Problem 9 - Physics

Question

Speakers A and B are vibrating in phase. They are directly facing each other, are \(7.80 \mathrm{~m}\) apart, and are each playing a \(73.0\) - \(\mathrm{Hz}\) tone. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker \(\mathrm{A}\) ?

Step-by-Step Solution

Answer

The points are at distances 0.65 m, 3.90 m, and 7.15 m from speaker A.

Step 1: Calculate Wavelength

First, let's calculate the wavelength of the sound wave using the formula: \( \lambda = \frac{v}{f} \), where \( v = 343 \, \text{m/s}\) is the speed of sound and \( f = 73.0 \, \text{Hz} \) is the frequency of the sound. Therefore, the wavelength \( \lambda = \frac{343}{73} \approx 4.70 \, \text{m} \).

Step 2: Identify Conditions for Constructive Interference

For constructive interference to occur, the path difference between the waves from Speakers A and B must be an integer multiple of the wavelength \( \lambda \). The condition can be expressed as \( d_1 + d_2 = n\lambda \), where \( n \) is an integer and \( d_1 \) and \( d_2 \) are distances from the point of interest to Speakers A and B, respectively.

Step 3: Express Distances in Terms of One Variable

Since the total distance between the speakers is \( D = 7.80 \, \text{m} \), we have \( d_1 + d_2 = 7.80 \). Solving for \( d_2 \) in terms of \( d_1 \), we find \( d_2 = 7.80 - d_1 \). Substituting this into the constructive interference condition gives the equation: \( d_1 + (7.80 - d_1) = n\lambda \) or simply \( 7.80 = n\lambda \).

Step 4: Calculate Positions of Constructive Interference

Solve for \( d_1 \) using the condition \( 7.80 = n\lambda \), which can be rewritten as \( n = \frac{7.80}{\lambda} \) to find all integer values of \( n \). Substitute each value of \( n \) back into \( d_1 = n \frac{\lambda}{2} \):1. For \( n = 1 \), \( d_1 = \frac{4.70}{2} \times 1 = 2.35 \, \text{m} \)2. For \( n = 3 \), \( d_1 = \frac{4.70}{2} \times 3 = 7.05 \, \text{m} \)3. For \( n = 5 \), \( d_1 = \frac{4.70}{2} \times 5 = 11.75 \, \text{m} \)The third point is out of range, so for the range \( 0 \leq d_1 \leq 7.80 \), the positions are \( 2.35 \, \text{m}, \) and \( 7.05 \, \text{m} \). Only the center between the two as a third point \( 3.90 \, \text{m} \) is applicable.

Key Concepts

Sound Waves

Sound waves are a type of mechanical wave that travel through a medium such as air, water, or solids. They are caused by the vibration of particles in the medium, which leads to the propagation of energy through the medium. When sound waves travel, they create areas of compression and rarefaction, causing the pressure in the air or other medium to fluctuate. This fluctuation is what our ears interpret as sound.
Because sound waves are mechanical waves, they need a medium through which to travel, unlike electromagnetic waves which can travel through a vacuum.

• Sound waves are longitudinal waves, meaning that the particle displacement is parallel to the direction of wave propagation.
• They can vary in frequency and amplitude, which influence the pitch and volume of the sound, respectively.
• The frequency of a sound wave is measured in hertz (Hz), and it represents the number of wave cycles that pass a point in one second.

Wavelength Calculation

Calculating the wavelength of sound is crucial for understanding how sound waves behave. The wavelength, often represented by the Greek letter lambda ( \( \lambda \) ), is the distance over which the wave's shape repeats.
We calculate wavelength using the formula: \( \lambda = \frac{v}{f} \) , where \(v\) is the speed of sound in the medium and \(f\) is the frequency of the wave. Wavelength provides a measure of how "long" each wave cycle is, and it is typically measured in meters.

• The speed of sound varies with the medium and environmental conditions (such as temperature and pressure), and is approximately 343 m/s in air at room temperature.
• In the given problem: the sound frequency is 73 Hz, and we use the formula to find: \( \lambda = \frac{343}{73} \approx 4.70 \, \text{m} \).
• This means each cycle of the sound wave from the speakers spans approximately 4.70 meters.

Speed of Sound

The speed of sound refers to how fast a sound wave travels through a medium. It is influenced by the medium's properties, such as density and elasticity. In particular, sound travels faster in solids than in liquids, and faster in liquids than in gases, because particles in solids are more tightly packed compared to those in liquids and gases.
In the given exercise, the speed of sound in air is assumed to be 343 m/s.

• This speed can change based on environmental conditions. For instance, warmer temperatures increase the speed of sound because the air particles move more quickly and transmit the waves more rapidly.
• The formula used to calculate the wavelength, which involves the speed of sound, helps us understand the relationship between frequency and wavelength: as the speed of sound increases or decreases, the wavelength changes respectively if the frequency of the sound remains the same.
• Knowing the speed of sound is essential in real-world applications like sonar technology, musical acoustics, and even meteorology.

Understanding these aspects of sound enables deeper insights into wave behavior, interference patterns, and sound propagation in various environments.