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Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
Page 2


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Page 3


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Q.4. A 160 g rope 4 m long is fixed at one end and tied to a light string of
the same length at the other end. Its tension is 400 N.
(a) What are the wavelengths of the fundamental and the first two
overtones?
(b) What are the frequencies of these standing waves?
[Hint: In this case, the fixed end is a node and the end tied with the light
string is antinode.]
Sol.
Page 4


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Q.4. A 160 g rope 4 m long is fixed at one end and tied to a light string of
the same length at the other end. Its tension is 400 N.
(a) What are the wavelengths of the fundamental and the first two
overtones?
(b) What are the frequencies of these standing waves?
[Hint: In this case, the fixed end is a node and the end tied with the light
string is antinode.]
Sol.
(a) 1 = ?1/4
Now
Similarly f2 and f3.
Q.5. A string fastened at both ends has successive resonances with
wavelengths of 0.54m for the nth harmonic and 0.48 m for the (n +1)th
harmonic.
(a) Which harmonics are these?
(b) What is the length of the string?
(c) What is the wavelength of the fundamental frequency?
Sol.
or
Page 5


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Q.4. A 160 g rope 4 m long is fixed at one end and tied to a light string of
the same length at the other end. Its tension is 400 N.
(a) What are the wavelengths of the fundamental and the first two
overtones?
(b) What are the frequencies of these standing waves?
[Hint: In this case, the fixed end is a node and the end tied with the light
string is antinode.]
Sol.
(a) 1 = ?1/4
Now
Similarly f2 and f3.
Q.5. A string fastened at both ends has successive resonances with
wavelengths of 0.54m for the nth harmonic and 0.48 m for the (n +1)th
harmonic.
(a) Which harmonics are these?
(b) What is the length of the string?
(c) What is the wavelength of the fundamental frequency?
Sol.
or
...(i)
Similarly ,
or
... (ii)
From Eqs. (i) and (ii) we have,
(b) From Eq. (i)
l = (0.27)n = 0.27 × 8 = 2.16 m
Q.6. If the frequencies of the second and fifth harmonics of a string
differ by 54 Hz. What is the fundamental frequency of the string?
Sol.
5f - 2f=54
3f = 54 f = 18 Hz
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FAQs on DC Pandey Solutions: Superposition of Waves - Physics Class 11 - NEET

1. What is the concept of superposition of waves?
Ans. Superposition of waves is a principle in physics that states when two or more waves meet, the resulting wave is the algebraic sum of the individual waves. This means that the displacement of the medium at any point is the sum of the displacements caused by each individual wave.
2. How does superposition of waves help in understanding interference?
Ans. Superposition of waves helps in understanding interference by explaining how waves interact with each other. When two waves of the same frequency and amplitude meet, they can interfere constructively, resulting in a wave with greater amplitude, or destructively, resulting in a wave with smaller or zero amplitude. This interference pattern can be observed in various phenomena, such as the interference of light waves in the double-slit experiment.
3. What is the difference between constructive and destructive interference?
Ans. Constructive interference occurs when two waves of the same frequency and amplitude meet in such a way that they reinforce each other, resulting in a wave with a greater amplitude. Destructive interference, on the other hand, occurs when two waves of the same frequency and amplitude meet in such a way that they cancel each other out, resulting in a wave with a smaller or zero amplitude.
4. How is superposition of waves applied in real-life scenarios?
Ans. The superposition of waves is applied in various real-life scenarios. For example, in noise-canceling headphones, a microphone detects the ambient noise and creates a wave with the same amplitude but opposite phase. When this wave is combined with the incoming sound wave, they cancel each other out, reducing the overall noise. Superposition is also utilized in the field of seismology to study earthquake waves and in the analysis of ocean waves.
5. Can the superposition of waves be applied to all types of waves?
Ans. Yes, the superposition of waves can be applied to all types of waves, including electromagnetic waves, acoustic waves, and water waves. The principle of superposition holds true as long as the waves are linear, meaning that they obey the principles of superposition and do not interact in a nonlinear manner. This allows for the analysis and understanding of wave phenomena in various scientific and engineering fields.
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