(a)
The amplitude and phase constant of the sum of the given function by using trigonometry identity.
(a)
Answer to Problem 39AP
The amplitude of the sum of the given function by trigonometry identity is
Explanation of Solution
Given info: The mechanical vibration of first wave is
Write the expression for the sum of two wave functions.
Here,
Substitute
Further solve the equation,
Conclusion:
Therefore, the amplitude of the sum of the given function by trigonometry identity is
(b)
The amplitude and phase constant of the sum of the given function by representing the oscillation as phasors.
(b)
Answer to Problem 39AP
The amplitude of the sum of the given function by phasor representation is
Explanation of Solution
Given info: The mechanical vibration of first wave is
Write the expression for the phasor of a first oscillation.
Write the expression for the phasor of a second oscillation.
Write the expression for the sum of two wave functions.
Substitute
Thus, the phasor representation of the sum of two wave functions is
Formula to calculate the amplitude of the resultant wave is,
Here,
Substitute
Thus, the amplitude of the resultant wave is
Formula to calculate the angle of the resultant wave makes with the first wave is,
Substitute
Thus, phase difference between the resultant and the
Conclusion:
Therefore, the amplitude of the sum of the given function by phasor representation is
(c)
The result by compare the answer to part (a) and part (b).
(c)
Answer to Problem 39AP
The result of part (a) and part (b) are identical.
Explanation of Solution
Given info: The mechanical vibration of first wave is
Since from the trigonometry identities the amplitude and the phase angle of the sum of two waves are identical to the amplitude and the phase angle of the sum of two waves by phasor representation, hence the both the method is valid to estimate the amplitude and the phase angle of the resultant wave.
Conclusion:
Therefore, the result of part (a) and part (b) are identical.
(d)
The amplitude and phase constant of the sum of the given function by represent the oscillation as phasors.
(d)
Answer to Problem 39AP
The amplitude of the sum of the given function by phasor representation is
Explanation of Solution
Given info: The mechanical vibration of first wave is
Write the expression for the phasor of a first oscillation.
Write the expression for the phasor of a second oscillation.
Write the expression for the phasor of a third oscillation.
Write the expression for the sum of two wave functions.
Substitute
Thus, the phasor representation of the sum of three wave functions is
Formula to calculate the amplitude of the resultant wave is,
Here,
Substitute
Thus, the amplitude of the resultant wave is
Formula to calculate the angle of the resultant wave is,
Substitute
Write the expression for the angle with the first wave.
Substitute
Conclusion:
Therefore, the amplitude of the sum of the given function by phasor representation is
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Chapter 32 Solutions
Physics for Scientists and Engineers with Modern Physics
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