P-waves, or primary waves, are a type of seismic wave that are generated by earthquakes and travel through the Earth's interior. The best model for their motion is a longitudinal wave model.
In a longitudinal wave, particles of the medium (like rock or sediment) oscillate back and forth in the same direction as the wave is traveling. This means that as a P-wave moves through the Earth's crust, it compresses and expands the material in the same direction of its propagation.
If a more detailed mathematical or physical model is needed, one might use the principles of elasticity and wave propagation in solids to describe the motion of P-waves, often involving equations derived from Hooke's law and the wave equation for compressional waves.
In summary, the motion of P-waves is best modeled as longitudinal waves, which can be analyzed further using principles of wave mechanics within the context of geophysics.