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Waves and Sounds - Honors Physics: A word on resonance and beats

Resources to guide research for 3/2014 Take Home Assignment in Ms. Ormond's Honors Physics class.

Resonance

 musical instruments are set into vibrational motion at theirnatural 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 patternsby 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 anotherinterconnected 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.

From The Physics Classroom

Beats

Beats

When two (or more) sounds are present having a frequency difference of less than about 20 or 30 Hz, you will hear "beats." The frequency of the beats will be at the difference frequency. If the frequency difference is larger than about 20 or 30 Hz, a tone is usually perceived rather than distinct beats. For complex sounds, beats can arise from any of the partials of the sounds.

graph showing beatsMathematically beats can be understood from a mathematical identity for sine waves. For sounds with (angular) frequencies of w1 and w2 added together the identity gives 

sin(w1t) + sin(w2t) = 2 sin(w3t)cos( w4t)
where w3 is the average of w1 and w2
and w4 is one half of their difference.

If the difference frequency is small, the sound is the sound of a single tone at the average frequency, but with a volume which varies in time at the difference frequency. This is illustrated on the graphic to the right. The sum is a sine wave with a variable amplitude and the pattern repeats every 0.05s. That is, there are 1/0.05 = 20 repeats per second.

You can hear beats using this Sound File (mp3) ( wav). There are two pure tones near middle C which are 6 Hz apart. First each tone is played separately using then left and right stereo channels, then they are played together. As an interesting experiment, listen to the sound file using a good pair of earphones. You will hear the beats even if the tones are sent separately to your two ears. This means your hearing system actually performs a (phase coherent) addition of the sounds somewhere in your brain, after the sound has been detected by your ear. The last sound in the file has the two tones played equally in both ears. The beats sound similar, but note also that it does not sound the same "spatially." See the wikipedia article on binaural beats for more information.

Beats can be used anytime you have periodic behavior. Since it is easy to hear beats faster than about 1 per second, it is relatively easy to match two signals with an accuracy of 1 Hz (or better) even if the original frequencies are microwave signals at 1 GHz = 1,000,000,000 or above. That corresponds to more than 12 digits of accuracy!

Thanks to Michigan Technical University