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Chapter 16: Problem 18

When two pure tones with similar frequencies combine to produce beats, the result is a train of wave packets. That is, the sinusoidal waves are partially localized into packets. Suppose two sinusoidal waves of equal amplitude A, traveling in the same direction, have wave numbers \(\kappa\) and \(\kappa+\Delta \kappa\) and angular frequencies \(\omega\) and \(\omega+\Delta \omega\), respectively. Let \(\Delta x\) be the length of a wave packet, that is, the distance between two nodes of the envelope of the combined sine functions. What is the value of the product \(\Delta x \Delta \kappa ?\)

Short Answer

Expert verified
Answer: The value of the product ΔxΔκ is 2nπ, where n is an integer representing the nth node.

Step by step solution

01

Define the waves

Let's represent the two sinusoidal waves as follows: $$y_1 = A \cos (\kappa x - \omega t)$$ $$y_2 = A \cos ((\kappa+\Delta \kappa) x - (\omega+\Delta \omega) t)$$ where \(y_1\) and \(y_2\) represent the displacements of the waves, \(A\) is the amplitude, \(\kappa\) and \(\kappa+\Delta\kappa\) are the respective wave numbers, and \(\omega\) and \(\omega+\Delta\omega\) are the respective angular frequencies.
02

Add the two waves

We need to find the total displacement \(y_{total}\), which is the sum of the displacements of the two individual waves: $$y_{total} = y_1 + y_2$$
03

Use the trigonometric identity for the cosine of the sum of angles

Applying the identity \(\cos(A+B) = \cos A \cos B - \sin A \sin B\), we can rewrite the expression for \(y_{total}\) as: $$y_{total} = 2A\cos\left(\frac{\Delta \kappa x}{2}-\frac{\Delta \omega t}{2}\right)\cos\left(\!\left(\kappa x-\omega t\!\right)\!+\!\left(\!\frac{\Delta \kappa x}{2}-\frac{\Delta \omega t}{2}\right)\!\!\right)\!$$
04

Find the envelope function

To find the envelope function for the combined sine functions (i.e., the wave packet), we focus on the term that does not oscillate with respect to position and time: $$y_{env} = 2A\cos\left(\frac{\Delta \kappa x}{2}-\frac{\Delta \omega t}{2}\right)$$
05

Find the length of a wave packet and product

The length of a wave packet, \(\Delta x\), is the distance between two nodes of the envelope function, that is the distance between two points where the cosine term equals zero. To find this length, we should set the cosine term to zero: $$\frac{\Delta \kappa x}{2} = n\pi$$ $$\Delta x = \frac{2n\pi}{\Delta \kappa}$$ where \(n\) is an integer representing the \(n^{th}\) node. Now, we can find the product \(\Delta x \Delta \kappa\): $$\Delta x \Delta \kappa = 2n\pi$$ The value of the product \(\Delta x \Delta \kappa\) is \(2n\pi\), where \(n\) is an integer representing the \(n^{th}\) node.

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Most popular questions from this chapter

You are standing between two speakers that are separated by \(80.0 \mathrm{~m}\). Both speakers are playing a pure tone of \(286 \mathrm{~Hz}\). You begin running directly toward one of the speakers, and you measure a beat frequency of \(10.0 \mathrm{~Hz}\). How fast are you running?

A string of a violin produces 2 beats per second when sounded along with a standard fork of frequency \(400 . \mathrm{Hz}\). The beat frequency increases when the string is tightened. a) What was the frequency of the violin at first? b) What should be done to tune the violin?

A police siren contains at least two frequencies, producing the wavering sound (beats). Explain how the siren sound changes as a police car approaches, passes, and moves away from a pedestrian.

Two vehicles carrying speakers that produce a tone of frequency \(1000.0 \mathrm{~Hz}\) are moving directly toward each other. Vehicle \(\mathrm{A}\) is moving at \(10.00 \mathrm{~m} / \mathrm{s}\) and vehicle \(\mathrm{B}\) is moving at \(20.00 \mathrm{~m} / \mathrm{s}\). Assume the speed of sound in air is \(343.0 \mathrm{~m} / \mathrm{s}\), and find the frequencies that the driver of each vehicle hears.

You are driving along a highway at \(30.0 \mathrm{~m} / \mathrm{s}\) when you hear a siren. You look in the rear-view mirror and see a police car approaching you from behind with a constant speed. The frequency of the siren that you hear is \(1300 \mathrm{~Hz}\). Right after the police car passes you, the frequency of the siren that you hear is \(1280 \mathrm{~Hz}\). a) How fast was the police car moving? b) You are so nervous after the police car passes you that you pull off the road and stop. Then you hear another siren, this time from an ambulance approaching from behind. The frequency of its siren that you hear is \(1400 \mathrm{~Hz}\). Once it passes, the frequency is \(1200 \mathrm{~Hz}\). What is the actual frequency of the ambulance's siren?
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