Please help I cant breath with this qn, due today exam pending, thanks for taking your time to help.
The human outer ear contains a more-or-less cylindrical cavity called the auditory canal that behaves like a resonant tube to aid in the hearing process. One end terminates at the eardrum (tympanic membrane), while the other opens to the outside. (See the figure (Figure 1) .) Typically, this canal is approximately 2.21cm long.
part A
At what frequencies would it resonate in its first two harmonics?
Express your answers separated by a comma.
Part B
What are the corresponding sound wavelengths in part A?
Express your answers separated by a comma.
Question 2
In a certain home sound system, two small speakers are located so that one is 70.0cm closer to the listener than the other.
For what frequencies of audible sound will these speakers produce constructive interference at the listener? (Find only the three lowest audible frequencies).
Please if you know how to solve this answer it all without leaving it like do the rest the same way. God bless!
qn 2
speed of sound divide by 0.70
so 344/0.70 is freq 1
freq 2 answer in freq 1 times 2
freq 3 answer in freq 1 times 3
To solve these questions, we need to understand the concept of resonance in a resonant tube and constructive interference in a sound system.
Question 1:
Part A: To find the frequencies at which the auditory canal resonates in its first two harmonics, we can use the formula for the resonant frequency of a tube that is closed at one end:
f = (n * v) / (4 * L)
where:
f is the frequency of the harmonic,
n is an integer representing the harmonic number (1 for the fundamental frequency, 2 for the second harmonic, etc.),
v is the speed of sound in air (approximately 343 m/s at room temperature),
L is the length of the auditory canal.
For the first harmonic (n = 1), we can substitute the given values:
f1 = (1 * 343) / (4 * 0.0221)
Simplifying the equation gives us the resonance frequency for the first harmonic.
Similarly, for the second harmonic (n = 2), we substitute n = 2 into the formula to find the resonance frequency for the second harmonic.
Part B: To find the corresponding sound wavelengths, we can use the formula:
λ = v / f
where:
λ is the wavelength of the sound wave,
v is the speed of sound in air,
f is the frequency of the harmonic.
Using the values of speed of sound and the frequencies found in Part A, we can calculate the corresponding wavelengths for each frequency.
Question 2:
For constructive interference in a sound system, we need to consider the path length difference between two speakers and the listener. The path length difference should be in sync with the wavelength of the sound wave.
To find the frequencies at which the speakers produce constructive interference, we can use the formula:
Δx = (m * λ) / 2
where:
Δx is the path length difference,
m is an integer representing the interference order (1 for the first-order, 2 for the second-order, etc.),
λ is the wavelength of the sound wave.
By substituting the given path length difference and rearranging the equation, we can calculate the wavelength corresponding to each interference order.
Then, we can find the frequencies by using the formula:
f = v / λ
where:
f is the frequency of the sound wave,
v is the speed of sound in air,
λ is the wavelength of the sound wave.
By applying this formula to the wavelengths obtained in the previous step, we can calculate the frequencies at which the speakers produce constructive interference for the three lowest audible frequencies.
To summarize, for Question 1, you need to calculate the resonant frequencies and corresponding wavelengths of the auditory canal's harmonics. For Question 2, you need to calculate the frequencies at which the speakers produce constructive interference at the listener.