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Chapter 16: Problem 5

Find the minimum and maximum wavelengths of sound in water that is in the audible range \((20-20000 \mathrm{~Hz})\) for an average human ear. Speed of sound in water \(=1450 \mathrm{~m} \mathrm{~s}^{-1}\).

Short Answer

Expert verified
The minimum wavelength is 0.0725 m and the maximum is 72.5 m.

Step by step solution

01

Understand the Relationship Between Speed, Frequency, and Wavelength

Sound travels in waves and the speed of sound in a medium, the frequency of sound, and its wavelength are related by the equation: \( v = f \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. In this problem, \( v = 1450 \mathrm{~m} \mathrm{~s}^{-1} \).
02

Determine the Wavelength for the Minimum Frequency

To find the wavelength for the minimum frequency, rearrange the formula to \( \lambda = \frac{v}{f} \). Here, the minimum audible frequency \( f = 20 \mathrm{~Hz} \). Substitute these values: \( \lambda_{\text{min}} = \frac{1450}{20} = 72.5 \mathrm{~m} \).
03

Determine the Wavelength for the Maximum Frequency

Using the same formula, find the wavelength for the maximum frequency. Here, the maximum audible frequency \( f = 20000 \mathrm{~Hz} \). Substitute these values: \( \lambda_{\text{max}} = \frac{1450}{20000} = 0.0725 \mathrm{~m} \).
04

Conclusion

The minimum wavelength of sound in water for the audible range is \( 72.5 \mathrm{~m} \) and the maximum wavelength is \( 0.0725 \mathrm{~m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Sound
The speed of sound indicates how quickly a sound wave moves through a certain medium. This speed can change depending on various factors like the medium's density and temperature. In our specific context, we are considering sound traveling through water, where the speed is given as \(1450 \, \mathrm{m/s}\). Compared to air, sound moves faster in water. This is due to water's higher density, which allows for quicker transmission of sound vibrations.
Sound speed is crucial for understanding how sound behaves in different environments, as it directly affects the wavelength when the frequency is fixed. Knowing the speed of sound helps us predict how sound travels in oceans or how marine creatures communicate over vast distances.
Audible Frequency Range
The audible frequency range determines which frequencies can be detected by the average human ear. Typically, this range lies between \(20 \, \text{Hz}\) and \(20000 \, \text{Hz}\). Frequencies below or above this range may not be heard by humans but can be detected by some animals.
Within this range, lower frequencies produce deeper sounds, while higher frequencies lead to higher-pitched sounds. This is the reason a bass drum sounds distinctly different from a high-pitched whistle. Understanding the audible frequency range assists in designing audio equipment and ensuring that sounds fall within a perceptible spectrum for humans.
Wavelength Calculation
Calculating the wavelength of sound involves determining how long one complete cycle of a sound wave is, given its speed and frequency. The formula \(v = f \lambda\) links these three elements: where \(v\) represents speed of sound, \(f\) denotes frequency, and \(\lambda\) is wavelength.
To find the wavelength, you rearrange the formula to \(\lambda = \frac{v}{f}\). For instance, using a frequency of \(20 \, \text{Hz}\), you find the minimum wavelength in water to be \(72.5 \, \text{m}\), and for \(20000 \, \text{Hz}\), the maximum wavelength becomes \(0.0725 \, \text{m}\). This calculation illustrates how changes in frequency influence wavelength, with higher frequencies leading to shorter wavelengths. Such insights are vital in applications like underwater acoustics where specific wavelengths are chosen based on environmental needs.

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Most popular questions from this chapter

An electronically driven loudspeaker is placed near the open end of a resonance column apparatus. The length of air column in the tube is \(80 \mathrm{~cm}\). The frequency of the loudspeaker can be varied between \(20 \mathrm{~Hz}\) and \(2 \mathrm{kHz}\). Find the frequencies at which the column will resonate. Speed of sound in air \(=320 \mathrm{~m} \mathrm{~s}^{-1}\).

Two electric trains run at the same speed of \(72 \mathrm{~km} / \mathrm{h}\) along the same track and in the same direction with a separation of \(2 \cdot 4 \mathrm{~km}\) between them. The two trains simultaneously sound brief whistles. A person is situated at a perpendicular distance of \(500 \mathrm{~m}\) from the track and is equidistant from the two trains at the instant of the whistling. If both the whistles were at \(500 \mathrm{~Hz}\) and the speed of sound in air is \(340 \mathrm{~m} / \mathrm{s}\), find the frequencies heard by the person.

An open organ pipe has a length of \(5 \mathrm{~cm}\). (a) Find the fundamental frequency of vibration of this pipe. (b) What is the highest harmonic of such a tube that is in the audible range ? Speed of sound in air is \(340 \mathrm{~m} \mathrm{~s}^{-1}\) and the audible range is \(20-20,000 \mathrm{~Hz}\).

A car moves with a speed of \(54 \mathrm{~km} \mathrm{~h}^{-1}\) towards a cliff. The horn of the car emits sound of frequency \(400 \mathrm{~Hz}\) at a speed of \(335 \mathrm{~m} \mathrm{~s}^{-1}\). (a) Find the wavelength of the sound emitted by the horn in front of the car. (b) Find the wavelength of the wave reflected from the cliff. (c) What frequency does a person sitting in the car hear for the reflected sound wave? (d) How many beats does he hear in 10 seconds between the sound coming directly from the horn and that coming after the reflection?

Two submarines are approaching each other in a calm sea. The first submarine travels at a speed of \(36 \mathrm{~km} \mathrm{~h}^{-1}\) and the other at \(54 \mathrm{~km} \mathrm{~h}^{-1}\) relative to the water. The first submarine sends a sound signal (sound waves in water are also called sonar \()\) at a frequency of \(2000 \mathrm{~Hz}\). (a) At what frequency is this signal received by the second submarine? (b) The signal is reflected from the second submarine. At what frequency is this signal received by the first submarine. Take the speed of the sound wave in water to be \(1500 \mathrm{~m} \mathrm{~s}^{-1}\).
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