Reflection/transmission coefficients at small angles and magnitude

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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


(3.9a)

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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Reflection/transmission coefficients at small angles and magnitude
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