Wave propagation and modes of vibration

In Chapter 4, we considered the problem of free vibrations of an elastic body of finite dimensions, considering the vibrations of a string and a rod. Since the Earth has a finite radius and is bounded by a free surface, we must consider its free oscillations. Up to this point, we have treated wave propagation in the Earth without considering its finite dimensions. We find body waves traveling in its interior and the presence of its free surface generates surface waves. Body waves have relatively high frequencies (larger than 0.07 Hz) or short periods (less than 15 s). For an average velocity of 10 km s-1 they correspond to wave lengths of about 150 km, which is small compared with the Earth's radius (6370 km). Hence, body waves' trajectories, traveling times and energy transport can be studied using the wave propagation approach and ray theory approximation in a flat or spherical medium, depending on distances. Surface waves extend to very low frequencies or large periods and their wave lengths reach values of the order of the Earth's dimensions. For example, waves of 400 s period, for a velocity of 4.5 km s-1, have a wave length of 1800 km, about a third of the Earth's radius. For wave lengths of this order, the problem must be treated in the form of free oscillations or vibrations.