To show that the head wave is predicted by Fermat\'s principle, consider a layer
ID: 159305 • Letter: T
Question
To show that the head wave is predicted by Fermat's principle, consider a layer of thickness h with velocity v0 overlying a halfspace with a higher velocity, v1. (a) Derive the travel time to distance x for a wave that is incident on the boundary at a distance y from the source, travels for some distance just below the boundary, and then returns to the surface at the same incidence angle at which it went down. (b) Find the y value giving an extremal travel time, show that it correspnds to the critical angle of incidence.
Explanation / Answer
Fermat principle states that the ray paths between the two points are those for which the travel time is extreme. , minimum or maximum depending on the near by paths.
The time as a function of x is given by
T(x) = (a^2+x^2)^1/2/V1 + ((b-x)^2+c^2)1/2/v2
So after differentiation the equation will finally go to
T(x) = Sin i1/V1 - Sin i2/V2
which will finally yields to snells law
v1/ sin i1 = v2/ sin i2
as seismic wave propagation follows the ray otpics so the fermat principle tries to explain the same thing and give these equation to calculate the travel time
If the wave reaches directly to the end point then it will be the shortest distance but as it has been reflected back so now the equation will be
dL/dX= 1/2(2x/square root (a^2+y^2) +1/2 (2 (d-x)(-1)/ square root (b^2+(d-y)^2 = 0
where a and b are the two heights and y is the distance at which the wave got reflected from the surface
this will then sin theta 1 = sin theta 2 , which again proves the snells law
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