from the sketch above

according to  snell’s law,

but  

Where

for the direct wave the velocity is given from the relation

where X is the distance covered by the wave, V the velocity and

for the  , the refracted wave , time is give by

from the picture above

 but

therefore

NOW SUBSTITUTING THE VALUE OF AB, BC and CD into the equation below we have

but  therefore

but

factorizing the above equation we have

but

substituting into the equation below we have

this is the equation for the intercept time.

derivation for the depth of seismic wave

 from this equation

remember  

also from trigonometrical identities

substituting this into the equation below we have

at critical distance , the time for direct wave and reflected time are equal.

Multiplying throughout by

MAKING h THE SUBJECT OF THE RELATION

H IS the height downward  with is equal to depth , therefore the  is given by