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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

Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by \(y(x, t)=1.00 \cdot 10^{-2} \sin (25 x) \cos (1200 t) .\) The string has a linear mass density of \(0.01 \mathrm{~kg} / \mathrm{m},\) and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.

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 \(t=0\).

Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

Fans at a local football stadium are so excited that their team is winning that they start "the wave" in celebration. Which of the following four statements is (are) true? I. This wave is a traveling wave. II. This wave is a transverse wave. III. This wave is a longitudinal wave. IV. This wave is a combination of a longitudinal wave and a transverse wave. a) I and II c) III only e) I and III b) II only d) I and IV

A string with a mass of \(30.0 \mathrm{~g}\) and a length of \(2.00 \mathrm{~m}\) is stretched under a tension of \(70.0 \mathrm{~N}\). How much power must be supplied to the string to generate a traveling wave that has a frequency of \(50.0 \mathrm{~Hz}\) and an amplitude of \(4.00 \mathrm{~cm} ?\)
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