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Chapter 16: Problem 51

Find the resonance frequency of the ear canal. Treat it as a half-open pipe of diameter \(8.0 \mathrm{~mm}\) and length \(25 \mathrm{~mm}\). Assume that the temperature inside the ear canal is body temperature \(\left(37^{\circ} \mathrm{C}\right)\).

Short Answer

Expert verified
Answer: The resonance frequency of the ear canal is approximately 3445 Hz.

Step by step solution

01

Convert temperature to speed of sound

Using the given temperature, which is the body temperature of \(37^{\circ} \mathrm{C}\), we can find the speed of sound \(v\) in the medium by plugging the values into the equation: \(v = 331.4\sqrt{1+\dfrac{37}{273.15}}\) Calculate the value using a calculator: \(v \approx 331.4\sqrt{1.1355} \approx 344.5 \mathrm{~m/s}\)
02

Convert length and diameter to meters

We need to convert the given dimensions of ear canal, diameter (D) and length (L), into meters: \(D = 8.0 \mathrm{~mm} = 8.0 \times 10^{-3} \mathrm{~m}\) \(L = 25 \mathrm{~mm} = 25 \times 10^{-3} \mathrm{~m}\)
03

Compute the resonance frequency

Now that we have the speed of sound \(v\) and the length \(L\) of the half-open pipe, we can calculate the resonance frequency \(f\) using the formula: \(f = \dfrac{v}{4L}\) Substitute the values for \(v\) and \(L\): \(f = \dfrac{344.5}{4 \times 25 \times 10^{-3}}\) Calculate the value using a calculator: \(f \approx \dfrac{344.5}{0.1} \approx 3445 \mathrm{~Hz}\) The resonance frequency of the ear canal is approximately \(3445 \mathrm{~Hz}\).

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Most popular questions from this chapter

The sound level in decibels is typically expressed as \(\beta=10 \log \left(I / I_{0}\right),\) but since sound is a pressure wave, the sound level can be expressed in terms of a pressure difference. Intensity depends on the amplitude squared, so the expression is \(\beta=20 \log \left(P / P_{0}\right),\) where \(P_{0}\) is the smallest pressure difference noticeable by the ear: \(P_{0}=2.00 \cdot 10^{-5} \mathrm{~Pa}\). A loud rock concert has a sound level of \(110 . \mathrm{dB}\), find the amplitude of the pressure wave generated by this concert.

A sound meter placed \(3 \mathrm{~m}\) from a speaker registers a sound level of \(80 \mathrm{~dB}\). If the volume on the speaker is then turned down so that the power is reduced by a factor of 25 , what will the sound meter read? a) \(3.2 \mathrm{~dB}\) c) \(32 \mathrm{~dB}\) e) \(66 \mathrm{~dB}\) b) \(11 \mathrm{~dB}\) d) \(55 \mathrm{~dB}\)

A police siren contains at least two frequencies, producing the wavering sound (beats). Explain how the siren sound changes as a police car approaches, passes, and moves away from a pedestrian.

Two identical half-open pipes each have a fundamental frequency of \(500 .\) Hz. What percentage change in the length of one of the pipes will cause a beat frequency of \(10.0 \mathrm{~Hz}\) when they are sounded simultaneously?

Two sources, \(A\) and \(B\), emit a sound of a certain wavelength. The sound emitted from both sources is detected at a point away from the sources. The sound from source \(\mathrm{A}\) is a distance \(d\) from the observation point, whereas the sound from source \(\mathrm{B}\) has to travel a distance of \(3 \lambda .\) What is the largest value of the wavelength, in terms of \(d\), for the maximum sound intensity to be detected at the observation point? If \(d=10.0 \mathrm{~m}\) and the speed of sound is \(340 \mathrm{~m} / \mathrm{s}\), what is the frequency of the emitted sound?
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