The Physics

The so-called deep-water equation for the speed of water waves is $$v = \sqrt \frac{g \lambda}{2 \pi}$$ where \(g\) is gravitational accelecation and \(\lambda\) is the wavelength of the water waves.

This equation is good when the water is deeper that about 10 wavelengths.

The water waves will have a wavelength half that of the radar waves, \( \lambda_{water} = \frac{1}{2} \lambda_{radar} \). $$v = \sqrt { \frac{g}{2 \pi} \frac {\lambda_{radar}}{2}}$$ so in terms of the radar frequency the equation becomes $$ v = \sqrt \frac{g c}{2 \pi \times 2 f}$$ with \(f\) the radar fequency and \(c\) the speed of light. $$ v = \sqrt \frac{g c}{4 \pi f}$$ In knots this reduces to [f in MHz]. $$ v = \frac{30}{\sqrt{f}}(knots)$$
The radar waves will reflect from waves advancing toward the radar and also those receding from the radar. This will produce a positively Doppler shifted set of echoes and a negatively Doppler shifted set.

These will form a pair of lines extending in range, called Bragg lines in honour of the father/son scientists who worked in Adelaide on X-ray crystallography, William Henry Bragg [the father] and William Lawrence Bragg [the son].




The above equation can be used to calculate the expected Doppler shift of the Bragg lines, which comes out as ... \( D = \pm .1 \sqrt f (Hz)\) with \(f\) in \(MHz\).