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Chapter 13: Problem 54

\(\bullet\bullet\) The range of sound frequencies audible to the human ear extends from about \(20 \mathrm{~Hz}\) to \(20 \mathrm{kHz}\). If the speed of sound in air is \(345 \mathrm{~m} / \mathrm{s},\) what are the wavelength limits of this audible range?

Short Answer

Expert verified
The wavelength limits are from 0.01725 m to 17.25 m.

Step by step solution

01

Understand the Relationship Between Frequency and Wavelength

The wavelength \( \lambda \) of a sound wave is related to its frequency \( f \) and the speed of sound \( v \) by the formula: \[ \lambda = \frac{v}{f} \]. In this problem, we know the speed of sound \( v = 345 \, \text{m/s} \) and we need to find the wavelength for the limits of the audible frequency range, which is from \(20 \, \text{Hz}\) to \(20,000 \, \text{Hz} \).
02

Calculate Wavelength for the Lowest Frequency

Using the formula \( \lambda = \frac{v}{f} \), substitute the values for the lowest audible frequency (\( f = 20 \, \text{Hz} \)): \[ \lambda_{\text{low}} = \frac{345 \, \text{m/s}}{20 \, \text{Hz}} = 17.25 \, \text{m} \]. This is the wavelength corresponding to the lowest frequency in the audible range.
03

Calculate Wavelength for the Highest Frequency

Using the same formula \( \lambda = \frac{v}{f} \), substitute the values for the highest audible frequency (\( f = 20,000 \, \text{Hz} \)): \[ \lambda_{\text{high}} = \frac{345 \, \text{m/s}}{20,000 \, \text{Hz}} = 0.01725 \, \text{m} \]. This is the wavelength corresponding to the highest frequency in the audible range.
04

Determine the Range of Wavelengths

Combine the results from Steps 2 and 3: the range of wavelengths audible to the human ear is from \( 0.01725 \, \text{m} \) to \( 17.25 \, \text{m} \). This specifies the limits in terms of wavelength for the given speed of sound in air.

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Key Concepts

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

Audible Frequency Range
Sound waves encompass a wide spectrum of frequencies, but the human ear is adept at hearing only a select band. This band is known as the **audible frequency range**. Most humans can detect sounds from approximately 20 Hz to 20,000 Hz, commonly denoted as 20 kHz. Frequencies below 20 Hz are called infrasound, while those above 20 kHz fall into the ultrasound category.
  • **20 Hz** refers to deep bass sounds, like a drum beat.
  • **20 kHz** refers to high-pitched sounds, comparable to a dog whistle frequency often beyond human hearing.
Understanding this range is crucial because it determines what wavelengths are perceptible.
The span from low to high audible frequencies impacts how sound waves interact with environments, such as air, which our ears adapt to for communication.
Wavelength Calculation
The relationship between frequency and wavelength is pivotal in understanding sound waves. Wavelength refers to the distance over which the wave's shape repeats, a fundamental property in physics.
Using the formula \( \lambda = \frac{v}{f} \), we can calculate the wavelength \( \lambda \) if we know the speed of sound \( v \) and frequency \( f \).
  • The **lowest frequency** (20 Hz) results in a longer wavelength. With a speed of sound of 345 m/s, this calculates to \( 17.25 \) meters. Such long waves can seem less directional.
  • The **highest frequency** (20,000 Hz) results in a shorter wavelength of \( 0.01725 \) meters. These shorter waves can have more direct paths.
By understanding wavelength calculations, you can better grasp how different sound frequencies behave and travel through mediums like air.
Speed of Sound
The **speed of sound** is a critical component in determining how sound waves propagate in various environments. In our case, the speed of sound in air is specified as 345 m/s. This value can change depending on factors like temperature, medium, and density of the environment.
  • **In air**, sound travels at around 345 m/s at room temperature (about 25掳C or 77掳F), which simplifies many calculations for everyday problems.
  • **In water**, sound is much faster due to water's density, traveling around 1,480 m/s.
Understanding the speed of sound helps us calculate other related properties such as wavelength and frequency. It also influences acoustic applications like musical instruments, communication systems, and research into environmental sciences. Knowing how fast sound moves allows scientists and engineers to design structures and tools that need precise sound wave management.

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Most popular questions from this chapter

\(\bullet\bullet\) Assume that \(P\) and \(S\) (primary and secondary) waves from an earthquake with a focus near the Earth's surface travel through the Earth at nearly constant but different average speeds. A monitoring station that is \(1000 \mathrm{~km}\) from the epicenter detected the S wave to arrive at \(42 \mathrm{~s}\) after the arrival of the P wave. If the P wave has an average speed of \(8.0 \mathrm{~km} / \mathrm{s},\) what is the average speed of the \(\mathrm{S}\) wave?

Spring \(\mathrm{A}(50.0 \mathrm{~N} / \mathrm{m})\) is attached to the ceiling. The top of spring \(\mathrm{B}(30.0 \mathrm{~N} / \mathrm{m})\) is hooked onto the bottom of spring A. Then a 0.250-kg mass is then attached to the bottom of Spring B. (a) How far will the object fall until it reaches equilibrium? (b) What is the period of the resulting oscillation?

A piece of steel string is under tension. (a) If the tension doubles, the transverse wave speed (1) doubles, (2) halves, (3) increases by \(\sqrt{2},\) (4) decreases by \(\sqrt{2}\). Why? (b) If the linear mass density of a 10.0 -m length of string is \(0.125 \mathrm{~kg} / \mathrm{m}\) and it is under a tension of \(9.00 \mathrm{~N}\), what is the transverse wave speed in the string? (c) What are its waves' natural frequencies?

\(\bullet\) An object of mass \(0.50 \mathrm{~kg}\) is attached to a spring with spring constant \(10 \mathrm{~N} / \mathrm{m}\). If the object is pulled down \(0.050 \mathrm{~m}\) from the equilibrium position and released, what is its maximum speed?

A standing wave is formed in a stretched string that is \(3.0 \mathrm{~m}\) long. What are the wavelengths of (a) the first harmonic and (b) the second harmonic?
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