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Chapter 12: Problem 1

Bats emit ultrasonic waves with a frequency as high as $1.0 \times 10^{5} \mathrm{Hz} .$ What is the wavelength of such a wave in air of temperature \(15^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
Answer: The wavelength of the ultrasonic wave is approximately \(3.4\,\text{mm}\).

Step by step solution

01

Calculate the speed of sound in air at the given temperature

To calculate the speed of sound in the air at 15掳C, we use the formula \(v = 331.4 + 0.6 \cdot T,\) where \(v\) is the speed of sound and \(T\) is the temperature in Celsius. Plugging in the given temperature, we have: \(v = 331.4 + 0.6 \cdot 15 = 331.4 + 9\) \(v = 340.4 \,\text{m/s}\) The speed of sound in air at 15掳C is 340.4 m/s.
02

Calculate the wavelength of the ultrasonic wave

The ultrasonic wave frequency \(f\) is given as \(1.0 \times 10^5 \,\text{Hz}\). To calculate the wavelength \(\lambda\), we can use the speed of sound \(v\) and the frequency \(f\) in the following formula: \(\lambda = \frac{v}{f}\) Plugging in the speed of sound and frequency, we have: \(\lambda = \frac{340.4}{1.0 \times 10^5} = 3.404 \times 10^{-3} \,\text{m}\) The wavelength of the ultrasonic wave in air at 15掳C is \(3.404 \times 10^{-3}\,\text{m}\) or approximately \(3.4\,\text{mm}\).

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Most popular questions from this chapter

The length of the auditory canal in humans averages about \(2.5 \mathrm{cm} .\) What are the lowest three standing wave frequencies for a pipe of this length open at one end? What effect might resonance have on the sensitivity of the ear at various frequencies? (Refer to Fig. 12.12 Note that frequencies critical to specch recognition are in the range 2 to \(5 \mathrm{kHz}\) )
A cello string has a fundamental frequency of \(65.40 \mathrm{Hz}\) What beat frequency is heard when this cello string is bowed at the same time as a violin string with frequency of \(196.0 \mathrm{Hz} ?\) [Hint: The beats occur between the third harmonic of the cello string and the fundamental of the violin. \(]\)
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A source of sound waves of frequency \(1.0 \mathrm{kHz}\) is traveling through the air at 0.50 times the speed of sound. (a) Find the frequency of the sound received by a stationary observer if the source moves toward her. (b) Repeat if the source moves away from her instead.
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