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When an earthquake occurs, energy radiates out from the point of rupture in the form of seismic waves. These waves are what we feel as shaking ground, and they are classified into two main types: body waves and surface waves. Body waves travel through the Earth's interior and include P-waves (Primary waves) and S-waves (Secondary waves).

P-waves, the fastest type of seismic waves, compress and expand the Earth's material, making them very important for seismic monitoring as they provide the first signals that help determine the earthquake's magnitude and location. They are akin to sound waves in air; they can travel through solids, liquids, and gases.

S-waves follow after P-waves and exhibit a different type of motion, causing particles to oscillate perpendicular to the wave's direction. Due to this transverse motion, S-waves can only move through rigid, elastic materials, which is only possible in solids and not in fluids like the Earth's outer core or air.

The propagation of earthquake waves is a dynamic process that offers critical insights into the seismic activity and the materials they travel through. P-waves, being faster, traverse the Earth's crust and mantle, arriving first on seismographs. Their velocity, which ranges from 4-7 km/s in the crust, is influenced by the density and elasticity of the materials they pass through.

Conversely, S-waves follow with a slower speed (2-5 km/s) and a notable restriction: they are unable to penetrate the Earth's liquid outer core. This phenomenon not only impacts the time at which these waves are observed on seismograms in comparison to P-waves but also provides a basis for the creation of shadow zones where certain waves are absent. The analysis of the propagation patterns and velocities of these waves helps seismologists understand the earthquake's source and characteristics, as well as the Earth's internal structure.

The way that these waves propagate also helps scientists to pinpoint the location of the earthquake’s epicenter. By measuring the time difference between the arrival of P-waves and S-waves at various seismographic stations, the distance to the earthquake can be calculated - a fundamental technique in earthquake seismology.

The study of seismic waves is crucial for understanding the Earth's interior structure. The ability of P-waves to move through all materials and S-waves to be impeded by liquids reveals much about the Earth's make-up. For instance, the fact that S-waves are absorbed by the outer core signifies that it is liquid. When these waves encounter interfaces between different materials inside the Earth, such as the crust-mantle boundary or the core-mantle boundary, some of the wave energy is reflected or refracted, leading to complex wave paths.

Seismologists analyze these interactions to deduce the internal layering of the planet. For example, the presence of the P-wave shadow zone, an area on the Earth’s surface where P-waves are not directly observed, suggests that the Earth's inner core is solid. These investigative techniques help scientists build models of the Earth’s interior, leading to discoveries like the thin, brittle crust; the mantle with varying viscosity; and the solid inner and liquid outer core.

The study of the Earth's interior structure through seismic waves has been transformative, leading to the development of the field of seismology and enhancing our understanding of not only Earth’s composition but also geological processes such as plate tectonics and volcanic activity.