Waves in Fourier space

Arbitrary functions can be made from the superposition of sinusoids. Sinusoids and complex exponentials often occur. One reason they occur is that they are the solutions to linear partial differential equations (PDEs) with constant coefficients. The PDEs arise because most laws of physics are expressible as PDEs.

Using Fourier integrals on time functions we encounter the Fourier kernel $\ \exp ( -i \omega t )$.Specializing the arbitrary function in equation (4) to be the real part of the function $\exp [ - i \omega (t-t_0 ) ]$ gives  

$$\hbox{moving cosine wave} \ \ =\ \ \cos \left[ \ \omega \lef... ...a \ +\ {z \over v }\ \cos \, \theta \ -\ t \ \right) \ \right]$$ (5)

To use Fourier integrals on the space-axis x the spatial angular frequency must be defined. Since we will ultimately encounter many space axes (three for shot, three for geophone, also the midpoint and offset), the convention will be to use a subscript on the letter k to denote the axis being Fourier transformed. So k_x is the angular spatial frequency on the x-axis and $\exp ( i k_x x )$ is its Fourier kernel. For each axis and Fourier kernel there is the question of the sign of i. The sign convention used here is the one used in most physics books, namely, the one that agrees with equation (5). With this convention, a wave moves in the positive direction along the space axes. Thus the Fourier kernel for (x , z , t)-space will be taken to be  

$$\hbox{Fourier kernel} \ =\ e^{ i \, k_x x} \ e^{ i \, k_z z... ...\omega t} \ = \ \exp [ i ( k_x x \ +\ k_z z \ -\ \omega t ) ]$$ (6)

Now for the whistles, bells, and trumpets. Equating (5) to the real part of (6), physical angles and velocity are related to Fourier components. These relations should be memorized!  

$$\vbox{\offinterlineskip \hrule \halign {&\vrule ...$$ (7)

Equally important is what comes next. Insert the angle definitions into the familiar relation $\sin^2 \theta + \cos^2 \theta = 1$.This gives a most important relationship, seen earlier as the dispersion relation of the scalar wave equation.

 

$$k_x^2 \ +\ k_z^2 \ \ =\ \ { \omega^2 \over v^2 }$$ (8)

The importance of (8) is that it enables us to make the distinction between an arbitrary function and a chaotic function that actually is a wavefield. Take any function p(t , x , z). Fourier transform it to $P( \omega , k_x , k_z )$.Look in the $( \omega , k_x , k_z )$-volume for any nonvanishing values of P. You will have a wavefield if and only if all nonvanishing P have coordinates that satisfy (8). Even better, in practice the (x , t)-dependence at z = 0 is usually known, but the z-dependence is not. Then the z-dependence is found by assuming P is a wavefield, so the z-dependence is inferred from (8).