# Directivity of a harmonic source plus ghost

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

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

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

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 ${\displaystyle z=c\lambda }$.