Standing Waves and Resonance
One fundamental type of motion in the physical world is termed periodic motion. Periodic motion is motion repeats itself in a given cycle and is described by the following quantities: the period (T) of the motion, defined as the time required to complete a full cycle (in units of seconds per cycle), the frequency (ƒ) which is the reciprocal of the period or 1/T (in units of Hertz), and the amplitude (A) which is the maximum displacement from the equilibrium position. A common type of periodic motion is that of the traveling wave, which propagates in the form of a sine wave. Traveling waves are observed when the waves are not confined to a specific region of the medium. The most commonly observed traveling wave is a water wave. A traveling wave has a velocity of propagation (v) equal to the product of the frequency and the wavelength (&lgr;) which is the crest-to-crest distance of the wave, or the distance between two subsequent amplitudes. This fundamental wave relationship is expressed as: v = ƒ&lgr;. Another parameter is also used to describe traveling waves, the angular wave frequency, which is related to the frequency as follows: = 2ƒ. The mechanism by which a traveling wave propagates itself through a medium is described by a wave equation of the type y(x,t) = A sin(2/&lgr;)(x-vt).
The concept of a standing, or stationary, wave can be understood by considering a wave propagating on a string whose ends are held some distance apart. For a string of length L which is fixed at both ends, the solution of the wave equation can take the form of a standing wave: y (x,t) = A sint sin(nx/L). The wave will propagate from one end of the string to the other end, and will then reflect and travel back in the opposite direction. The reflected wave will then interfere with the portion of the wave incoming from the opposite direction. This type of wave interference of an incident sine wave with a reflected sine wave produces an irregular and non-repeating wave propagation pattern leading to a likewise irregular and non-repeating motion of the string.
If the string is vibrated at a harmonic frequency, a sine wave pattern can be produced whose amplitude will change over time. Such frequencies can be selected so as to have the interference of the incident and reflected waves occur in such a manner that there are specific points, or nodes, along the string which seem to be immobile. Since the observed wave pattern is characterized by nodes which appear to be standing still, such a wave pattern is referred to as a standing wave pattern. Other points along the vibrating string change position over time, but in a regular manner. These points, or antinodes, vibrate back and forth between a large positive and large negative displacement. A standing wave pattern always consists of alternating nodes and antinodes and since antinode displacement occurs at regular time intervals, the motion of the string is also regular and repeating.
The harmonic frequencies required to generate standing waves are also called the natural frequencies of the string, or vibrating object, and they are those vibrations which result in the highest amplitude vibrations with the least input of energy. The lowest possible frequency which can generate a standing wave is referred to as the fundamental frequency. Such a wave has no nodes (i.e., it spans from one end of the medium to the other). The first overtone, or harmonic, of a fundamental is a wave with one node, which is exactly at the midpoint of the fundamental. The frequency of each harmonic is proportional to the speed of wave propagation and to the wavelength. The propagation speed is a function of properties such as string tension, diameter, and composition. The wavelength of the harmonic is dependent upon the length of the string and the harmonic number.
An object can be induced into vibrating at one of its harmonic frequencies under the influence of an externally applied oscillatory force provided by another object vibrating at one of those harmonic frequencies. This phenomenon is referred to as resonance (i.e., when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion). An example of resonance is the vibration induced in a piano wire of a given pitch when a musical note of the same pitch is played close by. Thus, all objects have resonant frequencies which are determined by the physical properties of the vibrating object. It is easy to induce an object to vibrate at one of its resonant frequencies, and much harder hard to induce it to vibrate at other frequencies. A vibrating object will also select its resonant frequencies from a complex excitation pattern and easily vibrate at those frequencies.
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