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Chapter 15: Problem 13

If two traveling waves have the same wavelength, frequency, and amplitude and are added appropriately, the result is a standing wave. Is it possible to combine two standing waves in some way to give a traveling wave?

Short Answer

Expert verified
Answer: No, it is not possible to combine two standing waves to create a traveling wave.

Step by step solution

01

Understand the properties of traveling and standing waves

Traveling waves are characterized by their ability to transport energy from one point to another through space. The equation for a traveling wave is given by: y(x, t) = A\cdot\sin(kx - \omega t) Where A is the amplitude, k is the wavenumber, x is the position in space, t is time, and \omega is the angular frequency. Standing waves, on the other hand, are characterized by the lack of energy transport through space, with energy oscillating back and forth in place between potential and kinetic energies. The equation for a standing wave is given by: y(x, t) = 2A\cdot\sin(kx)\cdot\cos(\omega t) The fundamental difference between these two types of waves lies in their spatial and temporal dependencies.
02

Analyze the conditions for combining two standing waves

To investigate the possibility of combining two standing waves to create a traveling wave, we need to analyze the conditions and mathematical expressions necessary to create such a transformation. Let's consider two standing waves: y_1(x, t) = 2A\cdot\sin(kx)\cdot\cos(\omega t) y_2(x, t) = 2A\cdot\sin(k(x - d))\cdot\cos(\omega t) Where d is some distance between the positions of the two standing waves. To obtain a traveling wave, we need to sum these two standing waves: y(x, t) = y_1(x, t) + y_2(x, t)
03

Simplify the sum of the two standing waves

By summing the two standing waves, we get: y(x, t) = 2A\cdot\sin(kx)\cdot\cos(\omega t) + 2A\cdot\sin(k(x - d))\cdot\cos(\omega t) Now we have to try and simplify this expression to get it into the form of a traveling wave. However, due to the product of sin and cos terms in both standing waves, and the additional phase shift in the second wave, it is impossible to simplify this sum into the form of a traveling wave equation.
04

Conclusion

Since we cannot simplify the sum of two standing waves into the form of a traveling wave equation, it is not possible to combine two standing waves to create a traveling wave.

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Most popular questions from this chapter

One of the main things allowing humans to determine whether a sound is coming from the left or the right is the fact that the sound will reach one ear before the other. Given that the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\) and that human ears are typically \(20.0 \mathrm{~cm}\) apart, what is the maximum time resolution for human hearing that allows sounds coming from the left to be distinguished from sounds coming from the right? Why is it impossible for a diver to be able to tell from which direction the sound of a motor boat is coming? The speed of sound in water is \(1.50 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).

Write the equation for a sinusoidal wave propagating in the negative \(x\) -direction with a speed of \(120 . \mathrm{m} / \mathrm{s}\), if a particle in the medium in which the wave is moving is observed to swing back and forth through a \(6.00-\mathrm{cm}\) range in \(4.00 \mathrm{~s}\). Assume that \(t=0\) is taken to be the instant when the particle is at \(y=0\) and that the particle moves in the positive \(y\) -direction immediately after \(t=0\).

A wave traveling on a string has the equation of motion \(y(x, t)=0.02 \sin (5.00 x-8.00 t)\) a) Calculate the wavelength and the frequency of the wave. b) Calculate its velocity. c) If the linear mass density of the string is \(\mu=0.10 \mathrm{~kg} / \mathrm{m}\), what is the tension on the string?

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