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Chapter 18: Problem 14

Two sound waves, from two different sources with the same frequency, \(540 \mathrm{~Hz}\), travel in the same direction at \(330 \mathrm{~m} / \mathrm{s}\). The sources are in phase. What is the phase difference of the waves at a point that is \(4.40 \mathrm{~m}\) from one source and \(4.00 \mathrm{~m}\) from the other?

Short Answer

Expert verified
The phase difference between the sound waves is 4.11 radians.

Step by step solution

01

- Calculate the wavelength

Use the formula for the wavelength: \[ \lambda = \frac{v}{f} \] where \(v\) is the speed of sound in meters per second (\(330 \mathrm{~m} / \text{~s}\)), and \(f\) is the frequency (\(540 \mathrm{~Hz}\)). \text{ Let's calculate \(\lambda\) :} \[ \lambda = \frac{330 \text{~m} / \text{s}}{540 \text{~Hz}} = 0.6111 \text{m} \]
02

- Calculate the path difference

The path difference is given by: \[ \Delta d = \text{path from source 2} - \text{path from source 1} \] Given: \( \text{path from source 2} = 4.40 \text{~m} \ \text{path from \source 1} = 4.00 \text{~m} \) \Using the values: \[ \Delta d = 4.40 \text{~m} - 4.00 ~m = 0.40 ~m \]
03

- Calculate the Phase Difference

Use the formula to find \( \Delta \phi \) \[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta d \] Substituting the values from Steps 1 \and \2: \[ \ \text{ \Delta \phi} = \frac{2\pi}{0.6111\text{~m}} \cdot 0.40 ~m\= \4.11 \text{~radians} \]

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Key Concepts

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

Wavelength Calculation
To start, let's understand the concept of wavelength. Wavelength (\( \text{位} \)) is the distance between consecutive crests or troughs in a wave. For sound waves, it can be calculated using the formula: \[ 位 = \frac{v}{f} \] where: \(\text{v} \) is the speed of sound, and \( \text{f} \) is the frequency of the wave. \

For example, with a sound frequency of 540 Hz and speed of sound at 330 m/s: \[ 位 = \frac{330 \text{ m/s}}{540 \text{ Hz}} = 0.6111 \text{ meters} \] This means each wave cycle is 0.6111 meters long.
Path Difference
Path difference refers to the difference in distance traveled by two waves from their sources to a common point. It's crucial in determining how waves interact with each other, such as through constructive or destructive interference.

To calculate it, you can use the simple formula: \[ \text{鈭哾} = \text{path from source 2} - \text{path from source 1} \] In our example, where one path is 4.40 meters and the other is 4.00 meters: \[ 鈭哾 = 4.40 \text{ meters} - 4.00 \text{ meters} = 0.40 \text{ meters} \]

This tells us there's a 0.40 meter difference in how far each wave has traveled.
Phase Difference Formula
Phase difference between waves tells us how much one wave is shifted relative to another. This is important in understanding wave interference patterns.

The formula to find the phase difference (\text{螖蠁\radians}) is: \[ \text{螖蠁} = \frac{2蟺}{位} \times 鈭哾 \] Plugging in the earlier calculated values: \[ 鈭喯� = \frac{2蟺}{0.6111 \text{ meters}} \times 0.40 \text{ meters} = 4.11 \text{radians} \]

This result means the waves have a phase difference of 4.11 radians at the given point, affecting how they superimpose and interfere.

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Most popular questions from this chapter

Pipe \(A\), which is \(1.2 \mathrm{~m}\) long and open at both ends, oscillates at its third lowest harmonic frequency. It is filled with air for which the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). Pipe \(B\), which is closed at one end, oscillates at its second lowest harmonic frequency. These frequencies of pipes \(A\) and \(B\) happen to match. (a) If an \(x\) axis extends along the interior of pipe \(A\), with \(x=0\) at one end, where along the axis are the displacement nodes? (b) How long is pipe \(B ?\) (c) What is the lowest harmonic frequency of pipe \(A\) ?

A tuning fork of unknown frequency makes three beats per second with a standard fork of frequency \(384 \mathrm{~Hz}\). The beat frequency decreases when a small piece of wax is put on a prong of the first fork. What is the freauency of this fork?

A certain sound source is increased in sound level by \(30 \mathrm{~dB}\). By what multiple is (a) its intensity increased and (b) its pressure amplitude increased?

A \(2000 \mathrm{~Hz}\) siren and a civil defense official are both at rest with respect to the ground. What frequency does the official hear if the wind is blowing at \(12 \mathrm{~m} / \mathrm{s}\) (a) from source to official and (b) from official to source?

Trooper \(B\) is chasing speeder \(A\) along a straight stretch of road. Both are moving at a speed of \(160 \mathrm{~km} / \mathrm{h}\). Trooper \(B\), failing to catch up, sounds his siren again. Take the speed of sound in air to be \(343 \mathrm{~m} / \mathrm{s}\) and the frequency of the source to be \(500 \mathrm{~Hz}\). What is the Doppler shift in the frequency heard by speeder \(A\) ?
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