Concept explainers
(a)
The range ofwavenumber.
(a)
Answer to Problem 67P
Explanation of Solution
Introduction:
A sound wave form originated by tuning fork of frequency centered on
Figure 1:A sound waveform created by a tuning fork
Basically, the wave packet is the result of the superposition of the two or more waves, hence, the resultant of the super-position of the wave results in the spread of frequencies,
Let say, the v is the speed of the pulse. Therefore, the pulse spread in space can be written as
The range of frequencies
As it is well known relation between the velocity, frequency and the wave number is
Putting the value of v from equation 1 in equation 2,
Comparing equation 1 and 4, we get
k is the wave number.
Conclusion:
The wave number range is
(b)
The average value of the wavelength.
(b)
Answer to Problem 67P
Explanation of Solution
Introduction:
A wavelength
N number of cycles are there in a wave packet of length
The distance for one cycle is basically equal to the wavelength and will be
Hence, one can write
Conclusion:
The average wavelength will be
(c)
The average value of the wave number k
(c)
Answer to Problem 67P
Explanation of Solution
Introduction:
As calculated before:
N number of cycles are there in a wave packet of length
The distance for one cycle is basically equal to the wavelength and will be
Hence,
The relation between
From equation 3:
Conclusion:
The average value of the wave number kis
(d)
Therange in angular frequencies.
(d)
Answer to Problem 67P
Explanation of Solution
Introduction:
A sound wave form originated by tuning fork of frequency centered on
Figure 1:A sound waveform created by a tuning fork
Basically, the wave packet is the result of the superposition of the two or more waves, hence, the resultant of the super-position of the wave results in the spread of frequencies,
The relation between the range of frequencies
Conclusion:
The range in angular frequencies
(e)
The frequency in terms of N and
(e)
Answer to Problem 67P
Explanation of Solution
Introduction:
A sound wave form originated by tuning fork of frequency centered on
Figure 1:A sound waveform created by a tuning fork
Basically, the wave packet is the result of the superposition of the two or more waves, hence, the resultant of the super-position of the wave results in the spread of frequencies,
N number of cycles are there in a wave packet of length
The time required to complete one cycle is basically equal to the frequency and will be
Hence, one can write
Conclusion:
The frequency will be
(f)
The uncertainty in N.
(f)
Answer to Problem 67P
There is an uncertainty of
Explanation of Solution
Introduction:
A sound wave form originated by tuning fork of frequency centered on
Figure 1:A sound waveform created by a tuning fork
Basically, the wave packet is the result of the superposition of the two or more waves, hence, the resultant of the super-position of the wave results in the spread of frequencies,
As it is clear from the Figure 1, there is a cycle which is not a complete one. Hence, the cycle can be either not present or may be present in the wave packet.
Therefore, there is an error or uncertainty of
Conclusion:
There is an uncertainty of
(g)
The uncertainty in wave number k
(g)
Answer to Problem 67P
Explanation of Solution
Introduction:
A sound wave form originated by tuning fork of frequency centered on
Figure 1:A sound waveform created by a tuning fork
Basically, the wave packet is the result of the superposition of the two or more waves, hence, the resultant of the super-position of the wave results in the spread of frequencies,
As calculated before:
N number of cycles are there in a wave packet of length
The distance for one cycle is basically equal to the wavelength and will be
Hence, we can write
The relation between
From equation 3 we obtain
As calculated by the previous section that the uncertainty in N is
Conclusion:
The uncertainty in k will be,
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Chapter 16 Solutions
Physics for Scientists and Engineers
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- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage LearningUniversity Physics Volume 1PhysicsISBN:9781938168277Author:William Moebs, Samuel J. Ling, Jeff SannyPublisher:OpenStax - Rice University