Angular wavenumber

A very interesting thing I re-discovered today is the angular wavenumber. Although it seems like I had understood that pretty well long back, I had forgotten quite some things about it.

The rumination came from my continuous learning process of Fourier Transform. While I seem to have gotten a hang of temporal Fourier transform, spacial transform just wasn’t becoming that obvious to me. I have summarised Fourier transform here.

Basics

“In the physical sciences, the wavenumber (also wave number) is the spatial frequency of a wave, either in cycles per unit distance or radians per unit distance. It can be envisaged as the number of waves that exist over a specified distance (analogous to frequency being the number of cycles or radians per unit time).” -wikipedia

k_x=\frac{2\pi}{\lambda_x}
k_x=\frac{\omega}{c_x}

Wavenumber in imaging

In temporal FT every signal is decomposed into monochromatic signals of different amplitudes and phase (that add up to give the signal). Similarly spacial FT decomposes a signal into its constituent plane wave sinusoids.

F(u,v)=\int\int f(x,y) e^{i2\pi(ux+vy)}dx dy

1
1 k_x=50 k_y=30
2
k_x=50 k_y=30
3
k_x=50 k_y=30
4
4 A sum of all the wavefields

Few more examples from here:

Screenshot 2017-07-12 00.38.13

Screenshot 2017-07-12 00.42.58Screenshot 2017-07-12 00.44.10.pngScreenshot 2017-07-12 00.45.20.png

Now there is a big significance of decomposing a signal in its constituent wavenumbers, one of the biggest one is that energy travelling in plane wave stays preserved in depth make propagation operators computationally less complicated to use.

 

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