Please show your work! Thank you! The clarinet is a roughly cylindrical instrume
ID: 1476532 • Letter: P
Question
Please show your work! Thank you!
The clarinet is a roughly cylindrical instrument of length L that is open to the outside air at the bell, but closed by the mouthpiece, the reed, and the player's mouth at the other end. The lowest note played by a standard clarinet is 146.83 Hz.
(a) Find the effective length of the clarinet. Hint: You will need the speed of sound waves in air, which may require a bit of research.
(b) Find the next four frequencies that can be played without making any change to the instrument.
Explanation / Answer
(a)
The speed of sound in air at a temperature of 20 0C is
v = 340 m/s
Clarinet is similar to a closed organ pipe. It only consists of odd harmonics.
The frequency of the clarinet is
fn = n v / 4L
Here, effective length of the clarinet is L and n = 1,3,5,7,9...
The lowest or fundamental note is when n = 1.
Thus, the fundamental frequency is
f1 = v / 4L
L = v / 4f1
Substitute 146.83 Hz for f1 and 340 m/s for v in the above equation,
L = v / 4f1
= (340 m/s) / 4 (146.83 Hz)
= 0.57890 m
Therefore, the effective length of the clarinet is 0.57890 m.
(b)
It is given that the fundamental frequency or lowest note is
fn = nf1
f1 = n 146.83 Hz)
Since the closed organ pipe or clarinet consists of odd harmonics, the next four frequencies are n = 3,5,7,9.
Thus,
f3 = 3(146.83 Hz)
= 440.49 Hz
f5 = 5(146.83 Hz)
= 734.15 Hz
f7 = 7(146.83 Hz)
= 1027.8 Hz
f9 = 9(146.83 Hz)
= 1321.5 Hz
Therefore, the next four frequencies that can be played by clarinet are 440.49 Hz, 734.15 Hz, 1027.8 Hz and 1321.5 Hz.