several efficient methods for scalarizing electromagnetic problems in the orthogonal coordinates ${\displaystyle x_{1},x_{2},x_{3}}$ were discussed by Borisov in Ref.^[10] The most important conditions of their applicability are ${\displaystyle h_{3}=1}$ and ${\displaystyle \partial _{3}(h_{1}/h_{2})=0}$, where ${\displaystyle h_{i},i=1,2,3}$ are the metric (Lamé) coefficients (so that the squared length element is ${\displaystyle ds^{2}=h_{1}^{2}dx_{1}^{2}+h_{2}^{2}dx_{2}^{2}+h_{3}^{2}dx_{3}^{2}}$). 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.

Cartesian and cylindrical coordinates

In the Cartesian ${\displaystyle \left\{x_{1}=x,\;x_{2}=y,\;x_{3}=z\right\}}$ and general-type cylindrical coordinates ${\displaystyle \left\{x_{1}=x_{1}(x,y),\;x_{2}=x_{2}(x,y),\;x_{3}=z\right\}}$ separation of the spatial variables result in the initial value problem for a hyperbolic PDE known as the 1D Klein–Gordon equation (KGE)

${\displaystyle {\begin{aligned}&\left(\partial _{\tau }^{2}-\partial _{z}^{2}+k^{2}\right)\psi (\tau ,z)=f(\tau ,z)\\&\psi (\tau ,z)=0{\text{ for }}\tau <0\end{aligned}}}$

Here ${\displaystyle \tau }$ is the time variable expressed in units of length using some characteristic velocity (e.g., speed of light or sound), ${\displaystyle k}$ is a constant originated from the separation of variables, and ${\displaystyle f(\tau ,z)}$ 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).

The above problem possesses known Riemann function

${\displaystyle R_{k}(\tau ,z;\tau ',z')=J_{0}\left(k{\sqrt {(\tau -\tau ')^{2}-(z-z')^{2}}}\right),}$

where ${\displaystyle J_{0}(\cdot )}$ is the Bessel function of the first kind of order zero.

Canonical variables ξ, η.

Initial variables z, τ.

The simplest STTD representing a triangle integration domain resulted from the Riemann–Volterra integral formula.

Passing to the canonical variables ${\displaystyle z,\tau \to \xi ,\eta }$ 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).

Rotation of the STTD 45° counter clockwise yields more common form of the STTD in the conventional spacetime ${\displaystyle z,\tau }$.

For the homogeneous initial conditions the (unique^[8]) solution of the problem is given by the Riemann formula

${\displaystyle \psi (\tau ,z)={\frac {1}{2}}\iint \limits _{\triangle MPQ}\,d\tau '\,dz'R(\tau ,z;\tau ',z')f(\tau ',z').}$

Evolution of the wave process can be traced using a fixed observation point (${\displaystyle z={\text{const}}}$) successively increasing the triangle height (${\displaystyle \tau }$) or, alternatively, taking "momentary picture" of the wavefunction ${\displaystyle \psi }$ by shifting the spacetime triangle along the ${\displaystyle z}$ axis (${\displaystyle \tau ={\text{const}}}$).

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.

Static limitations to the source area^[10]

STTD for a source limited from left by plane ${\displaystyle z=0}$, i.e. ${\displaystyle f(\tau ,z)=0{\text{ for }}z<0}$, which is the case, e.g., for a travelling source propagating along a semi-infinite radiator ${\displaystyle l}$.

STTD for a source limited from right by plane ${\displaystyle z=l}$, i.e. ${\displaystyle f(\tau ,z)=0{\text{ for }}z>l.}$

STTD for a source limited from both sides, i.e. ${\displaystyle f(\tau ,z)=0{\text{ for }}z\notin [0,l]}$, which is the case, e.g., for a travelling source propagating along a radiator of finite length ${\displaystyle l}$.

Combined action of limitations of different type, see Refs.^{[1][10][11][12][13]} for details and more complicated examples

STTD for a semi-infinite travelling source pulse.

STTD for a finite travelling source pulse.

STTD for a finite travelling source pulse propagating along a semi-infinite radiator ${\displaystyle z\in \left[0,\infty \right)}$.

A sequence of generic STTDs for a "short", finite source pulse of duration ${\displaystyle T}$ propagating along a finite radiator ${\displaystyle z\in [0,l]}$ with a constant velocity ${\displaystyle \beta }$. In this case the source can be expressed in the form

${\displaystyle f(\tau ,z)=f_{0}(\tau ,z)\times h(z)h(l-z)h\left(\tau -{\frac {z}{\beta }}\right)h\left(T-\tau +{\frac {z}{\beta }}\right),}$

    where ${\displaystyle h(\cdot )}$ is the Heaviside step function.

The same STTD sequence for a "long" pulse.

Spherical coordinates

In the spherical coordinate system — which in view of the General considerations must be represented in the sequence ${\displaystyle \left\{x_{1}=\theta ,\;x_{2}=\varphi ,\;x_{3}=r\right\}}$, assuring ${\displaystyle h_{3}=1}$ — one can scalarize problems for the transverse electric (TE) or transverse magnetic (TM) waves using the Borgnis functions, Debye potentials or Hertz vectors. Subsequent separation of the angular variables ${\displaystyle \theta ,\varphi }$ via expansion of the initial wavefunction ${\displaystyle \psi \left(\tau ,\theta ,\varphi ,r\right)}$ and the source

${\displaystyle \;f\left(\tau ,\theta ,\varphi ,r\right)}$ in terms of

${\displaystyle \left({\begin{array}{c}\sin m\varphi \\\cos m\varphi \end{array}}\right)P_{n}^{(m)}(\cos \theta ),}$

where ${\displaystyle P_{n}^{(m)}\!\left(\cdot \right)}$ is the associated Legendre polynomial of degree ${\displaystyle n}$ and order ${\displaystyle m}$, results in the initial value problem for the hyperbolic Euler–Poisson–Darboux equation^[3][10]

${\displaystyle {\begin{aligned}&\left(\partial _{\tau }^{2}-\partial _{r}^{2}+{\frac {n(n+1)}{r^{2}}}\right)\psi _{nm}(\tau ,r)=f_{nm}(\tau ,r)\\&\psi _{nm}(\tau ,r)=0{\text{ for }}\tau <0\end{aligned}}}$

known to have the Riemann function

${\displaystyle R(\tau ,r;\tau ',r')={P_{n}}\left({\frac {r^{2}+{r'}^{2}-(\tau -\tau ')^{2}}{2rr'}}\right),}$

where ${\displaystyle P_{n}\!\left(\cdot \right)}$ is the (ordinary) Legendre polynomial of degree ${\displaystyle n}$.