Concept explainers
In a first experiment, a sinusoidal sound wave is sent through a long tube of air. transporting energy at the average rate of Pavg, 1· In a second experiment, two other sound waves, identical to the first one, are to be sent simultaneously through the tube with a phase difference ϕ of either 0, 0.2 wavelength, or 0.5 wavelength between the waves, (a) With only mental calculation, rank those choices of ϕ according to the average rate at which the waves will transport energy, greatest first, (b) For the first choice of ϕ, what is the average rate in terms of Pavg, 1?
To find:
a) The rank of phase difference according to the average rate of transport of energy by the waves, greatest first.
b) The average rate of energy transport for the first choice in (a).
Answer to Problem 1Q
Solution:
a) The rank of phase difference according to the average rate of transport of energy by the waves, greatest first, is
b) The average rate of energy transport for (a)
Explanation of Solution
1) Concept:
The rate of energy transported by a travelling wave depends on the intensity of the wave as well as the area to which the energy is transported. The intensity of a resultant wave depends on the phase difference between the two superposing waves.
2) Formula:
3) Given:
i) The average rate of energy transported by a single wave =
ii) The phase difference between the two waves sent through the pipe are
4) Calculations:
a) The rate of energy transported is given by
When two waves are sent with phase difference
Thus, the rate of energy transport for
When two waves are sent with phase difference
Hence, the resultant intensity will be more than the single wave but less than that for
When two waves are sent with phase difference
Hence, the ranking of the situations will be
b) Since the amplitude of the resultant wave is twice the single wave, intensity is four fold. Hence, the rate of energy transported will be
Conclusion:
The energy transported by a wave can be calculated by using the intensity of the wave. Here, the intensity changes according to the phase difference between the two superposing waves. Hence the rate of energy transported changes as the phase difference changes.
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