# Difference between revisions of "Directivity of a harmonic source plus ghost"

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 6 181 - 220 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 6.9

Show that equation (6.7c) gives the directivity diagrams shown in Figure 6.9a.

### Solution

The directivity is given by equation (6.7c). We take $\displaystyle 2A=1$ , and $\displaystyle c={\rm depth}/\lambda =0.1$ , 0.5, and 1.0 for the three parts of Figure 6.9a. Then equation (6.7c) gives

\displaystyle \begin{align} A^{*} ={\rm \; sin\; }(2\pi c{\rm \; cos\; }\theta ). \end{align}

Substituting the three values of $\displaystyle c$ , we have:

\displaystyle \begin{align} a):c=0.1\;,\; A^{*} ={\rm \; sin\; }(0.63{\rm \; cos\; }\theta), \end{align} \displaystyle \begin{align} b):c=0.5\;,\; A^{*} ={\rm \; sin\; }(3.1{\rm \; cos\; }\theta), \end{align} \displaystyle \begin{align} c):c=1.0,\; A^{*} ={\rm \; sin\; }(6.3{\rm \; cos\; }\theta). \end{align}

The results of the calculations are shown in Tables 6.9a,b. Figure 6.9a.  Directivity of a harmonic source at depth $\displaystyle z=c\lambda$ .

Ignoring the minus signs (which indicate phase reversals), the curves for $\displaystyle \varphi _{a}$ and $\displaystyle \varphi _{b}$ , shown in Figure 6.9b, conform closely to Figure 6.9a. However, we need more points to plot the $\displaystyle \varphi _{c}$ -curve properly and Table 6.9b shows calculated values for intermediate points. The $\displaystyle \Psi _{c}$ -curve in Figure 6.9b also conforms closely to Figure 6.9a.

Table 6.9a. Values for $\displaystyle \Psi _{a}$ , $\displaystyle \Psi _{b}$ , $\displaystyle \Psi _{c^{'} }$ .
$\displaystyle \theta ^{\circ}$ $\displaystyle \Psi _{a}$ $\displaystyle \Psi _{b}$ $\displaystyle \Psi _{c}$
0 0.59 0.00 0.00
15 0.57 0.15 −0.20
30 0.52 0.44 −0.74
45 0.43 0.81 −0.97
60 0.31 1.00 −0.01
75 0.16 0.72 1.00
90 0.00 0.00 0.00
Table 6.9b. Intermediate values for $\displaystyle \Psi _{c^{'} }$ .
$\displaystyle \theta ^{\circ}$ $\displaystyle \Psi _{c}$ $\displaystyle \theta ^{\circ}$ $\displaystyle \Psi _{c}$
5 −0.01 50 −0.79
10 −0.08 55 −0.46
20 −0.35 65 0.46
25 −0.54 70 0.83
35 −0.90 80 0.89
40 −0.99 85 0.52 Figure 6.9b.  Calculated directivity at source depth $\displaystyle z=c\lambda$ .