Problem 54 - College Physics

Question

\(\bullet\bullet\) 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?

Step-by-Step Solution

Answer

The wavelength limits are from 0.01725 m to 17.25 m.

Step 1: 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 \(20,000 \, \text{Hz} \).

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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°C or 77°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.