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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°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°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°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

(a) What is the pressure amplitude of a sound wave with an intensity level of \(120.0 \mathrm{dB}\) in air? (b) What force does this exert on an eardrum of area \(0.550 \times 10^{-4} \mathrm{m}^{2} ?\)
A piano tuner sounds two strings simultancously. One has been previously tuned to vibrate at \(293.0 \mathrm{Hz}\). The tuner hears 3.0 beats per second. The tuner increases the tension on the as-yet untuned string, and now when they are played together the beat frequency is \(1.0 \mathrm{s}^{-1} .\) (a) What was the original frequency of the untuned string? (b) By what percentage did the tuner increase the tension on that string?
Your friend needs advice on her newest "acoustic sculpture." She attaches one end of a steel wire, of diameter \(4.00 \mathrm{mm}\) and density $7860 \mathrm{kg} / \mathrm{m}^{3},$ to a wall. After passing over a pulley, located \(1.00 \mathrm{m}\) from the wall, the other end of the wire is attached to a hanging weight. Below the horizontal length of wire she places a 1.50 -m-long hollow tube, open at one end and closed at the other. Once the sculpture is in place, air will blow through the tube, creating a sound. Your friend wants this sound to cause the steel wire to oscillate at the same resonant frequency as the tube. What weight (in newtons) should she hang from the wire if the temperature is \(18.0^{\circ} \mathrm{C} ?\)
A sound wave with an intensity level of \(80.0 \mathrm{dB}\) is incident on an eardrum of area \(0.600 \times 10^{-4} \mathrm{m}^{2} .\) How much energy is absorbed by the eardrum in 3.0 min?
When playing fortissimo (very loudly), a trumpet emits sound energy at a rate of \(0.800 \mathrm{W}\) out of a bell (opening) of diameter \(12.7 \mathrm{cm} .\) (a) What is the sound intensity level right in front of the trumpet? (b) If the trumpet radiates sound waves uniformly in all directions, what is the sound intensity level at a distance of \(10.0 \mathrm{m} ?\)
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