Resonance

The goal of this Unit is to develop an understanding of the nature, properties, behavior, and mathematics of sound and to apply this understanding to the analysis of music and musical instruments. Thus far in this unit, applications of sound wave principles have been made towards a discussion of beats, musical intervals, concert hall acoustics, the distinctions between noise and music, and sound production by musical instruments. The focus will be upon the application of mathematical relationships and standing wave concepts to musical instruments. Three general categories of instruments will be investigated: string instruments (which would include guitar strings, violin strings, and piano strings), open-end air column instruments (which would include the brass instruments such as the flute and trombone and woodwinds such as the saxophone and oboe),and closed-end air column instruments (which would include the clarinet). A fourth category - vibrating mechanical systems (which includes all the percussion instruments) - will not be discussed. These instrument categories may be unusual to some; they are based upon the commonalities among their standing wave patterns and the mathematical relationships between the frequencies which the instruments produce.

Musical instruments are set into vibrational motion at their natural frequency when a person hits, strikes, strums, plucks or somehow disturbs the object. Each natural frequency of the object is associated with one of the many standing wave patterns by which that object could vibrate. The natural frequencies of a musical instruments are sometimes referred to as the harmonics of the instrument. An instrument can be forced into vibrating at one of its harmonics (with one of its standing wave patterns) if another interconnected object pushes it with one of those frequencies. This is known as resonance - when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion.

The word resonance comes from Latin and means to "resound" - to sound out together with a loud sound. Resonance is a common cause of sound production in musical instruments. In class, one of our models of resonance in a musical instrument included the resonance tube (a hollow cylindrical tube) immersed in a cylinder of water and forced into vibration by a tuning fork. The tuning fork was the object which forced the air inside of the resonance tube into resonance. As the tines of the tuning fork vibrated at their own natural frequency, they created sound waves which impinged upon the opening of the resonance tube. These impinging sound waves produced by the tuning fork forced air inside of the resonance tube to vibrate at the same frequency. Yet, in the absence of resonance, the sound of these vibrations is not loud enough to discern. Resonance only occurs when the first object is vibrating at the natural frequency of the second object. So if the frequency at which the tuning fork vibrates is not identical to one of the natural frequencies of the air column inside the resonance tube, resonance will not occur and the two objects will not sound out together with a loud sound. But the resonance tube can be moved up and down within the water, thus decreasing or increasing the length of the air column. An increase in the length of a vibrational system (here, the air in the tube) increases the wavelength and decreases the natural frequency of that system. Conversely, a decrease in the length decreases the wavelength and increases the natural frequency. So by moving the resonance tube up and down within the water, the natural frequency of the air in the tube could be matched to the frequency at which the tuning fork vibrates. When the match is achieved, the tuning fork forces the air column inside of the resonance tube to vibrate at its own natural frequency and resonance is achieved. And always, the result of resonace is a big vibration - that is, a loud sound.

The familiar "sound of the sea" which is heard when a seashell is placed up to your ear is also explained by resonance. Even in an apparently quiet room, there are sound waves with a range of frequencies. These sounds are mostly inaudible due to their low intensity. This so-called background noise fills the seashell, causing vibrations within the seashell. But the seashell has a set of natural frequencies at which it will vibrate. If one of the frequencies in the room forces air within the seashell to vibrate at its natural frequency, a resonance situation is created. And always, the result of resonace is a big vibration - that is, a loud sound. In fact, the sound is loud enough to hear. So the next time you hear the "sound of the sea" in a seashell, remember that all that you are hearing is the amplification of one of the many background frequencies in the room.

Musical instruments produce their selected sounds in the same manner. Brass instruments typically consist of a mouthpiece attached to a long tube filled with air. The tube is often curled in order to reduce the size of the instrument. The metal tube merely serves as a container for a column of air; it is the vibrations of this column which produces the sounds which we hear. The length of the vibrating air column inside the tube can be adjusted either by sliding the tube to increase and decrease its length or by opening and closing holes located along the tube in order to control where the air enters and exits the tube. Brass instruments involve the blowing of air into a mouthpiece. The vibrations of the lips against the mouthpiece produce a range of frequencies. One of the frequencies in the range of frequencies matches one of the natural frequencies of the air column inside of the brass instrument. This forces the air inside of the column into resonance vibrations. And always, the result of resonace is a big vibration - that is, a loud sound.

Woodwind instruments operate in a similar manner. Only, the source of vibrations is not the lips of the musician against a mouthpiece, but rather the vibration of a reed or wooden strip. The operation of a woodwind instrument was modeled in class using a plastic straw. The ends of the straw were cut with a scissors, forming a tapered reed. When air is blown through the reed, the reed vibrates producing turbulence with a range of vibrational frequencies. When the frequency of vibration of the reed matches the frequency of vibration of the air column in the straw, resonance occurs. And once more, the result of resonance is a big vibration - the reed and air column sound out toegether to produce a loud sound. As if this weren't silly enough, the teacher then began shortening the length of the straw by cutting small pieces off its opposite end. As the straw (and the air column which it contained) was shortened, the wavelength was decreased and the frequency was increased. Higher and higher pitches were observed as the straw was shortened. Woodwind instruments produce their sounds in a manner similar to the straw demonstration. A vibrating reed forces an air column to vibrate at one of its natural frequencies. Only for wind instruments, the length of the air column is controlled by opening and closing holes within the metal tube (since the tubes are a little difficult to cut and a to expensive to replace every time they are cut).