Homework 3 |

- In class, we derived the following acoustic wave equation for
pressure
:

- Using the connection between the pressure and displacement, derive the
acoustic wave equation for the displacement vector
:

- Consider a geometrical wave representation in the vicinity of a wavefront

and derive partial differential equations for the traveltime function and the vector amplitude . - Assuming that the geometrical wave propagates in the direction
of the traveltime gradient

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

where and are the corresponding geometrical spreading factors.

- Using the connection between the pressure and displacement, derive the
acoustic wave equation for the displacement vector
:
- Consider a medium with a constant gradient of slowness squared

(6) - 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

where and are the components of and find the corresponding transformation of the matrices and .

- 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 .
- 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 .

- 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
- 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**Geometry of reflection in a constant-velocity medium with a curved reflector.

Figure 1.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):

- Apply Fermat's principle to specify in the equation

required for finding the reflection point . - Newton's method (successive linearization) solves nonlinear
equations like (11) iteratively by starting with some
and repeating the iteration

forSpecify for the problem of finding the reflection point.

- Consider the special case of a dipping-plane reflector

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

- Apply Fermat's principle to specify in the equation

Homework 3 |

2013-10-10