Lesson
1: The Nature of a Sound Wave
Lesson 2: Sound Properties and Their Perception
Lesson 3: Behavior of Sound Waves
The Doppler Effect and Shock Waves
Reflection, Refraction, and Diffraction
Lesson 4: Resonance and Standing Waves
Fundamental Frequency and Harmonics
Lesson 5: Musical Instruments
Lesson 4: Resonance and Standing Waves
Forced Vibration
Musical instruments and other objects are set into vibration at their natural frequency when a person hits, strikes, strums, plucks or somehow disturbs the object. For instance, a guitar string is strummed or plucked; a piano string is hit with a hammer when a pedal is played; and the tines of a tuning fork are hit with a rubber mallet. Whatever the case, a person or thing puts energy into the instrument by direct contact with it. This input of energy disturbs the particles and forces the object into vibrational motion - at its natural frequency.
If you were to take a guitar string and stretch it to a given length and a given tightness and have a friend pluck it, you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar. On the other hand, if the string is attached to the sound box of the guitar, the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a forced vibration. In the case of the guitar string mounted to the sound box, the fact that the surface area of the sound box is greater than the surface area of the string, means that more surrounding air particles will be forced into vibration. This causes an increase in the amplitude and thus loudness of the sound.
This
same principle of a forced vibration was demonstrated in
class using a tuning fork. If the tuning fork is held in
your hand and hit with a hammer, a sound is produced as the
tines of the tuning fork set surrounding air particles into
vibrational motion. The sound produced by the tuning fork is
barely audible to students in the back rows of the room.
However, if the tuning fork is set upon the whiteboard panel
or the glass panel of the overhead projector, the panels
begin vibrating at the same natural frequency of the tuning
fork. The tuning fork forces surrounding glass (or vinyl)
particles into vibrational motion. The vibrating whiteboard
or overhead projector panel in turn forces surrounding air
particles into vibrational motion and the result is an
increase in the amplitude and thus
loudness of the sound. This principle of forced
vibration explains why the classroom tuning fork is mounted
on a sound box, why a commercial music box mechanism is
mounted on a sounding board, why a guitar utilizes a sound
box, and why a piano string is attached to a sounding board
- a louder sound is always produced.
Now consider a
related situation which resembles another classroom
demonstration. Suppose that a tuning fork is mounted on a
sound box and set upon the table; and suppose a second
tuning fork/sound 
box
system having the same natural frequency (say 256 Hz) is
placed on the table near the first system. Neither of the
tuning forks is vibrating. Then the first tuning fork is
struck with a rubber mallet and the tines begin vibrating at
its natural frequency - 256 Hz. These vibrations set the
sound box and the air inside the sound box vibrating at the
same natural frequency of 256 Hz. Surrounding air particles
are set into vibrational motion at the same natural
frequency of 256 Hz and every student in the classroom hears
the sound. Then the tines of the tuning fork are grabbed to
prevent their vibration and remarkably the sound of 256 Hz
is still being heard. Only now the sound is being produced
by the second tuning fork - the one which wasn't hit with
the mallet. Amazing!! In fact, it is so amazing, that the
demonstration is repeated to assure that the same surprising
results are observed. They are! What is happening?
In this demonstration, one tuning fork forces another tuning fork into vibrational motion at the same natural frequency. The two forks are connected by the surrounding air particles. As the air particles surrounding the first fork (and its connected sound box) begin vibrating, the pressure waves which it creates begin to impinge at a periodic and regular rate of 256 Hz upon the second tuning fork (and its connected sound box). The energy carried by this sound wave through the air is tuned to the frequency of the second tuning fork. Since the incoming sound waves share the same natural frequency as the second tuning fork, the tuning fork easily begins vibrating at its natural frequency. This is an example of resonance - when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion.
The result of resonance is always a large
vibration. Regardless of the vibrating system, if resonance
occurs, a large vibration results. This was demonstrated in
class with an odd-looking mechanical system resembling an
inverted pendulum. Three sets of two identical plastic
bobs
were mounted on a very elastic metal pole, which were in
turn mounted to a metal bar. Each metal pole and attached
bob had a different length, thus giving it a different
natural frequency of vibration. The bobs were color coded to
distinguish between them - they were colored red, blue and
green (this will be significant later in the course). The
red bobs were mounted on the longer poles and they had the
lowest natural frequency of vibration. The blue bobs were
mounted on the shorter poles and they had the highest
natural frequency of vibration. (Note the length-wavelength-frequency
relationship that was discussed earlier.) When the red
bob was disturbed, it began vibrating at its natural
frequency, which in turn caused the attached bar to vibrate
at the same frequency; this in turn set the other attached
red bob into vibrating at the same natural frequency. This
is resonance - one bob vibrating at a given frequency
forcing a second object with the same natural frequency into
vibrational motion. While the green and the blue bobs were
disturbed by the vibrations transmitted through the metal
bar, only the red bob would resonate. This is because the
frequency of the first red bob was tuned to the frequency of
the second red bob; they share the same natural frequency.
The result was that the second red bob begins vibrating with
a huge amplitude.
Another
classroom demonstration of resonance involved a plastic tube
containing an air column. The length of the air column was
adjusted by raising and lowering a reservoir of water (dyed
red). The raising and lowering of the reservoir would adjust
the height of water in the open-air tube, and thus adjust
the length of the air column inside the tube. As the length
of the air column was decreased, the natural frequency of
the air column is increased. (Again note the length-wavelength-frequency
relationship that was discussed earlier.) While
adjusting the height of the liquid in the tube, a vibrating
tuning fork is held above the air column of the tube. When
the natural frequency of the air column is tuned to
the frequency of the vibrating tuning fork, resonance occurs
and a loud sound results. Quite amazingly, the vibrating
tuning fork forces air particles within the air column into
vibrational motion. Once more in this resonance situation,
the tuning fork and the air column share the same
vibrational frequency.
In conclusion, resonance occurs when two interconnected objects share the same vibrational frequency. When one of the objects is vibrating, it forces the second object into vibrational motion. The result is a large vibration, and if a sound wave within the audible range of human hearing is produced, a loud sound is heard.