Lesson 35: Resonance

     How do musical instruments make sound? To understand this, we must understand what happens when a sound wave encounters matter or another sound wave, a situation that is called interference. Interference is the defining characteristic of all waves, including sound. One form of interference is called reflection and it occurs when a wave (in our case, sound) encounters a boundary between two mediums and "bounces" back the way it came. The boundary can be rigid or open, and the nature of the boundary affects how the wave will reflect.

     First, let us define a rigid boundary. A rigid boundary is where the wave encounters a medium with a higher density than what it is currently traveling through. A good example of this is a sound wave encountering a solid wall - the wall is the rigid boundary. When the wave reaches this boundary, it is inverted, or flipped upside down. For a sound wave, all of the compressions become rarefactions; all of the rarefactions become compressions. Waves traveling on strings, like on a violin or guitar, or percussion instruments encounter rigid boundary reflection when they reach the end of the string.

     An open boundary is the exact opposite of the rigid boundary. An open boundary is where the wave encounters a medium with a lower density than what it is currently traveling through. Sound passing through a tunnel encounters open boundary reflection. When the wave reaches the open boundary, it remains erect, or unchanged. Waves traveling through wind instruments (booth woodwinds and brass) encounter open boundary reflection.

     Now, when those sound waves reflect, they encounter other sound waves going in the opposite direction. If the length of the string or the air column is just right, the reflections and new sound waves will meet in the same place every time. When this occurs a standing wave is produced (so named because it looks like the wave is "standing still") and instead of transmitting energy from one end to the other, that energy is multiplied over and over again by the interference between the waves themselves. This type of interference is called "constructive interference." (There is destructive interference too - when this happens, the musical instrument goes mute!) That constructive interference makes the entire musical instrument vibrate at the same frequency as the sound wave. This effect is called resonance. All matter resonates at certain frequencies: maybe you have been at a rock concert near the speakers and it feels like you are shaking all over your body - even on the inside. This occurs when the note being played is the same as your resonant frequency.

     A famous demonstration of resonance is the opera singer who sings such a high note that a crystal goblet is shattered. This is because the pitch that the singer sings matches the crystal's resonant frequency. Resonance is a powerful thing because the sound waves are constantly increasing their energy. Resonance can be seen at sporting events when the crowd stomps its feet in unison - the stadium can begin to vibrate just like a musical instrument. A choir uses resonance to amplify their voices many times louder than the sum of each individual member voice can produce. This is also why it is very easy to hear a choir that is not synchronized because the resonance is destroyed by the singers that are off key.

     Using resonance, the musical instrument creates a sound loud enough to be heard a good distance away. So the question becomes - how big does the instrument need to be? The answer depends on the type of reflection that occurs. Brass and woodwinds are classified as "open pipe resonators" because the pipes and tubes that the musician blows into is (surprise) open at the other end. The musical instruments are constructed with the knowledge that each note has an individual wavelength l. To make this note on an open pipe resonator, the air column must be L =l/2. Strings and percussion (and the clarinet) are classified as "closed pipe resonators". The length of the string necessary to make the same note is L =l/4.

     Each of the resonant lengths above represents the minimum length necessary for an instrument to make a note. If the length of an instrument is increased by a half wavelength, the instrument will create the same note with added harmonics. A harmonic simply represents how many standing waves are present on the string or in the air column. The number of standing waves depends on the frequency - to double the number of standing waves means to double the frequency. Therefore if the instrument is twice as long, it has twice as many standing waves. The blend of harmonics is what is responsible for the quality of the sound and the timbre of an instrument A table of harmonics is listed below for both types of resonators:

Harmonic	Open Resonator Length	Closed Resonator Length	Musical Interval
First	l/ 2	l/ 4	Fundamental - lowest frequency for this note
Second	l	Does not exist	One octave above fundamental
Third	3l/ 2	3l/ 4	One octave plus one fifth (one twelfth) above fundamental
Fourth	2l	Does not exist	Two octaves above fundamental
Fifth	5l/ 2	5l/ 4	Two octaves plus a third above fundamental
Sixth			Two octaves plus a fifth above fundamental
Seventh			Dissonant note - source of a lot of "squeaky" clarinets
Eighth			Three octaves above fundamental
Ninth			Three octaves plus a second above fundamental
Tenth			Three octaves plus a third above fundamental

     Each successive harmonic has a lower amplitude - realistically the fifth harmonic represents the last harmonic audible to humans. The specifics about music theory are not important - what is important is knowing the sequence of events inside the instrument.

1. The sound wave is reflected at the end of the instrument (open or rigid boundary)
2. The reflection interferes with other sound waves traveling in opposite direction
3. A standing wave is created where the interference occurs at the same place in the instrument
4. The standing wave produces constructive interference that increases the sound wave's amplitude
5. The constructive interference creates resonance that makes the instrument vibrate at the sound wave's frequency, amplifying the sound
6. The amplified sound is heard by other people

Makeup Lab Report
Read the situation and answer the questions below with COMPLETE SENTENCES. Failure to do this will result in a zero.
You should already have copied the lecture above, word for word, before completing the lab report.

     Solve the following problems.

1. A note has a 256.00 Hz frequency. What is its wavelength of the speed of sound is 343.00 m/s?
2. What is the fundamental length of a closed pipe resonator if this note is to be produced?
3. What is the fundamental length of an open pipe resonator if this note is to be produced?
4. What is the length of an open resonator necessary to play the third harmonic of this note?
5. What is the length of a closed resonator necessary to play the third harmonic of this note?
6. Complete the harmonic table above (it should already be copied…)

Homework Assignment
Answer each of the following questions with a complete sentence. Show any applicable calculations.

1. How does the slide of a trombone change the pitch of the notes being played?
2. Why are grand pianos shaped the way they are?
3. Why does a cello produce lower notes than a violin?
4. Why does a "jingle bell" produce a higher pitched sound than a church bell?
5. Blow across the nozzle of an empty 2 Liter soda bottle and an empty 20 ounce soda bottle. Explain the reason for the difference in sound produced.
6. Give an example of how percussion instruments follow the relationships of resonance.