Modes of Standing Waves - I

A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching the building's natural frequency of vibration; this produces a resonance that results in one building collapsing while the neighboring buildings do not. Often, buildings of a certain height are devastated, while other taller buildings remain intact. This phenomenon occurs because the building height matches the condition for setting up a standing wave for that particular height. As the earthquake waves travel along the surface of the Earth and reflect off denser rocks, constructive interference occurs at specific points. Often, areas closer to the epicenter are not damaged, while areas farther away are damaged.

On the other hand, standing waves on strings have frequencies related to the propagation speed of the disturbance on the string. The distance between the points where the string is fixed determines the wavelength. The symmetrical boundary conditions—a node at each end—dictate the possible frequencies that can excite standing waves. Starting from a frequency of zero and slowly increasing the frequency, the first mode, n = 1, appears. The first mode, also called the fundamental mode or the first harmonic, corresponds to half a wavelength, so the wavelength is equal to twice the length between the nodes. The successive normal modes are called overtones: the first overtone corresponds to the second harmonic, and so on. In this case, the mathematical pattern reveals that the successive harmonic frequencies are integral multiples of the first harmonic.