Directionality of electromagnetic radiation from fractures

Abstract

The directionality of electromagnetic radiation from tensile fracturing is calculated within our previously proposed model and shown to agree with experimental observations in the field. The best locations and orientations of measuring antennas are presented.

Appendix

Let t be the time since the beginning of the crack and assume the crack propagates at a constant velocity, v. Consider the crack at time t when its length is $x=vt$. The contribution to the dipole moment, p at this time from a strip of length $d{x}^{\prime }$ and width b located at ${x}^{\prime }$ and having a dipole moment of a constant density $\uprho$ (Fig. 4) is:

$$\begin{aligned}&dp=Re\{2b\rho \hbox {exp}[-\mu (x-{x}^{\prime })/ \hbox {v}]\nonumber \\&\qquad \quad \times \,\hbox {exp}\left[ i\omega \left( {x-{x}^{\prime }} \right) / v \right] \}d{x}^{\prime } \end{aligned}$$

(7)

The factor of 2 comes from the two crack sides; $\mu$ is the decaying constant (in S$^{-1}$) and the time elapsed between its creation and the “present” is $\left( {x-{x}^{\prime }} \right) / v$ . The total oscillating dipole moment as seen from a distant location is the integral on ${x}^{\prime }$ of dpfrom ${x}^{\prime }=0$ to x:

$$\begin{aligned}&p=Re\left\{ 2b\rho \exp \left( -\frac{\mu x}{v}+i\omega t \right) \right. \nonumber \\&\qquad \quad \times \left. \mathop {\int }\nolimits _0^x \hbox {exp}[(\mu -i\omega ){x}^{\prime }/ v ]d{x}^{\prime }\right\} \end{aligned}$$

(8)

Yielding

$$\begin{aligned} p=Re\left\{ {\frac{2b\rho v}{(\mu -i\omega }\left[ {1-\exp \left( {-\frac{\left( {\mu -i\omega } \right) x}{v}} \right) } \right] } \right\} \end{aligned}$$

(9)

Or

$$\begin{aligned} p=Aexp\left( {-\mu t} \right) \sin (wt+\varphi )+B \end{aligned}$$

(10)

where

$$\begin{aligned} A=\frac{2b\rho v}{\sqrt{(\mu ^{2}+\omega ^{2}}} B=\frac{2b\rho \mu v}{\upmu ^{2}+\omega ^{2}} \end{aligned}$$

(11)

And $\varphi =arctg\left( {\mu /w} \right)$.

Fig. 4

Schematic one wall of the crack at time $t (= x/v)$ showing x and ${x}^\prime$

At distances where r $\gg \lambda$ the only contributions to the radiation comes from$\ddot{p}$, the second time derivative of the total dipole moment. Thus B is of no consequence and the oscillation is only modified by a change of amplitude and a constant phase addition.