Lesson 5: Musical
Instruments
Resonance
The goal of Unit 11 of The Physics
Classroom 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: 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.
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 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.
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 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.
Resonance was also
modeled in class by the demonstration with the famous
"singing rod." A long hollow aluminum rod was held by the
teacher at its center. Being a trained musician, he/she
reached in the rosin bag to prepare for the event. Then with
great enthusiasm, he/she slowly slid her hand across the
length of the aluminum rod, causing it to sound out with a
loud sound. This once more was an example of resonance. As
the hand is slid 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 forces the air column inside of the rod to
vibrate at its natural frequency. The match between the
vibrations of the rod and one of the natural frequencies of
the singing rod causes resonance. 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).
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 specfic frequencies.
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