Problem 17 - Pearson Physics

Question

Two tuning forks have frequencies of \(278 \mathrm{~Hz}\) and \(292 \mathrm{~Hz}\). What is the beat frequency if both tuning forks are sounded simultaneously?

Step-by-Step Solution

Answer

The beat frequency is 14 Hz.

Step 1: Understanding Beat Frequency

The beat frequency is the result of the interference of two sound waves with different frequencies. It can be calculated using the formula \( f_{beat} = |f_1 - f_2| \), where \( f_1 \) and \( f_2 \) are the frequencies of the tuning forks.

Step 2: Apply the Formula

Let's identify the frequencies: \( f_1 = 278 \, \mathrm{Hz} \) and \( f_2 = 292 \, \mathrm{Hz} \). Substitute them into the formula: \( f_{beat} = |278 - 292| \).

Step 3: Calculate the Beat Frequency

Perform the absolute subtraction: \( f_{beat} = |278 - 292| = |-14| = 14 \, \mathrm{Hz} \).

Key Concepts

Interference of Sound Waves

When two sound waves meet, they interfere with each other. This interference can be either constructive or destructive, depending on how the waves align. Constructive interference occurs when the crest of one wave aligns with the crest of another, amplifying the sound.
Destructive interference happens when the crest of one wave aligns with the trough of another, diminishing the sound.

• Constructive Interference: When two sound waves are perfectly in phase, their amplitudes add up, resulting in a louder sound. This is because the maximum displacement of the waves coincides.
• Destructive Interference: In this case, when out-of-phase waves meet, they cancel each other out. As a result, the sound becomes quieter or completely silent.

The combination of these interferences results in a pulsating sound, known as a beat. The beat frequency, which is the focus of the exercise, is a direct consequence of these alternating interferences of sound waves.

Frequency Difference

The concept of frequency difference is crucial in understanding beats. It describes how much one sound frequency lacks or exceeds another. This difference results in beats, which are periodic fluctuations in loudness or amplitude heard when two sounds are played together.
To determine the beat frequency, we calculate the absolute difference between the two frequencies involved.

• The formula used is: \[ f_{beat} = |f_1 - f_2| \]
• Where:
  • \( f_1 \) is the frequency of the first sound wave (first tuning fork)
  • \( f_2 \) is the frequency of the second sound wave (second tuning fork)

In the context of the exercise, you identify the two frequencies, apply the formula, and calculate the resulting absolute value of their difference. This number gives you the beat frequency.

Tuning Forks

Tuning forks are devices used to produce a specific pitch and are commonly used in musical settings and scientific experiments. They consist of a handle and two tines that vibrate when struck.

• Each fork resonates at a specific frequency, determined by its size and material.
• Tuning forks are effective for demonstrating sound wave principles because they produce a clear, single frequency when vibrating.

In the exercise given, we are dealing with two tuning forks with frequencies of 278 Hz and 292 Hz.
These specific values emphasize how slight variations in frequency can lead to noticeable changes in sound perception, such as the creation of beats.

• Using these forks together creates beats, where listeners hear a fluctuating sound that results from the interference of their waves.

This demonstrates the beauty and practicality of understanding tuning forks in studying wave interference and beat frequencies.