
Concept explainers
Consider the following wave function in SI units:
Explain how this wave function can apply to a wave
(a)

Whether the wave move toward right or the left.
Answer to Problem 54AP
The wave does not move toward right or the left while the wave moves outward equally in all directions.
Explanation of Solution
The given wave function is,
The standard form wave function for the standing wave is,
Here,
If
The wave moves outward equally in all directions because of the negative sign in
Conclusion:
Therefore, the wave does not move toward right or the left while the wave moves outward equally in all directions
(b)

The effect on its amplitude as it moves away from the source.
Answer to Problem 54AP
The amplitude of the wave will be decreased as it moves away from the source because amplitude is inversely proportional to the distance.
Explanation of Solution
From equation (1), the given wave function is,
From equation (2), the standard form wave function for the standing wave is,
From equation (1) and (2), it is clear that the amplitude is inversely proportional to its distance from the center. The amplitude of the wave will be decreased as it moves away from the source because amplitude is inversely proportional to the distance.
Conclusion:
Therefore, the amplitude of the wave will be decreased as it moves away from the source because amplitude is inversely proportional to the distance
(c)

The effect on its speed as it moves away from the source.
Answer to Problem 54AP
The speed of the wave is constant as it moves away from the source.
Explanation of Solution
The given wave function is,
The standard form wave function for the standing wave is,
Formula to calculate the speed of the wave is,
Here,
Substitute
The calculated value of the speed of the wave is equal to the speed of the wave in the water at
Conclusion:
Therefore, the speed of the wave is constant as it moves away from the source.
(d)

The effect on its frequency as it moves away from the source.
Answer to Problem 54AP
The frequency of the wave is constant as wave moves away from the source.
Explanation of Solution
The given wave function is,
The standard form wave function for the standing wave is,
Formula to calculate the frequency of the wave is,
Here,
Substitute
The frequency of the wave is constant at
Conclusion:
Therefore, the frequency of the wave is constant as the wave moves away from the source.
(e)

The effect on its wavelength as it moves away from the source.
Answer to Problem 54AP
The wavelength of the wave is constant as wave moves away from the source.
Explanation of Solution
The given wave function is,
The standard form wave function for the standing wave is,
Formula to calculate the wavelength of the wave is,
Here,
Substitute d
The wavelength of the wave is constant at
Conclusion:
Therefore, the wavelength of the wave is constant as the wave moves away from the source.
(f)

The effect of its power as it moves away from the source.
Answer to Problem 54AP
The power of the source and the net power of the wave at all distance as wave moves away from the source.
Explanation of Solution
The given wave function is,
The standard form wave function for the standing wave is,
Formula to calculate the intensity of the wave is,
Here,
Substitute d
Formula to calculate the power of the source and the net power of the wave at all distance is,
Here,
Substitute
Thus, the power of the source and the net power of the wave at all distance will be same because the wave moves outward equally in all directions
Conclusion:
Therefore, the power of the source and the net power of the wave at all distance as the wave moves away from the source.
(e)

The effect of its intensity as it moves away from the source.
Answer to Problem 54AP
The intensity of the source and the intensity of the wave at all distance as wave moves away from the source.
Explanation of Solution
The given wave function is,
The standard form wave function for the standing wave is,
The intensity of the wave is,
The intensity of the wave follows the inverse square law at
Substitute
Thus, the intensity of the source and the intensity of the wave are same as the wave moves away from the source because the wave moves outward equally in all directions.
Conclusion:
Therefore, the intensity of the source and the intensity of the wave are same as the wave moves away from the source.
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