Homework 3

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# Theoretical part

1. In class, we derived the following acoustic wave equation for pressure :
 (1)

1. Using the connection between the pressure and displacement, derive the acoustic wave equation for the displacement vector :
 (2)

2. Consider a geometrical wave representation in the vicinity of a wavefront
 (3)

and derive partial differential equations for the traveltime function and the vector amplitude .

3. Assuming that the geometrical wave propagates in the direction of the traveltime gradient
 (4)

show that the amplitude continuation along a ray is given by equation
 (5)

where and are the corresponding geometrical spreading factors.

2. Consider a medium with a constant gradient of slowness squared
 (6)

1. In the 2-D case, the ray-family coordinate system can be specified by , where goes along the ray, and is the initial ray angle. A family of rays starts from the source point which each ray traveling in the direction . Show that the coordinate transformation matrix changes along the ray as
 (7)

where and are the components of and find the corresponding transformation of the matrices and .
 (8) (9)

2. Find the one-point geometrical spreading from a point source in the 2-D case as a function of , , the source location , the initial ray direction , and the ray coordinate .

3. Using analytical ray tracing solutions, find the two-point geometrical spreading from a point source in the 2-D case as a function of , , the source location and the receiver location .

3. Consider the source point and the receiver point at the surface above a 2-D constant-velocity medium and a curved reflector defined by the equation with a twice differentiable function (Figure 1).

curve
Figure 1.
Geometry of reflection in a constant-velocity medium with a curved reflector.

Note that the two-point ray trajectory can be parametrized by the reflection point with the following expression for the reflection traveltime (using the Pythagoras theorem):

 (10)

1. Apply Fermat's principle to specify in the equation
 (11)

required for finding the reflection point .

2. Newton's method (successive linearization) solves nonlinear equations like (11) iteratively by starting with some and repeating the iteration
 (12)

for

Specify for the problem of finding the reflection point.

3. Consider the special case of a dipping-plane reflector
 (13)

where is the dip angle. Show that, in this case, equation (11) reduces to a linear equation for . Find and substitute it into (10) to define the reflection traveltime.

4. (EXTRA CREDIT) Find the reflection traveltime for a circle reflector
 (14)

 Homework 3

Next: Computational part Up: Homework 3 Previous: Homework 3

2013-10-10