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What is a [seismic] wave front?
A seismic wave front emanates from an energy source like ripples on a pond, but in three dimensions. It is the surface connecting points of equal travel time from the source.
In a homogeneous medium, a surface drawn through points of equal travel time is spherical, as depicted in the animation below.
The seismic trace, which is what is recorded by the seismograph, represents particle motion vs. time. In a homogeneous medium, the wave front can also be described as a surface of constant phase