The frequency of the modes depends on the Earth's composition and shape, but some are more (or only) sensitive to the solid mantle, while other to the outer and/or the inner core. They are therefore very important to infer properties of the Earth's deep interior.
To generate a wave, there must be restoring forces giving rise to oscillations when the disturbed particles overshoot their equilibrium position. For the seismic fundamental mode and its uppertones, these forces are due to the elasticity of the medium and to the molecular cohesion, which are described by the rigidity and bulk modulus.
The fluid mixture which constitutes the Earth's core being stratified, gravitating, rotating and pervaded by a magnetic field, the possible additional restoring forces are, correspondingly, the Archimedean force, the Coriolis force and the Lorentz force. As these forces are weak compared to the elastic force responsible for the seismic free oscillations, the Archimedean, Coriolis and magnetic oscillations will be much slower than the fundamental normal mode. This is why these long period oscillations are often called "undertones".
The oscillations resulting from Archimedean buoyancy force are called internal (or gravity) waves; those resulting from the Coriolis force are called inertial modes, and those resulting from the Lorentz force are the magnetic or Alfvèn waves. A combination of all three restoring forces would generate MAC waves (Magnetic, Archimedean, Coriolis).
The undertones, mostly restricted to the fluid core, are also called "core modes". Observing the Archimedean modes would be very useful in order to determine whether the outer core stratification is stable or not. A stable stratification allows internal gravity waves. The stratification is characterised by the Brünt-Väisälä frequency (see e.g. Melchior, 1986) that depends upon the density gradient. This quantity is related to the temperature gradient and is not directly observable; therefore this profile is poorly controlled, uncertain and varies from one model of the Earth to another. The core mode spectrum is still not completely mapped but enough is known to suggest that this is dense. Unexplained peaks in the gravity spectrum at periods between 12 and 24 hours have sometimes been attributed to such modes but in fact, they are still not detected.
There are particular modes that are sometimes considered as "core modes": the "Slichter modes". They correspond to inner core translation and have periods around 5 hours.
Calculations using Earth's models indicate that undertones would produce accelerations at the surface of the Earth less than 0.01 nm/s², which is very weak. Attempts to observe those modes remain unsuccessful. However, an international global network of superconducting gravimeters, set up by the Global Geodynamics Project (GGP), should help to contribute to detect and characterise Slichter frequencies and the core modes.