The diameter of a trumpet bell is roughly 10 cm. When the trumpet plays middle-C (f=262 Hz), does the sound come out in a narrow beam or spread in all directions? Give clear physical justification for your answer.
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A trumpet consists mainly of a long brass tube with a mouthpiece at one end and a flared opening or "bell" at the other end. This tube is coiled to make the trumpet more compact. The trumpet also has three valves that splice additional segments of tubing into the main tube, thus extending its overall length. When a musician seals his lips together and presses them against the mouthpiece of the trumpet, he creates a resonant system (a system in which a standing wave can exist). The air can oscillate in and out of the open bell while the air pressure in the mouthpiece oscillates up and down. a.) To sustain the oscillation, the musician blows small bursts of high-pressure air into the mouthpiece. Why are these bursts timed to coincide with the moments when the pressure inside the mouthpiece is already at its maximum? When you think about this, imagine a spring that is oscillating up and down. If the spring eventually slows down due to dissipative forces, how would you have to push the spring to keep it going? b.) As the musician presses the valves and lengthens the trumpet tube, the oscillation slows and the trumpet's pitch goes down. c.) Even without touching the valves, a good trumpet player can produce a variety of higher pitched notes using higher-order vibrational modes (overtones). What is the air inside the trumpet doing while it's oscillating in one of these higher-order modes? d.) The best way for a musical instrument to project sound is to move a large amount of air back and forth at low speeds rather than a small amount of air back and forth at high speeds. That's because room air can't respond quickly enough to the high-speed air near an instrument to extract its energy. Instead, that energy stays "inside the instrument" instead of going forth into the room. If the trumpet had no bell, the air at its open end would move back and forth very rapidly. Describe how the bell helps the trumpet project its sound.
Adi S.
French horn. The French horn, one of the most beautiful-sounding instruments in the orchestra, consists of about 3.7 $\mathrm{m}$ (roughly 12 $\mathrm{ft}$ ) of thin tubing, rolled into a spiral shape (although sizes do vary). The player blows into the mouthpiece, which can be treated as a closed end, and places his hand in the opposite end, which has a large flared opening. In brass instruments, the fundamental note is not normally playable. Instead the first overtone is the lowest playable note. (a) If the player's hand keeps the large end open, what is the frequency of the lowest playable note? (b) If the player now closes the large end with his hand, what is the frequency of the lowest playable note? See the answer to Problem $28 .$ (Note: The physics of the French horn is much more complex than is indicated here. The player's hand in the open end also changes the effective length of the tube, which then affects the frequency of the sound.)
Suppose you have a 66 cm long tube that can produce the sound of note C4 (261 Hz). How far down the end from where you are blowing in the air would you drill a large hole to produce a C# note (277 Hz)? Assume that you are only exciting the fundamental mode.
Madhur L.
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