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Chapter 16: Problem 43

Two identical loudspeakers \(2.0 \mathrm{m}\) apart are emitting \(1800 \mathrm{Hz}\) sound waves into a room where the speed of sound is \(340 \mathrm{m} / \mathrm{s} .\) Is the point \(4.0 \mathrm{m}\) directly in front of one of the speakers, perpendicular to the line joining the speakers, a point of maximum constructive interference, perfect destructive interference, or something in between?

Short Answer

Expert verified
After performing the above steps, compare the path difference with whole and half multiples of the wavelength. Depending on the comparison, the point is a location of either constructive interference, destructive interference, or something in between.

Step by step solution

01

Identify the given values

We have the following given values - Distance between the speakers, D = 2.0 m; Frequency of the sound waves, f = 1800 Hz; Speed of sound, v = 340 m/s; Distance of the point from one of the speakers, R = 4.0 m.
02

Calculate the wavelength

We know that the wavelength of a wave can be calculated by the formula, \(\lambda = v / f\). So, let's calculate the wavelength (位).
03

Find the path difference

Next, we calculate the path difference. The path difference is the difference in distances of the point from each speaker. If we let the speaker to the left be speaker 1 and to the right be speaker 2, then the path difference is \(\Delta l = \sqrt{R^{2} + (\frac{D}{2})^{2}} - \sqrt{R^{2} + (\frac{D}{2})^{2}}\). Calculate this path difference.
04

Find the type of interference

If the path difference is a multiple of the wavelength (\(\Delta l = n \lambda\)), there will be constructive interference. If the path difference is an odd multiple of half of the wavelength (\(\Delta l = (n + \frac{1}{2}) \lambda\)), there will be destructive interference. Otherwise, it will be something in between.

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Key Concepts

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

Constructive Interference
Constructive interference occurs when two sound waves meet and combine to create a wave with a larger amplitude. This happens when the waves are in phase, meaning their peaks and troughs align perfectly.

To have constructive interference, the path difference between two sound waves must be an integer multiple of the wave's wavelength. For instance, if the path difference is zero, or one full wavelength, the waves reinforce each other. This results in a louder sound. With constructive interference, listeners may notice enhanced sound strength at certain points, like the maximum volume of a music note.
  • Two waves must have the same frequency.
  • Condition: Path difference = 0, 位, 2位, 3位, ... (integer multiples of 位)
Destructive Interference
Destructive interference happens when two waves meet and cancel each other out, resulting in a reduced or completely nullified wave amplitude. This occurs when the waves are out of phase, where the crest of one wave aligns with the trough of another.

For perfect destructive interference, the path difference should be an odd multiple of half the wavelength. This means the waves are doing the opposite actions at the same time, like pushing and pulling against each other simultaneously. This results in a significantly softer sound or complete silence.
  • Condition: Path difference = 位/2, 3位/2, 5位/2, ... (odd multiples of 位/2)
  • Waves cancel each other's energy.
Path Difference
Path difference is a critical factor in sound wave interference. It refers to the difference in distance each sound wave travels before reaching a specific point. Path difference is crucial in determining whether the interference will be constructive or destructive.

In the example of the two speakers, calculating the exact path difference between waves from Speaker 1 and Speaker 2 helps us predict the interference at a certain point. It is the basis for understanding how sound waves interact in space.
  • Path Difference = Distance difference between point from two sources.
  • Formula for calculation often involves geometry principles.
Wave Frequency
Wave frequency is the number of complete wave cycles passing a point in one second. Measured in hertz (Hz), it is fundamental in determining wave properties like wavelength and energy.

In this exercise, the frequency provided is 1800 Hz, meaning each wave cycle completes 1800 times per second. Knowing the frequency helps in calculating the wavelength, which further aids in understanding interference patterns.
  • Higher frequency: short wavelength, higher energy.
  • Lower frequency: long wavelength, lower energy.
Wavelength Calculation
Wavelength is the physical distance between corresponding points of consecutive cycles in a wave, such as crest to crest or trough to trough.

Calculating the wavelength involves using the formula \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound and \( f \) is the frequency.

For this exercise, plugging in the values gives:\[ \lambda = \frac{340 \, \text{m/s}}{1800 \, \text{Hz}} \approx 0.189 \text{ m} \].
  • Essential for understanding wave behavior and interference.
  • Helps predict interference patterns based on set distances.

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Most popular questions from this chapter

Musicians can use beats to tune their instruments. One flute is properly tuned and plays the musical note A at exactly \(440 \mathrm{Hz}\). A second player sounds the same note and hears that her instrument is slightly "flat" (that is, at too low a frequency). Playing at the same time as the first flute, she hears two loud-soft-loud beats per second. What is the frequency of her instrument?

The fundamental frequency of a standing wave on a \(1.0-\mathrm{m}-\) long string is \(440 \mathrm{Hz} .\) What would be the wave speed of a pulse moving along this string?

When a sound wave travels directly toward a hard wall, the incoming and reflected waves can combine to produce a standing wave. There is an antinode right at the wall, just as at the end of a closed tube, so the sound near the wall is loud. You are standing beside a brick wall listening to a \(50 \mathrm{Hz}\) tone from a distant loudspeaker. How far from the wall must you move to find the first quiet spot? Assume a sound speed of \(340 \mathrm{m} / \mathrm{s}\).

A student waiting at a stoplight notices that her turn signal, which has a period of \(0.85 \mathrm{s},\) makes one blink exactly in sync with the turn signal of the car in front of her. The blinker of the car ahead then starts to get ahead, but 17 s later the two are exactly in sync again. What is the period of the blinker of the other car?

A \(40-\mathrm{cm}\) -long tube has a \(40-\mathrm{cm}-\) long insert that can be pulled in and out, as shown in Figure \(\mathrm{P} 16.61 .\) A vibrating tuning fork is held next to the tube. As the insert is slowly pulled out, the sound from the tuning fork creates standing waves in the tube when the total length \(L\) is \(42.5 \mathrm{cm}, 56.7 \mathrm{cm},\) and \(70.9 \mathrm{cm} .\) What is the frequency of the tuning fork? The air temperature is \(20^{\circ} \mathrm{C}\).
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