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Chapter 12: Problem 59

At what frequency \(f\) does a sound wave in air have a wavelength of $15 \mathrm{cm},$ about half the diameter of the human head? Some methods of localization work well only for frequencies below \(f\), while others work well only above \(f\). (See Conceptual Questions 4 and 5 .)

Short Answer

Expert verified
Answer: The frequency of the sound wave is approximately 2287 Hz.

Step by step solution

01

Convert wavelength to meters

Since the given wavelength is 15 cm, we need to convert it to meters: \(\lambda = 15 \mathrm{cm} = 0.15 \,\mathrm{m}\). This will allow us to use the wavelength in the formula with the correct units.
02

Find the velocity of sound in air

The velocity of sound in air, at room temperature (20掳C), is approximately 343 m/s. Keep in mind that the velocity of sound depends on the temperature.
03

Use the formula to find the frequency

We know the velocity of sound in air, \(v=343 \, \mathrm{m/s}\), and the wavelength, \(\lambda = 0.15 \, \mathrm{m}\). We will plug in these values into the formula \(v = f\lambda\) and solve for \(f\): $$ f = \frac{v}{\lambda} = \frac{343 \, \mathrm{m/s}}{0.15 \, \mathrm{m}} \approx 2287 \, \mathrm{Hz} $$
04

Interpret the result

The sound wave in air has a frequency of approximately 2287 Hz when its wavelength is 15 cm (0.15 m). Some methods of localization work well only for frequencies below 2287 Hz, while others work well only above 2287 Hz.

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Most popular questions from this chapter

(a) What should be the length of an organ pipe, closed at one end, if the fundamental frequency is to be \(261.5 \mathrm{Hz} ?\) (b) What is the fundamental frequency of the organ pipe of part (a) if the temperature drops to \(0.0^{\circ} \mathrm{C} ?\)
A bat emits chirping sounds of frequency \(82.0 \mathrm{kHz}\) while hunting for moths to eat. If the bat is flying toward the moth at a speed of $4.40 \mathrm{m} / \mathrm{s}\( and the moth is flying away from the bat at \)1.20 \mathrm{m} / \mathrm{s},$ what is the frequency of the sound wave reflected from the moth as observed by the bat? Assume it is a cool night with a temperature of \(10.0^{\circ} \mathrm{C} .\) [Hint: There are two Doppler shifts. Think of the moth as a receiver, which then becomes a source as it "retransmits" the reflected wave. \(]\)
A violin is tuned by adjusting the tension in the strings. Brian's A string is tuned to a slightly lower frequency than Jennifer's, which is correctly tuncd to \(440.0 \mathrm{Hz}\) (a) What is the frequency of Brian's string if beats of \(2.0 \mathrm{Hz}\) are heard when the two bow the strings together? (b) Does Brian need to tighten or loosen his A string to get in tune with Jennifer? Explain.
In this problem, you will estimate the smallest kinetic energy of vibration that the human ear can detect. Suppose that a harmonic sound wave at the threshold of hearing $\left(I=1.0 \times 10^{-12} \mathrm{W} / \mathrm{m}^{2}\right)\( is incident on the eardrum. The speed of sound is \)340 \mathrm{m} / \mathrm{s}\( and the density of air is \)1.3 \mathrm{kg} / \mathrm{m}^{3} .$ (a) What is the maximum speed of an element of air in the sound wave? [Hint: See Eq. \((10-21) .]\) (b) Assume the eardrum vibrates with displacement \(s_{0}\) at angular frequency \(\omega ;\) its maximum speed is then equal to the maximum speed of an air element. The mass of the eardrum is approximately \(0.1 \mathrm{g} .\) What is the average kinetic energy of the eardrum? (c) The average kinetic energy of the eardrum due to collisions with air molecules in the absence of a sound wave is about \(10^{-20} \mathrm{J}\) Compare your answer with (b) and discuss.
A 30.0 -cm-long string has a mass of \(0.230 \mathrm{g}\) and is vibrating at its next-to-lowest natural frequency \(f_{2} .\) The tension in the string is \(7.00 \mathrm{N} .\) (a) What is \(f_{2} ?\) (b) What are the frequency and wavelength of the sound in the surrounding air if the speed of sound is $350 \mathrm{m} / \mathrm{s} ?$
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