Problem 3.3a
Derive Zoeppritz’s equations and Knott’s equations for a P-wave incident on a liquid/solid interface when the incident wave is (i) in the liquid and (ii) in the solid.
Background
Knott’s equations differ from the Zoeppritz equations in that they use potential functions instead of displacements. Knott’s equations can be derived directly from the Zoeppritz equations and vice versa [see equation (3.3
)]; however, we shall derive them from first principles. We use script letters,
and
for the amplitudes of the potential functions, reserving italic letters
and
for displacements.
To get Knott’s equations for a solid/solid interface, we start with the potential functions in equation (2.9c), the displacements being given by equations (2.9d,e), and apply the boundary conditions of problem 2.11, we write
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(3.3a)
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(3.3b)
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where
,
,
,
and
are given by equations (3.1d,e). Using equation (2.9e), the continuity of normal displacement requires that
be continuous at
. Using equations (3.2g), we obtain
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(3.3c)
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Continuity of tangential displacement requires the continuity of
[see equation (2.9d)]; this gives the equation
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(3.3d)
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The normal stress is given by
[see equation (2.11b)], so
But
,
(see equations (9,6) and (9,7) of Table 2.2a); using these relations plus equation (3.1a), we get
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(3.3e)
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The tangential stress is
from equation (2.11b), so we get
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(3.3f)
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We can write Knott’s equations in a more compact form by substituting
,
,
; the four equations now become
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(3.3g)
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(3.3h)
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(3.3i)
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(3.3j)
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To show the correspondence between Knott’s and Zoeppritz’s equations, we calculate the energy density in terms of the displacements and the potential functions used in Knott’s equations. In terms of displacements, the instantaneous kinetic energy density E for a harmonic P-wave
is equal to
The total energy density is the maximum kinetic energy density (see problem 3.7), that is,
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(3.3k)
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The energy density of a P-wave in terms of the potential function
[see equation (2.9a)], noting that
since there is no S-wave) is
from equations (2.9d,e). Taking the time factor as
and reinserting the factor
which was deleted to get equation (3.2g), we get
,
,
. Thus, we get for the total energy
Comparing with the expression for
in equation (3.3k), we see that
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(3.3l)
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where the second equation is obvious from symmetry.
Solution
Since we derived Zoeppritz’s equations in problem 3.2a, we derive Knott’s equations here and then get Zoeppritz’s equations from them using equation (3.31).
i) To derive Knott’s equations when the incident P-wave is in the liquid, the boundary conditions require the continuity of
and
and that
in the solid vanish at
. omitting the factor
, we have from equations (3.1b,c,d,e):
Continuity of
yields the first equation [note equation (3.2g)]:
Continuity of normal stress requires that
be continuous; since
in the liquid, this results in
Using equations (9,6) and (9,7) in Table 2.2a, also equation (3.1a), we have
Continuity of tangential stress
requires that
in the solid when
, so
Using the coefficients in equations (3.3g,h,i,j), the results become
These are Knott’s equations. We could derive Zoeppritz’s equations as we did equations (3.2e,f,h,i), or we can use equation (3.3
) to change the coefficients in Knott’s equations to Zoeppritz’s coeffiicients. Using this latter method, we have
,
. Substituting these, we get the Zoeppritz equations for a liquid-solid interface:
ii) When the incident wave is in the solid, we shall first derive Zoeppritz’s equations, then change them to Knott’s equations. We have
in the liquid,
and
are continuous, and
in the solid at
.
Equation (3.2e) gives for the normal displacement
The continuity of normal stress is expressed in equation (3.2h), which now becomes
Finally,
at
and equation (3.2i) becomes
Using equation (3.3
), we get the equivalent Knott’s equations:
Problem 3.3b
Calculate the amplitudes of the reflected and refracted P-and S-waves when an incident P-wave strikes the interface from a water layer
,
,
g/cm
) at
when the seafloor is (i) “soft”
,
m/s,
g/cm
, and (ii) “hard”
,
,
g/cm
).
Solution
i) Where the seafloor is “soft” and the P-wave is incident in the water, we have:
Thus Zoeppritz’s equations become
The solution is:
,
,
.
ii) When the seafloor is “hard”:
The equations are
The solution is
,
,
.
Problem 3.3c
Repeat part (b) for an angle of incidence of
.
Solution
i) For the “soft” bottom and
,
The equations are
The solution is
,
,
.
ii) For the “hard” bottom,
, so total reflection occurs.
The results are summarized in Table 3.3a. The table shows that
and
depend mainly on the hardness of the bottom and only moderately on
. However,
depends more on the angle of incidence than on the hardness.
Table 3.3a Reflected/transmitted amplitudes for soft/hard bottoms.
Bottom
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Soft
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0.431
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0.539
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0.244
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0.403
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0.512
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0.408
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Hard
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0.716
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0.227
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0.304
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Also in this chapter
External links
find literature about Reflection/refraction at a liquid/solid interface
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