Problem 3.9
Show that, when angles in the Zoeppritz equations (3.2e,f,h,i) are small (so that squares and products are negligible), equations (3.6a) and (3.6b) for reflection and transmission at normal incidence are still valid, and that the reflected and transmitted S-waves are given by
where
,
-
Solution
When the angle of incidence
is small,
and
and the same is true for
. In this case Snell’s law and the Zoeppritz equations (3.2e,f,h,i) become
In matrix notation, the Zoeppritz equations are now
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(3.9a)
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To get the amplitude ratios
and
, we solve this equation either by inverting the left-hand matrix [see Sheriff and Geldart, 1995, equation (15.20)] or by using Cramer’s rule (see Wylie, 1966, 453). Using the latter method, and neglecting squares and products of the angles, we first calculate the value of det(
), the determinant of the
matrix in equation (3.9a). We shall expand by elements in the first row [see Sheriff and Geldart, 1995, equation (15.2)]; when we do this we see that the 2nd and 4th
determinants in the expansion are multiplied by
and
, respectively, and since we are neglecting products and squares of angles, angles inside these two determinants have been replaced with zeros. The expansion about the first row becomes
Next we calculate the values of
and
,
, 2, where
is
with column 1 replaced with the elements of the right-hand matrix in equation (3.9a), etc. (see Cramer’s rule in Sheriff and Geldart, 1995, problem 15.2j). Expanding
about the first row and setting the angles in the 2nd and 4th
determinant equal to zero as before, the expansion becomes
The second and fourth determinants are zero, so
Dividing by
, we get
which is the same as equation (3.6a). Similarly, we find that
where
,
. Note that
Also, when
so q
and
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Also in this chapter
External links
find literature about Reflection/transmission coefficients at small angles and magnitude
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