New notations and definitions

Until now the proof was pure physics. But because in seismology the depth axis is ``special,'' we will change notations. Axis x₃ will be denoted by z, and the $\vec x$ will denote the surface position vector with the components (x₁, x₂). Also, the Laplacian will be the operator,

 

$$\Delta = \frac{{\partial ^2 }}{{\partial x_1 ^2 }} + \frac{{\partial ^2 }}{{\partial x_2 ^2 }}.$$ (23)

We will use the Fourier transform of the pressure field along the time axis:

 

$$P\left( {\vec x,z,\omega } \right) = F\left\{ {P\left( {\vec... ...\infty }^\infty {P\left( {\vec x,z,t} \right)e^{i\omega t} dt}.$$ (24)

The following property of the Fourier transform will be needed:

 

$$F\left\{ {\frac{{\partial ^2 P\left( {\vec x,z,t} \right)}}{... ...}}} \right\} = - \omega ^2 P\left( {\vec x,z,\omega } \right).$$ (25)

The spatial frequency is defined as:

 

$$m\left( {\vec x,z} \right) = \frac{\omega }{{v\left( {\vec x,z} \right)}}.$$ (26)

Let $\bar v$ be a spatial average of v in the medium, a known constant that does not depend on $\vec x$ or z. We also define

 

$$\bar m = \frac{\omega }{{\bar v}}$$ (27)

and the function

 

$$Q\left( {\vec x,z,\omega } \right) = P\left( {\vec x,z,\omega } \right)e^{ - i\bar mz}.$$ (28)

The index notation for derivatives will be used from now on. The symbol $\forall$ will denote the phrase ``for all.''