2.2.3. Electromagnetic Singularities

The electromagnetic coupling coefficient given in Equation (30), contains two singularities; either when *b*<sup>1</sup> or *b*<sup>2</sup> is equal to zero. The singularities lie on two circles in the *p*, *q* plane, shown in Figure 9; explicitly,

$$p^2 + q^2 = k\_0^2\tag{50}$$

and

$$(k\_0 \cos \varphi - p + k\_0)^2 + (k\_0 \sin \varphi - q)^2 = k\_0^2. \tag{51}$$

Each singularity will be most prominent when a frequency contour is tangential to the singular circle. In order to find the frequencies that this is true for, the solutions for *p* and *q* such that the gradient of the frequency contour expression of Equation (36) is equal to the gradient of the circle functions of Equations (50) and (51) are sought. Knowledge of the geometry of the contours and the radii of the circles is exploited to find the solutions for *p* and *q* and then the solutions are substituted into Equation (36) to give the tangential frequencies. The four solutions for *p* and *q* are

$$p = k\_0 \sin \varphi\_{bi} \text{ and } q = k\_0 \cos \varphi\_{bi\prime} \tag{52}$$

$$p = -k\_0 \sin q\_{\text{bi}} \text{ and } q = -k\_0 \cos q\_{\text{bi}}.\tag{53}$$

$$p = k\_0 \sin \varphi\_{bi} + 2k\_0 \sin^2 \varphi\_{bi} \text{ and } q = k\_0 \cos \varphi\_{bi} + 2k\_0 \sin \varphi\_{bi} \cos \varphi\_{bi} \tag{54}$$

and

$$p = -k\_0 \sin \varrho\_{bi} + 2k\_0 \sin^2 \varrho\_{bi} \text{ and } q = -k\_0 \cos \varrho\_{bi} + 2k\_0 \sin \varrho\_{bi} \cos \varrho\_{bi}.\tag{55}$$

Substituting the solutions for *p* and *q*, from Equations (52)–(55) into Equation (36) gives two distinct tangential frequencies:

$$\omega = 2^{3/4} \omega\_B \sqrt{\frac{\sqrt{1 \pm \sin \varrho\_{bi}}}{\cos \varrho\_{bi}}} \frac{\sqrt{\tanh(d \sqrt{2k\_0^2 (1 \pm \sin \varrho\_{bi})})}}{\sqrt{\tanh(2k\_0 \cos \varrho\_{bi} d)}},\tag{56}$$

where the solution with the + signs is for the *p* and *q* in Equations (52) and (53), and the − signs for the solutions of *p* and *q* in Equations (54) and (55).

**Figure 9.** Contours in the *p*, *q* plane defined by Equation (36) when *m* = *m* = 1. (**a**) Bistatic case with bistatic angle *ϕbi* = 45°, (**b**) monostatic case. The electromagnetic singularities are shown for both monostatic and bistatic radars. The yellow dashed circle shows the singularities defined by Equation (50) and the magenta dotted circle shows those defined by Equation (51). In the monostatic case, Equations (50) and (51) are equal and hence both circles are in the same location. The white contours highlight the frequencies tangential to the circles.

These values for *ω* are highlighted in Figure 7 by the white contours and are shown to be tangential to the circles expressed in Equations (50) and (51). Both singularities are highlighted in a simulated Doppler spectrum in Figure 10 by dashed vertical lines. A low amplitude Gaussian noise spectrum has been added as can be seen at the extremities of the plot. For deep water, or when *d* → ∞, the values for *ω* become

$$
\omega = 2^{3/4} \omega\_B \sqrt{\frac{\sqrt{1 \pm \sin q\_{bi}}}{\cos q\_{bi}}},
\tag{57}
$$

which agree with the results of Gill & Walsh [22].

**Figure 10.** A simulated bistatic Doppler spectrum showing the electromagnetic singularities.

#### 2.2.4. Currents

An ocean current affects how an ocean wave propagates, both in direction and speed. The change in speed means that, because of the Doppler effect, the entire spectrum is subject to an additional shift, *ω*. The additional shift is

$$
\triangle \omega = 2k\_0 \nu\_E(\varphi) \cos \varphi\_{\text{bi}\prime} \tag{58}
$$

where *νE*(*ϕ*) is the component of the current velocity in the elliptical normal direction for beam angle *ϕ*.
