From dip to no dip for non-diffracted multiple

In this Appendix, I show that the equations for the non-diffracted multiple from a dipping water-bottom reduce to the equations for a flat water-bottom when $\varphi=0$ as they should. Setting $\varphi=0$ in equations 39 through 42 we obtain

$$t_{s_2}=t_{r_2}=t_{r_1}=t_{s_1}=\frac{\tilde{Z}_s}{\cos\alpha_s}$$ (66)

and from equations 5 and 6 we get (as discussed at the end of Appendix A) $\tilde{t}_{r_2}=t_{r_2}$ and $\tilde{t}_{s_2}=t_{s_2}$.Therefore,

$$\begin{eqnarray} h_\xi&=&2h_D-V_1[t_{s_1}\sin\alpha_s+t_{r_1}\sin\alpha_r+\rho(\... ..._{s_1}\sin\alpha_s(1+\rho^2)\\ &=&2h_D-h_D(1+\rho^2)=h_D(1-\rho^2)\end{eqnarray}$$ (67) (68) (69) (70)

Similarly,

$$\begin{eqnarray} z_\xi&=&V_1(t_{r_1}+\rho\tilde{t}_{s_2}\sqrt{1-\rho^2\sin^2\alp... ...^2(1-\rho^2)}=Z_{wb}+\frac{\rho}{2}\sqrt{Z_{wb}^2+h_D^2(1-\rho^2)}\end{eqnarray}$$ (71) (72) (73) (74) (75)

Finally,

$$\begin{eqnarray} m_\xi&=&m_D+\frac{V_1}{2}(t_{s_1}\sin\alpha_s-t_{r_1}\sin\alpha... ...}\sin\alpha_s)+\rho^2(t_{s_2}\sin\alpha_s-t_{s_2}\sin\alpha_s)=m_D\end{eqnarray}$$ (76) (77)

E