Very simple but important property

If w=w₁w₂ and $A_1({\bf r})$ is a discontinuity amplitude of the field

$$U_1({\bf r},t) = {\bf P}^{(\pm)}_{w_1}(v)u_0({\bf r}_0,t)$$

(where ${\bf P}^{(-)}_{w_1}$ is the KO with the kernel w₁) then the discontinuity of the field

$$U({\bf r},t) = {\bf P}^{(\pm)}_w(v)u_0({P\bf r}_0,t)$$

has the amplitude

$$A({\bf r}) = A_1({\bf r})w_2.$$

In particular, it means that if

$$w=w_E \Psi({\bf r},{\bf r}_0)$$

and A_E is the amplitude of the field

$$u_E = {\bf P}^{(-)}_{w_E}(v)U_0(t),$$

then for field $u={\bf P}^{(-)}_w u_0$ has discontinuity amplitude:

$$A({\bf r}) = A_E \Psi({\bf r},{\bf r}^\ast_0),\ \ \ \ {\bf r}^\ast_0=\omega({\bf r}).$$

Here, dependence $\omega({\bf r})$ means that ${\bf r}^\ast_0$ belongs to the same ray of eiconal $\tau^{(-)}({\bf r})$.