Concept explainers
(a)
Wave function of two waves.
(a)
Explanation of Solution
Given:
The wave function of standing wave is given as
Introduction:
The superimposed traveling waves have same wave number and same angular frequency. However, the amplitude of superimposed waves is equal and half the amplitude of the standing wave.
Write expression for the wave function for the wave traveling in positive x-direction.
Write expression for the wave function for the wave traveling in negative x-direction.
Conclusion:
Thus, the wave function for superimposed waves in positive and negative x-direction is given above.
(b)
The distance between nodes of the standing wave.
(b)
Explanation of Solution
Given:
The wave function of standing wave is given as
Formula used:
Write expression for wave number.
Rearrange above expression for
Write expression for distance between adjacent nodes.
Substitute
Calculation:
Substitute
Conclusion:
Thus, the distance between two waves is
(c)
The maximum speed of the rope at
(c)
Explanation of Solution
Given:
The wave function of standing wave is given as
Formula used:
Write expression for standing wave.
Calculation:
Differentiate equation (1) with respect to
Substitute
Substitute
Conclusion:
Thus, the maximum speed of the wave at
(d)
The maximum acceleration of the rope at
(d)
Explanation of Solution
Given:
The wave function of standing wave is given as
Formula used:
Write expression for standing wave.
Calculation:
Differentiate equation (1) with respect to
Differentiate above expression with respect to
Substitute
Conclusion:
Thus, maximum acceleration at
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Chapter 16 Solutions
Physics for Scientists and Engineers
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