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Resonance If you have ever blown across the top of a bottle or other similar object, you may have heard it emit a particular sound. The air in the bottle is a column of air. It has a natural frequency (actually more than one) at which it will vibrate when set in motion. If you use a bottle that is a different size, you will hear a different frequency when you blow across it.

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Suppose that you use a tuning fork that has the same frequency as the natural frequency of the air columns. Since you are applying energy at the same frequency as the natural frequency of the air column, you are creating resonance. You are producing a large amplitude sound wave by constructive interference. This can be explained in terms of standing waves. By holding the tuning fork above the air column, you are sending sound waves down the air column. These sound waves reflect from the bottom and return up the air column. If the tuning fork is just the right frequency, this reflected wave will interfere with the wave from the tuning fork, creating a standing wave. Other frequencies might be sent down the tube, but if they dont constructively interfere, they die out.

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Closed Air Columns A sound wave consists of vibrating air molecules. If we were to try and picture a standing wave of these air molecules in a closed air column, there should be a node at the closed end of the air column (since the molecules would not be free to vibrate here at the fixed end) and an antinode at the open end of the air column (since they are free to vibrate here).

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Standing Waves

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A standing wave in a closed air column can be satisfied with a quarter of a wavelength. Three quarters of a standing wave would also satisfy the criteria of a node at one end and an antinode at the other end.

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Closed Air Columns

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Resonant Lengths Other lengths produce a standing wave for a particular frequency. Each time we would have to add half of a wavelength. The different air column lengths that give resonance at a particular frequency are called resonant (or resonance) lengths. These are actual lengths, measured in distance units (cm, m, etc.).

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Closed Tubes

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Calculate the first three resonant lengths for a 460 Hz tuning fork if the air temperature is 18 o C.

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A tuning fork vibrates above a vertical open tube filled with water. The water level is dropping slowly. As it does so the air in the tube above the water level resonates when the distance from the tube opening to the water level is 0.300 m and again at 0.500 m. If the air temperature is 15 o C, what is the frequency of the tuning fork?

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What is the length of a closed organ pipe that emits middle C (264 Hz) when the temperature is 15 o C?

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Fundamental Frequency & Overtones The frequency heard from the first resonant length (¼ λ) is called the fundamental frequency. This is usually the loudest frequency and is the one that determines the pitch. When we are told that an air column is resonating at a certain frequency we assume that it is resonating at the fundamental frequency (which corresponds to ¼ λ). Successive frequencies are referred to as overtones (first, second, third, etc). The fundamental and overtones are frequencies, measured in Hz.

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An organ pipe is 80.0cm long. If the air temperature is 23 o C, what is the fundamental and first three overtones if it is a closed pipe(at one end)?

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A 15.0 cm test tube is blown across so that it resonates. If the air temperature is 20.0 o C, calculate the fundamental frequency and the first two overtones (first three resonant lengths).

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Some instruments that behave as closed air columns include the clarinet, oboe, and bassoon. Some organ pipes can also be closed. The overtones are whole number multiples of the fundamental frequency. Whole number multiples of the fundamental frequency are referred to as harmonics. The fundamental frequency is referred to as the first harmonic. The first overtone in our example is three times as big as the fundamental frequency and is referred to as the third harmonic. The second overtone in our example is five times as big as the fundamental frequency and is referred to as the fifth harmonic. For closed air columns, only the odd number harmonics are present.

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Open Air Columns An open air column is one which is open at both ends. Because the air molecules are free to vibrate at each end, there must be an antinode at each end for a standing wave. Some musical instruments that behave as open air columns include the trumpet and the flute.

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Open Tube

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Assuming an air temperature of 20.0 o C, calculate the fundamental frequency and the first two overtones for a 30.0 cm long open air column.

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Fundamental Frequency & Overtones The overtones are all of the whole number multiples of the fundamental frequency, as they were with closed air columns. They are different multiples of the fundamental frequency. The first overtone in our example is two times as big as the fundamental frequency and is referred to as the second harmonic. The second overtone in our example is three times as big as the fundamental frequency and is referred to as the third harmonic. For open air columns, all of the harmonics are present

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(a) A flute (open at both ends) is designed to play middle C (264 Hz) as the fundamental frequency when all the holes are covered. Approximately how long should the distance be from the mouthpiece to end of the flute. (Note: this is only approximate since the antinode does not occur precisely at the mouthpiece.) Assume the temperature is 20 o C. (b) If the temperature is only 10 o C, what will be the frequency of the note played when all the openings are covered in the flute?

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How many overtones are present within the audible range for a 100 cm long organ pipe at 20 o C (a) if it is open (b) if it is closed ?

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