9: INTEGRAL OPERATORS OF WAVE-FIELD CONTINUATIONS

Kirchoff's type representations of the DKO for the homogeneous medium prompts the general form of an integral KO:

 

$$u^{(\pm)} ({\bf r},t) = \int_\Sigma w({\bf r},{\bf r}_0) {\b... ...\pm t} u_0 ({\bf r}_0,t \mp T({\bf r},{\bf r}_0)) d {\bf r_0} ,$$ (99)

where $w({\bf r},{\bf r}_0)$ - negative weight function, $T({\bf r},{\bf r}_0)$ - propagation time from ${\bf r}$ to ${\bf r}_0$ with accordance with a given eiconal equation

 

$$\Psi(\nabla \tau ; v) = 1 .$$ (100)

Without any lack of generality we can put n=0. We propose that

$$u_0 ({\bf r}_0,t) \stackrel{q}{\sim} A_0 ({\bf r}_0) R_q [t - \tau_0 ({\bf r}_0)] .$$

then

$${\bf D}^{q+1}u_0 ({\bf r}_0,t) \stackrel{-1}{\sim} A_0 ({\bf r}_0) \delta [t-\tau_0 ({\bf r}_0)]$$

and

 

$$U^{(\pm)} ({\bf r},t) = {\bf D}^{q+1} u({\bf r},t){\sim}\int... ...\delta[t-\tau_0({\bf r}_0)\mp T({\bf r},{\bf r}_0)]d {\bf r}_0.$$ (101)

It is well known that

$$\int_X f(x)\delta (x-x_0)dx =\left\{ \begin{array} {ll} f(x_0) & x_0 \in X \\ 0 & x_0 \not \in X. \end{array} \right.$$

It is easy to show that when point ${\bf r}$ is placed behind the front $\tau^{(-)} =t^\ast$ (where $\tau^{(-)}$ is the correspondent solution of equation (100) with condition $\tau \vert _\Sigma = \tau_0$), then argument of $\delta$-function at $t=t^\ast$ is always positive:

(see Figure ). It means that in this case, $U^{(-)}({\bf r},t^\ast) = 0$. It is also can be shown that when ${\bf r}$ is between the front $\tau^{(-)} =t^\ast$ and the surface $\Sigma$, that argument of $\delta$-function at $t=t^\ast$ takes both negative and positive meanings. So in this case, $U^{(-)}({\bf r},t^\ast) \neq 0$. This proves kinematically equivalence of the operator (99).