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Sound - Resonant Frequency and the Missing Fundamental

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Problem 6. Assume that the outer ear canal is a cylincrical pipe 3 cm long, closed at one end by the eardum. Calculate the resonant frequency (fundamental, v1) of this pipe. Our hearing should be especially sensitive for frequencies near this resonance.

Problem 7. A "tonic" chord in the musical key of A-major consists of tones with frequencies of 440, 550, and 660 Hz (A, C# and E). When such a chord is played on the piano or by three instruments, why is this not heard as a single tone with a pitch of 110 Hz (the "missing fundamental")?

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This solution is provided in 211 words and provided in both .doc and .pdf format. It includes diagrams and discusses the fundamental mode in order to calculate fundamental frequency. Fourier transformer is also discussed in relation to the missing fundamental problem.

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Multiple Choice

1. In order to cause the loudness of a sound to appear half its original value it must be: (a)increased by 50 dB (b) decreased by 50 dB (c) decreased by 10 dB (d) decreased by 3 dB (e) none of these. Explain your choice.

2. The flute and piccolo are in teh same family of instruments, and each functions as a pipe open at both ends. Both have the same fingering but different lengths. The flute is about 66-cm long and is an octave lower than the piccolo - that is, the piccolo plays at twice the frequency of the flute. It follows that the piccolo, in comparison tot he flute is about: (a) twice as long (b) half as long (c) eight times as long (d) the same length (e) none of these. Explain your choice.

What will happen to the fundamental frequency of an organ pipe open at both ends if a cap is placed over one end sealing it? The frequency will: (a) go up 20% (b) be halved (c) stay the same (d) double (e) none of these. Explain your choice.


1. Imagine a hypothetical piano with all strings made of the same material and all under the same tension. The piano extends from 27.5 Hz to 4186 Hz which is over seven octaves - that is, seven doublings of frequency. If the highest note corresponds to a string 15-cm long, how long will the lowest string have to be? What can you conclude about this approach to piano design?

2. A narrow glass tube is 0.50-m long and sealed at its bottom end is held vertically just below a loudspeaker that is connected to an audio generator and amplifier. A tone with a gradually increasing frequency is fed into the tube, and a loud resonance is first observed at 170 Hz. What is the speed of sound in the room?

3. A train whistle is blown by an engineer who hears it sound at 650 Hz. If the train is heading toward a station at 20 m/s, what will the whistle sound like to a waiting commuter? Take the speed of sound to be 340 m/s.

4. An ultrasonic wave at 80 000 Hz is emitted into a vein where the speed of sound is about 1.5 km/s. The wave reflects off the red blood cells moving toward the stationary receiver. If the frequency of the returning signal is 80 020 Hz, what is the speed of the blood flow?

5. A point source of sound on top of a police car emits a signal at 1000 Hz. If the car is traveling in a straight line at 30 m/s, what will be the wavelength percieved by people standing on the road both directly in front of and behind the car? Take the speed of sound to be 335 m/s. What is the wavelength as measured in the car?

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