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Chapter 9: Problem 27

Which of the following represents a standing wave? (A) \(y=A \sin (\omega t-k x)\) (B) \(y=A \sin k x \sin (\omega t-\theta)\) (C) \(y=A e^{-b x} \sin (\omega t-k x+\alpha)\) (D) \(y=(a x+b) \sin (\omega t-k x)\)

Short Answer

Expert verified
Option (B) \(y=A \sin k x \sin (\omega t-\theta)\) represents a standing wave, as it matches the general formula of a standing wave and has the product of two sinusoidal terms, one dependent on x and the other on t.

Step by step solution

01

Analyze option (A)

Option (A) gives the equation: \(y=A \sin (\omega t-k x)\). This is a standard equation of a simple harmonic wave travelling in the positive x-direction. It does not represent a standing wave as there is no additional wave travelling in the negative x-direction.
02

Analyze option (B)

Option (B) gives the equation: \(y=A \sin k x \sin (\omega t-\theta)\). We can rewrite it as: \(y(x,t) = A\sin(kx)\sin(\omega t - \theta)\). This equation has the product of two sinusoidal terms, one dependent on x and one on t. Comparing this with the general formula of a standing wave, we can see it matches the structure of the general formula. Thus, option (B) represents a standing wave.
03

Analyze option (C)

Option (C) gives the equation: \(y=A e^{-b x} \sin (\omega t-k x+\alpha)\). This represents a damped traveling wave, as the exponential term \(e^{-bx}\) represents damping. The sinusoidal term \(\sin(\omega t - kx + \alpha)\) represents a simple harmonic wave. Therefore, option (C) does not represent a standing wave.
04

Analyze option (D)

Option (D) gives the equation: \(y=(a x+b) \sin (\omega t-k x)\). This equation represents a wave traveling in the positive x-direction with an amplitude depending on x, as indicated by the factor \((ax+b)\). However, this equation does not represent a standing wave as it does not have the product of two sinusoidal terms with one dependent on x and the other dependent on t.
05

Conclusion

Based on our analysis of each option, we can determine that option (B) \(y=A \sin k x \sin (\omega t-\theta)\) represents a standing wave.

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Most popular questions from this chapter

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