Problem 14 - Physics

Question

A row of seats is parallel to a stage at a distance of \(8.7 \mathrm{~m}\) from it. At the center and front of the stage is a diffraction horn loudspeaker. This speaker sends out its sound through an opening that is like a small doorway with a width \(D\) of \(7.5 \mathrm{~cm} .\) The speaker is playing a tone that has a frequency of \(1.0 \times 10^{4} \mathrm{~Hz}\). The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the distance between two seats, located near the center of the row, at which the tone cannot be heard?

Step-by-Step Solution

Answer

The seats are 8.9 m apart where the tone cannot be heard.

Step 1: Identify the Relevant Formula

To find the angle at which the tone cannot be heard due to destructive interference, we need to use the formula for the angular position of the first minimum in single slit diffraction: \( \sin \theta = \frac{\lambda}{D} \), where \( \lambda \) is the wavelength and \( D \) is the width of the opening.

Step 2: Calculate the Wavelength

The wavelength is calculated using the formula: \( \lambda = \frac{v}{f} \), where \( v = 343 \text{ m/s} \) is the speed of sound and \( f = 1.0 \times 10^4 \text{ Hz} \) is the frequency. Substituting these values gives us \( \lambda = \frac{343}{1.0 \times 10^4} = 0.0343 \text{ m} \).

Step 3: Calculate the Angle of the First Minimum

Substitute the values into the formula for the angular position of the first minimum: \( \sin \theta = \frac{0.0343 \text{ m}}{7.5 \times 10^{-2} \text{ m}} = 0.4573 \). Solving for \( \theta \) gives us \( \theta = \sin^{-1}(0.4573) \approx 27.238^\circ \).

Step 4: Use Geometry to Find the Distance

The problem states that the row of seats is \(8.7\text{ m} \) from the stage. We can use the angle \( \theta \) we found and apply the tan function: \( \tan \theta = \frac{x}{8.7} \), where \( x \) is the distance from the center line where the tone cannot be heard. We find \( x = 8.7 \cdot \tan(27.238^\circ) = 4.450 \text{ m} \).

Step 5: Calculate the Total Distance Between the Seats

Since there are two positions, one on each side of the center line, where the sound cannot be heard, the total distance between the two seats is \( 2x = 2 \times 4.450 \text{ m} = 8.9 \text{ m} \).

Key Concepts

Destructive Interference

Destructive interference occurs when waves combine to produce a smaller amplitude. In the context of sound waves, this happens when two waves meet out of phase, effectively cancelling each other out. This is why there are certain spots where sound may not be heard.

In the setup of a single-slit diffraction through a loudspeaker, sound waves spread out as they pass through the slit. Some waves travel different distances to meet at a point. When they meet out of phase (crest of one wave aligns with the trough of another), they cancel, creating points of silence known as nodes.

These nodes are crucial to understanding the arrangement of seats in environments where loudspeakers are used, as they indicate areas of destructive interference where the sound will not be perceived.

Sound Wavelength

The wavelength of a sound wave is the distance over which the wave's shape repeats. It is key to understanding how sound propagates and where it can interfere destructively.

The wavelength \( \lambda \) can be calculated using the formula \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound and \( f \) is the frequency of the sound. In our exercise, the speed of sound is 343 m/s, and the frequency is \(1.0 \times 10^{4} \text{ Hz}\).

By substituting these values into the formula, we find that the wavelength for this setup is 0.0343 meters.

Understanding the wavelength helps predict where destructive interference occurs, as it plays a direct role in calculations involving diffraction patterns.

Single Slit Diffraction

Single slit diffraction causes waves to spread after passing through a narrow opening. For sound waves, like those from a loudspeaker, this principle is crucial for analysing how sound disperses in a room.

In the case of our exercise, the slit width is represented by the loudspeaker's opening, which is 7.5 cm wide. When sound travels through it, it diffracts or bends, leading to zones of varying sound intensity.

The angular position of these zones, especially where no sound is heard due to destructive interference, is predicted using the equation \( \sin \theta = \frac{\lambda}{D} \). This relationship helps to identify where these quiet zones appear relative to the path of the sound waves.

Comprehending single slit diffraction enables us to visualize how sound navigates around barriers, influencing room acoustics and speaker placement.

Angular Position of Minimum

The angular position of minimum is the angle at which minimum sound intensity occurs due to destructive interference. It's where the first quiet zone appears from the slit due to the path difference of waves equating to half a wavelength or its multiples.

This angle \( \theta \) can be found using \( \sin \theta = \frac{\lambda}{D} \), presenting where a minimum in sound intensity - or a node - is heard relative to the speaker.

In our example, substituting the calculated wavelength and slit width, we find \( \sin \theta \approx 0.4573 \). Solving for \( \theta \) provides an angle of approximately 27.238 degrees.

This angle helps determine the seating arrangement, identifying spots where the sound will be minimal due to diffraction, hence why certain distances from a sound source might experience reduced or no sound.