Marie Cornu, a physicist at the Polytechnic Institute in Paris, invented phasors in about 1880. This problem helps you see their general utility in representing oscillations. Two mechanical vibrations are represented by the expressions
and
where y1 and y2 are in centimeters and t is in seconds. Find the amplitude and phase constant of the sum of these functions (a) by using a trigonometric identity (as from Appendix B) and (b) by representing the oscillations as phasors. (c) State the result of comparing the answers to parts (a) and (b). (d) Phasors make it equally easy to add traveling waves. Find the amplitude and phase constant of the sum of the three waves represented by
where x, y1, y2, and y3, are in centimeters and t is in seconds.
(a)
Answer to Problem 39AP
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)
Answer to Problem 39AP
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)
Answer to Problem 39AP
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)
Answer to Problem 39AP
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
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