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8. The trumpet player now pushes both the 1ª and 2nd valves, opening both bits of extra tubing. What is the new length L of the horn? 9. What is the frequency of the 6™ harmonic produced in Problem 8?

8. The trumpet player now pushes both the 1ª and 2nd valves, opening both bits of extra tubing. What is the new length L of the horn? 9. What is the frequency of the 6™ harmonic produced in Problem 8?

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Trumpet Physics As we learned in the video lesson, an open pipe of length L can create standing waves at certain specific frequencies, given by: fn = n f1 2L = n fn = the frequency that creates a standing wave with n loops, in Hz = the "nth harmonic" fi = the frequency that creates a standing wave with 1 loop, in Hz = the "fundamental frequency" n = number of loops in the standing wave v = speed of the sound wave = 343 m/s (at 20° C) L= length of the pipe, in m %3D The fundamental frequency, f1, is the lowest frequency that makes a standing wave on the pipe. The higher frequencies or “harmonics" are multiples of the fundamental: f2 is twice as much as f1; f3 is three times as big as f1, and so on. A trumpet is essentially a long, curved open pipe. (Actually, its geometry is more complicated, but it is designed to produce the same set of frequencies that an open pipe would. So close enough.) It has three valves, each of which is connected to an extra bit of tubing. When no valves are pushed, the sound wave passes straight through the horn, travelling a distance L. When a valve is pushed, it opens up the extra bit of tubing, which increases the length L of the horn slightly. According to the equation above, increasing the length L lowers the frequency of the standing wave, which produces a lower tone or note. So, when a valve is pushed, it lowers the note the horn produces. Pushing a valve with a longer tube lowers the note even more. Opening more than one valve makes the sound travel through more than one extra bit of tubing, lowering the note even further. By picking the right combination of valves, the trumpet can make any note on the musical scale. Directions: Here are some practice problems to make you familiar with these concepts. Please review the following. Assume the speed of sound is 343 m/s.
Transcribed Image Text:Trumpet Physics As we learned in the video lesson, an open pipe of length L can create standing waves at certain specific frequencies, given by: fn = n f1 2L = n fn = the frequency that creates a standing wave with n loops, in Hz = the "nth harmonic" fi = the frequency that creates a standing wave with 1 loop, in Hz = the "fundamental frequency" n = number of loops in the standing wave v = speed of the sound wave = 343 m/s (at 20° C) L= length of the pipe, in m %3D The fundamental frequency, f1, is the lowest frequency that makes a standing wave on the pipe. The higher frequencies or “harmonics" are multiples of the fundamental: f2 is twice as much as f1; f3 is three times as big as f1, and so on. A trumpet is essentially a long, curved open pipe. (Actually, its geometry is more complicated, but it is designed to produce the same set of frequencies that an open pipe would. So close enough.) It has three valves, each of which is connected to an extra bit of tubing. When no valves are pushed, the sound wave passes straight through the horn, travelling a distance L. When a valve is pushed, it opens up the extra bit of tubing, which increases the length L of the horn slightly. According to the equation above, increasing the length L lowers the frequency of the standing wave, which produces a lower tone or note. So, when a valve is pushed, it lowers the note the horn produces. Pushing a valve with a longer tube lowers the note even more. Opening more than one valve makes the sound travel through more than one extra bit of tubing, lowering the note even further. By picking the right combination of valves, the trumpet can make any note on the musical scale. Directions: Here are some practice problems to make you familiar with these concepts. Please review the following. Assume the speed of sound is 343 m/s.
1. When no valves are pushed, the sixth harmonic of the horn (n = 6) produces a frequency of f6 = 698 Hz (an F). Use that information, and rearrange the equation above to calculate the length L of the trumpet's pipe when no valves are used. 2. When the first valve is pushed, the frequency of the sixth harmonic changes to fo = 622 Hz (an E-flat). Calculate the new length of the pipe L. 3. The new length is slightly longer than the original length, because of the added length of the tubing attached to the first valve. Subtract the two lengths to find the length of the tube attached to the first valve. 4. When the second valve (only) is pushed, the frequency of the 6™ harmonic changes to fo = 659 Hz (an E). What is the new length of the pipe L? 5. The difference in length between Step 4 and Step 1 is due to the length of the tubing attached to the second valve. Calculate the length of the tube attached to the second valve. 6. When the third valve (only) is pushed, the frequency of the 6th harmonic changes to fo = 585 Hz (a D). What is the new length of the pipe L? 7. The difference in length between Step 6 and Step 1 is due to the length of the tubing attached to the third valve. Calculate the length of the tube attached to the third valve. 8. The trumpet player now pushes both the 1" and 2nd valves, opening both bits of extra tubing. What is the new length L of the horn? 9. What is the frequency of the 6th harmonic produced in Problem 8? 10. The trumpet player now pushes both the 1st and 3ª valves, opening both bits of extra tubing. What is the new length L of the horn? What is the frequency of the 6th harmonic produced? 11. The trumpet player now pushes both the 2nd and 3d valves, opening both bits of extra tubing. What is the new length L of the horn? What is the frequency of the 6th harmonic produced?
Transcribed Image Text:1. When no valves are pushed, the sixth harmonic of the horn (n = 6) produces a frequency of f6 = 698 Hz (an F). Use that information, and rearrange the equation above to calculate the length L of the trumpet's pipe when no valves are used. 2. When the first valve is pushed, the frequency of the sixth harmonic changes to fo = 622 Hz (an E-flat). Calculate the new length of the pipe L. 3. The new length is slightly longer than the original length, because of the added length of the tubing attached to the first valve. Subtract the two lengths to find the length of the tube attached to the first valve. 4. When the second valve (only) is pushed, the frequency of the 6™ harmonic changes to fo = 659 Hz (an E). What is the new length of the pipe L? 5. The difference in length between Step 4 and Step 1 is due to the length of the tubing attached to the second valve. Calculate the length of the tube attached to the second valve. 6. When the third valve (only) is pushed, the frequency of the 6th harmonic changes to fo = 585 Hz (a D). What is the new length of the pipe L? 7. The difference in length between Step 6 and Step 1 is due to the length of the tubing attached to the third valve. Calculate the length of the tube attached to the third valve. 8. The trumpet player now pushes both the 1" and 2nd valves, opening both bits of extra tubing. What is the new length L of the horn? 9. What is the frequency of the 6th harmonic produced in Problem 8? 10. The trumpet player now pushes both the 1st and 3ª valves, opening both bits of extra tubing. What is the new length L of the horn? What is the frequency of the 6th harmonic produced? 11. The trumpet player now pushes both the 2nd and 3d valves, opening both bits of extra tubing. What is the new length L of the horn? What is the frequency of the 6th harmonic produced?
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