assume
that we are working over a layer of earth of uniform thickness d, with the next
layer of rock and/or soil parallel to the upper layer. See diagram.
A wave producing device is placed at point A and a receiver is placed is point B. The receiver is capable of recording times taken for the wave pulse to travel from point A to other points on the surface of the earth. When the device at A is activated, one wave travels along the surface, directly from A to B at a velocity v1. If is known to be less than the velocity,v2 , of the wave through the second layer, then a second wave will travel through the upper crust to the second layer along the path indicated by the arrows and back through the surface layer to point B. This is because travel in the second layer is faster than through the upper crust.
Let t denote the time for the second wave to travel from point A to point B through the surface layer to the top of the second layer and back through the surface layer along the path indicated by the arrows. First write t as a function of the angles α and β. Now use this function to find the value(s) of α and β that minimize the total “travel” time for the wave.
Caution: The first and third “legs” of the path are
in the surface layer, and the middle “leg” of the path is in the second layer.
Note that your
answers will be in terms of the velocities involved.
that we are working over a layer of earth of uniform thickness d, with the next
layer of rock and/or soil parallel to the upper layer. See diagram.
A wave producing device is placed at point A and a receiver is placed is point B. The receiver is capable of recording times taken for the wave pulse to travel from point A to other points on the surface of the earth. When the device at A is activated, one wave travels along the surface, directly from A to B at a velocity v1. If is known to be less than the velocity,v2 , of the wave through the second layer, then a second wave will travel through the upper crust to the second layer along the path indicated by the arrows and back through the surface layer to point B. This is because travel in the second layer is faster than through the upper crust.
Let t denote the time for the second wave to travel from point A to point B through the surface layer to the top of the second layer and back through the surface layer along the path indicated by the arrows. First write t as a function of the angles α and β. Now use this function to find the value(s) of α and β that minimize the total “travel” time for the wave.
Caution: The first and third “legs” of the path are
in the surface layer, and the middle “leg” of the path is in the second layer.
Note that your
answers will be in terms of the velocities involved.
