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Chapter 11: Problem 40

Two waves with identical frequency but different amplitudes $A_{1}=6.0 \mathrm{cm}\( and \)A_{2}=3.0 \mathrm{cm},$ occupy the same region of space (i.e., are superimposed). (a) At what phase difference will the resulting wave have the highest intensity? What is the amplitude of the resulting wave in that case? (b) At what phase difference will the resulting wave have the lowest intensity and what will its amplitude be? (c) What is the ratio of the two intensities?

Short Answer

Expert verified
Answer: The phase difference values for maximum intensity are \(0, 2\pi, 4\pi, \dots\) and for minimum intensity are \(\pi, 3\pi, 5\pi, \dots\). The amplitude of the resulting wave at maximum intensity is 9 cm and at minimum intensity is 3 cm. The ratio of the maximum to minimum intensities is 9.

Step by step solution

01

Use the principle of superposition

By the principle of superposition, the resulting wave's displacement y is the vector sum of the displacements of the two individual waves: \(y = y_1 + y_2 = A_1 \sin{(kx - \omega t)} + A_2 \sin{(kx - \omega t + \phi)}\)
02

Find phase difference for maximum intensity

For maximum intensity, the amplitude of the resulting wave should be maximum. It occurs when the two waves are in phase (\(\phi = 0, 2\pi, 4\pi, \dots)\). When the two waves add up constructively, the resulting amplitude is the sum of their amplitudes: \(A_{max} = A_1 + A_2 = 6 \,\text{cm} + 3\,\text{cm} = 9\, \text{cm}\)
03

Find phase difference for minimum intensity

For minimum intensity, the amplitude of the resulting wave should be minimum. It occurs when the two waves are out of phase (\(\phi = \pi, \3\pi, 5\pi, \dots)\). When the two waves have destructive interference, the resulting amplitude is the difference of their amplitudes: \(A_{min} = |A_1 - A_2| = |6\, \text{cm} - 3\, \text{cm}| = 3\, \text{cm}\)
04

Calculate intensities

Intensity is proportional to the square of the amplitude. To find the ratio of the intensities, calculate the intensities for maximum and minimum amplitude cases, and divide one by the other: \(I_{max} \propto A_{max}^2 = (9\, \text{cm})^2 = 81\,\text{cm}^2\) \(I_{min} \propto A_{min}^2 = (3\, \text{cm})^2 = 9\,\text{cm}^2\)
05

Calculate ratio of intensities

To find the ratio of the maximum and minimum intensities, divide \(I_{max}\) by \(I_{min}\): \(I_{ratio} = \frac{I_{max}}{I_{min}} = \frac{81\,\text{cm}^2}{9\, \text{cm}^2} = 9\) The resulting wave has the highest intensity when the phase difference is \(0, 2\pi, 4\pi, \dots\), and its amplitude is \(9\,\text{cm}\). It has the lowest intensity when the phase difference is \(\pi, 3\pi, 5\pi, \dots\), and its amplitude is \(3\,\text{cm}\). The ratio of the maximum to minimum intensities is \(9\).

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Most popular questions from this chapter

What is the speed of the wave represented by \(y(x, t)=\) $A \sin (k x-\omega t),\( where \)k=6.0 \mathrm{rad} / \mathrm{cm}\( and \)\omega=5.0 \mathrm{rad} / \mathrm{s} ?$
A string \(2.0 \mathrm{m}\) long is held fixed at both ends. If a sharp blow is applied to the string at its center, it takes \(0.050 \mathrm{s}\) for the pulse to travel to the ends of the string and return to the middle. What is the fundamental frequency of oscillation for this string?
Show that the amplitudes of the graphs you made in Problem 74 satisfy the equation \(A^{\prime}=2 A \cos (\omega t),\) where \(A^{\prime}\) is the amplitude of the wave you plotted and \(A\) is \(5.0 \mathrm{cm},\) the amplitude of the waves that were added together.
A longitudinal wave has a wavelength of \(10 \mathrm{cm}\) and an amplitude of \(5.0 \mathrm{cm}\) and travels in the \(y\) -direction. The wave speed in this medium is \(80 \mathrm{cm} / \mathrm{s},\) (a) Describe the motion of a particle in the medium as the wave travels through the medium. (b) How would your answer differ if the wave were transverse instead?
Write the equation for a transverse sinusoidal wave with a maximum amplitude of \(2.50 \mathrm{cm}\) and an angular frequency of 2.90 rad/s that is moving along the positive \(x\) -direction with a wave speed that is 5.00 times as fast as the maximum speed of a point on the string. Assume that at time \(t=0,\) the point \(x=0\) is at \(y=0\) and then moves in the \(-y\) -direction in the next instant of time.
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