Concept explainers
Rank the waves represented by the following functions from the largest to the smallest according to (i) their amplitudes, (ii) their wavelengths, (iii) their frequencies, (iv) their periods, and (v) their speeds. If the values of a quantity are equal for two waves, show them as having equal rank. For all functions, x and y are in meters and t is in seconds. (a) y = 4 sin (3x − 15t) (b) y = 6 cos (3x + 15t − 2) (c) y = 8 sin (2x + 15t) (d) y = 8 cos (4x + 20t) (e) y = 7 sin (6x + 24t)
(i)
Rank the waves in the decreasing order of their amplitudes.
Answer to Problem 3OQ
The order is (c) = (d) > (e) > (b) > (a).
Explanation of Solution
Write the general expression which denotes a sinusoidal wave travelling in +ve x-direction.
Here,
Wave can be represented as cosine function also.
Write the amplitude of
Write the amplitude of
Write the amplitude of
Write the amplitude of
Write the amplitude of
Conclusion:
Therefore, the order is (c) = (d) > (e) > (b) > (a).
(ii)
Rank the waves in the decreasing order of their wavelengths.
Answer to Problem 3OQ
The order is (c) > (a) = (b) > (d) > (e).
Explanation of Solution
From the general expression for wave in part (a), it is understood that the coefficient of
Write the relation between wavenumber and wavelength.
Here,
It means that higher the value of
Conclusion:
Consider
Substitute
Consider
Substitute
Consider
Substitute
Consider
Substitute
Consider
Substitute
Therefore, the order is (c) > (a) = (b) > (d) > (e).
(iii)
Rank the waves in the decreasing order of their frequencies.
Answer to Problem 3OQ
The order is (e) > (d) > (a) = (b) = (c).
Explanation of Solution
Write the relation between linear and angular frequency.
Here,
The coefficient of
Conclusion:
Consider
Substitute
Consider
Substitute
Consider
Substitute
Consider
Substitute
Consider
Substitute
Therefore, the order is order is (e) > (d) > (a) = (b) = (c).
(iv)
Rank the waves in the decreasing order of their periods.
Answer to Problem 3OQ
The order is (a) = (b) = (c) > (d) > (e).
Explanation of Solution
Write the relation between
Here,
Conclusion:
Therefore, the order is (a) = (b) = (c) > (d) > (e).
(v)
Rank the waves in the decreasing order of their speeds.
Answer to Problem 3OQ
The order is (c) > (a) = (b) = (d) > (e).
Explanation of Solution
Write the equation to find speed from wave number and angular frequency.
Conclusion:
Consider
Substitute
Consider
Substitute
Consider
Substitute
Consider
Substitute
Consider
Substitute
Therefore, the order is(c) > (a) = (b) = (d) > (e).
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Chapter 13 Solutions
Principles of Physics: A Calculus-Based Text
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