# Reinforcement depth in marine recording

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 3 47 - 77 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 3.5a

For a source at a depth ${\displaystyle h}$, show that the maximum amplitude of a downgoing incident wave and its reflection at the surface of the sea occurs at the depth ${\displaystyle \lambda /\left(4\cos \theta \right)}$, where ${\displaystyle \theta }$ is the angle of incidence, by expressing the pressure ${\displaystyle P}$ in the form used in equations (3.1b,d) and applying appropriate boundary conditions.

### Solution

Since the interface is liquid/vacuum, only two waves exist, the incident and reflected P-waves. Taking the z-axis positive downward, we take ${\displaystyle {\mathcal {P}}}$ in the form

{\displaystyle {\begin{aligned}{\mathcal {P}}=A_{0}e^{\mathrm {j} \omega p\left(x+z\cot \theta \right)}+A_{1}e^{\mathrm {j} \omega p\left(x-z\cot \theta \right)}.\end{aligned}}}

There is only one boundary condition, namely that ${\displaystyle {\mathcal {P}}=0}$ at ${\displaystyle z=0}$. This gives ${\displaystyle A_{1}=-A_{0}}$ Using Euler’s formulas (see Sheriff and Geldart, 1995, problem 15.12a), we get

{\displaystyle {\begin{aligned}{\mathcal {P}}&=A_{0}e^{\mathrm {j} \omega px}\left(e^{\mathrm {j} \omega pz\cot \theta }-e^{-\mathrm {j} \omega pz\cot \theta }\right)\\&=2\mathrm {j} A_{0}e^{\mathrm {j} \omega \left(px-t\right)}\sin \left(\omega pz\cot \theta \right)\end{aligned}}}

upon inserting the time factor. The amplitude of the combined incident and reflected waves is

{\displaystyle {\begin{aligned}2A_{0}\sin[\left(\omega pz\cot \theta \right]=2A_{0}\sin \left[\left(\omega z/\alpha \right)\cos \theta \right].\end{aligned}}}

It is a maximum when ${\displaystyle \left(\omega z/\alpha \right)\cos \theta =\pi /2}$, that is, when

{\displaystyle {\begin{aligned}z=\left(\pi /2\right)\alpha /\left(\omega \cos \theta \right)=\lambda /\left(4\cos \theta \right).\end{aligned}}}