The human ear canal is approximately 2.3 cm long. It is open to the outside and is closed at the other end by the eardrum.
Estimate the frequencies (in the audible range) of the standing waves in the ear canal. Express your answers using two significant figures. If there is more than one answer, enter them in ascending order separated by commas.
Since it one end is open and the other is shut, the ear canal is considered a closed pipe. We set L = 2.3cm = 0.023m. The frequency for a closed pipe is fn=n(v/4L)=v/wavelength where n=1,2,3,.. referring to the number of overtones/harmonics. L=(1/4)*wavelength so we solve for wavelength 0.023m=(1/4)*wavelength and get wavelength=0.092m. Since the audible frequency range is 20Hz-20,000Hz, we solve the equation fn=n(v/wavelength) using n=1,2,3,... until we reach 20,000Hz. Here, we f1=3728 Hz, f2=11185 Hz, f3=18642 Hz (f4 is higher than the audible range so we know to stop at n=3).
Oh and use the speed of sound of air (at 20 degrees C) to be v=343m/s.
To estimate the frequencies of the standing waves in the ear canal, we can use the formula:
f = v / (2L)
where f is the frequency, v is the speed of sound, and L is the length of the ear canal.
The speed of sound in air is approximately 343 m/s.
Converting the length of the ear canal to meters:
L = 2.3 cm = 0.023 m
Now we can calculate the frequencies:
f = 343 m/s / (2 * 0.023 m)
f = 7467 Hz
So the estimated frequency of the standing waves in the ear canal is 7467 Hz.
To estimate the frequencies of the standing waves in the ear canal, we can use the formula for the fundamental frequency of a closed, cylindrical tube. In this formula, the fundamental frequency is given by:
f1 = v / (2L)
Where:
f1 is the fundamental frequency of the standing wave
v is the speed of sound (approximately 343 m/s in air at room temperature)
L is the length of the tube
First, we need to convert the length of the ear canal from centimeters to meters.
2.3 cm = 0.023 m
Using the formula above, we can calculate the fundamental frequency:
f1 = 343 m/s / (2 * 0.023 m)
f1 = 7492.39 Hz
So, the estimated fundamental frequency of the standing wave in the ear canal is approximately 7492.39 Hz.
However, since the ear canal is open at one end, it can also support harmonics that are multiples of the fundamental frequency. The formula for the frequencies of the harmonics is:
fn = nf1
Where:
fn is the frequency of the nth harmonic
n is the harmonic number
f1 is the fundamental frequency
To find the frequencies of the harmonics, we can substitute different values of n into the formula. Let's calculate the first few harmonics (up to the 5th harmonic):
f2 = 2 * f1 = 2 * 7492.39 Hz = 14984.78 Hz
f3 = 3 * f1 = 3 * 7492.39 Hz = 22477.17 Hz
f4 = 4 * f1 = 4 * 7492.39 Hz = 29969.56 Hz
f5 = 5 * f1 = 5 * 7492.39 Hz = 37461.95 Hz
So, the estimated frequencies (in ascending order) of the standing waves in the ear canal are approximately: 7492.39 Hz, 14984.78 Hz, 22477.17 Hz, 29969.56 Hz, and 37461.95 Hz.