Resonance
The goal of Unit 11 of The Physics Classroom Tutorial 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. In Lesson 5, 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: instruments with vibrating strings (which would include guitar strings, violin strings, and piano strings), open-end air column instruments (which would include the brass instruments such as the trombone and woodwinds such as the flute and the recorder), and closed-end air column instruments (which would include some organ pipe and the bottles of a pop bottle orchestra). 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 that the instruments produce.As was mentioned in Lesson 4, 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 instrument 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. One of our best models of resonance in a musical instrument is a resonance tube (a
hollow
cylindrical tube) partially filled with water and forced
into vibration by a tuning fork. The tuning fork is the
object that forced the air inside of the resonance tube
into resonance. As the tines of the tuning fork vibrate at
their own natural frequency, they created sound waves that
impinge upon the opening of the resonance tube. These
impinging sound waves produced by the tuning fork force 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
location of the water level can be altered by raising and
lowering a reservoir of water, thus decreasing or increasing
the length of the air column. As
we have learned earlier, 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 of a
vibrational system decreases the wavelength and increases
the natural frequency. So by raising and lowering the water
level, 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. The result of
resonance is always a big vibration - that is, a loud
sound.Another common physics demonstration that serves as an excellent model of resonance is the famous "singing rod" demonstration. A long hollow aluminum rod is held at its center. Being a trained musician, teacher reaches in a rosin bag to prepare for the event. Then with great enthusiasm, he/she slowly slides her hand across the length of the aluminum rod, causing it to sound out with a loud sound. This is an example of resonance. As the hand slides across the surface of the aluminum rod, slip-stick friction between the hand and the rod produces vibrations of the aluminum. The vibrations of the aluminum force the air column inside of the rod to vibrate at its natural frequency. The match between the vibrations of the air column and one of the natural frequencies of the singing rod causes resonance. The result of resonance is always a big vibration - that is, a loud sound.
The familiar sound of the sea that 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 resonance 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 that produces the
sounds that 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.
The result of resonance is always 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 is often modeled in a
Physics class using a plastic straw. The ends of the straw
are 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 together to produce a loud
sound. As if this weren't silly enough, the length of the
straw is typically shortened by cutting small pieces off its
opposite end. As the straw (and the air column that it
contained) is shortened, the wavelength decreases and the
frequency was increases. Higher and higher pitches are
observed as the straw is 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 too expensive to
replace every time they are cut).Resonance is the cause of sound production in musical instruments. In the remainder of Lesson 5, the mathematics of standing waves will be applied to understanding how resonating strings and air columns produce their specific frequencies.
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