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Chapter 16: Problem 54

Two loudspeakers directly face each other \(30 \mathrm{m}\) apart, with the left speaker positioned at \(x=0 \mathrm{m}\). The pressure of the sound wave emitted by the left speaker is described by the equation \(\Delta p_{\mathrm{L}}=p_{0} \cos (1.90 x-630 t),\) while that from the right speaker is given by \(\Delta p_{\mathrm{R}}=p_{0} \cos (1.90 x+630 t),\) where \(x\) is measured in \(\mathrm{m}\) and \(t\) in \(\mathrm{s}\). What is the point nearest to the left speaker at which there is a node in the sound wave?

Short Answer

Expert verified
The nearest node to the left speaker is approximately at 0.83 meters from the speaker.

Step by step solution

01

Identify that you are looking for a Node

Recognize that a Node is the point where destructive interference occurs between two waves. This is the point where the amplitude of the resultant wave is zero. Be aware that this occurs when two waves have the same frequency and amplitude, but are out of phase with each other
02

Set the Wave Equations Equal to Each Other

We can set the two wave equations equal to each other combined with the condition for destructive interference since the amplitude part cancels out and left with the arguments (portion inside the cosine function) equal to each other. So the equation \(1.90x - 630t = 1.90x + 630t + \pi\) is obtained. The additional \(\pi\) in the second part is due to the phase difference for destructive interference. Note: Here the actual time \(t\) is irrelevant because we are not concerned with when the node appears, but where it appears.
03

Solve for Position x

Simplifying the equation yields \( 2 * 1.90x = \pi \). Therefore, \( x = \frac{\pi}{2 * 1.90} \) which equals approximately 0.83 meters.

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Key Concepts

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

Destructive Interference
When two sound waves meet, they can interfere with each other; interference can be constructive or destructive. Destructive interference happens when these waves are out of phase. This means their peaks and troughs don't match up, cancelling each other out and leading to a drop, or even complete absence, in amplitude.

In our exercise, the waves from the two speakers have the same frequency and amplitude but are out of phase, causing destructive interference. This occurs because one wave's peak coincides with the other's trough, effectively causing a zero amplitude at specific points.

Understanding how these waves interfere destructively helps us know where the nodes, or points of no sound, appear along the line between the speakers.
Wave Nodes
A node is a point along a medium, like air, where there is minimal to no movement because of destructive interference. In the context of sound, a node is a location where the sound is effectively silent due to the cancelling effect of the waves.

The nodes are the result of the superposition principle. The waves鈥� interaction leads to certain points where they extinguish each other. These occur consistently along the medium at points of destructive interference.
  • The exercise outlines that to find a node, we look for where the waveform equations align under the condition of phase opposition.
  • This process helps identify locations where sound output is effectively muted.
By mastering the concept of nodes, you'll better understand the distribution of sound in a space.
Sound Wave Equations
Wave equations describe the behavior and properties of waves mathematically. In our exercise, both speakers generate waves following specific equations: - The left speaker: \( \Delta p_{\mathrm{L}}=p_{0} \cos (1.90 x-630 t) \)- The right speaker: \( \Delta p_{\mathrm{R}}=p_{0} \cos (1.90 x+630 t) \)

These equations include:
  • Amplitude \(p_0\), which determines maximum displacement.
  • Frequency indicated by the constants in the cosine, governing how often the wave cycles.
  • Phase, which ultimately influences where interference occurs.
By analyzing these equations, you can determine where nodes form. Setting the arguments (the terms inside the cosine) equal 鈥� considering phase opposition 鈥� helps find the point of destructive interference and, thus, node formation. This approach ensures a deeper comprehension of the sound wave interactions and locations of silent points.

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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?

When you voice the vowel sound in "hat," you narrow the opening where your throat opens into the cavity of your mouth so that your vocal tract appears as two connected tubes. The first is in your throat, closed at the vocal cords and open at the back of the mouth. The second is the mouth itself, open at the lips and closed at the back of the mouth-a different condition than for the throat because of the relatively larger size of the cavity. The corresponding formant frequencies are \(800 \mathrm{Hz}\) (for the throat) and \(1500 \mathrm{Hz}\) (for the mouth). What are the lengths of these two cavities? Assume a sound speed of \(350 \mathrm{m} / \mathrm{s}\).

Two loudspeakers \(42.0 \mathrm{m}\) apart and facing each other emit identical 115 Hz sinusoidal sound waves in a room where the sound speed is \(345 \mathrm{m} / \mathrm{s} .\) Susan is walking along a line between the speakers. As she walks, she finds herself moving through loud and quiet spots. If Susan stands \(19.5 \mathrm{m}\) from one speaker, is she standing at a quiet spot or a loud spot?

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 \(20^{\circ} \mathrm{C}\).

The width of a particular microwave oven is exactly right to support a standing-wave mode. Measurements of the temperature across the oven show that there are cold spots at each edge of the oven and at three spots in between. The wavelength of the microwaves is \(12 \mathrm{cm} .\) How wide is the oven?
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