Wave on String | Physics Optional Notes for UPSC PDF Download

 Page 1


WAVE ON STRING  
WAVE AND ITS CHARACTERISTICS 
Introduction of Waves 
What is wave motion? 
? When a particle moves through space, it carries KE with itself. Wherever the 
particle goes, the energy goes with it. (One way of transporting energy from one 
place to another place) 
? There is another way (wave motion) to transport energy from one part of space to 
another without any bulk motion of material together with it. Sound is 
transmitted in the air in this manner. 
Ex. You (Haryana) want to communicate with your friend (Delhi) 
 
1
st 
 option involves the concept of particle & the second choice involves the concept of 
wave. 
 
When you say "Namaste" to your friend, no physical particles are sent from your lips to 
their ear. Instead, you create a disturbance in the air around your lips. This involves 
transferring energy to nearby air particles, either pushing them forward or pulling them 
back. This action temporarily changes the density of the air in that region. The disturbed 
air particles then affect the next layer of air, passing on the disturbance. This chain 
reaction continues until the air near your friend's ear is disturbed, enabling them to hear 
your greeting. 
 {Note: - In the above example air itself does not move.} 
A wave is a disturbance that propagates in space and transports energy and momentum 
from one point to another without the transport of matter. 
A few examples of waves: 
The ripples on a pond (water waves), the sound we hear, visible light, radio and TV 
signals, etc. 
Page 2


WAVE ON STRING  
WAVE AND ITS CHARACTERISTICS 
Introduction of Waves 
What is wave motion? 
? When a particle moves through space, it carries KE with itself. Wherever the 
particle goes, the energy goes with it. (One way of transporting energy from one 
place to another place) 
? There is another way (wave motion) to transport energy from one part of space to 
another without any bulk motion of material together with it. Sound is 
transmitted in the air in this manner. 
Ex. You (Haryana) want to communicate with your friend (Delhi) 
 
1
st 
 option involves the concept of particle & the second choice involves the concept of 
wave. 
 
When you say "Namaste" to your friend, no physical particles are sent from your lips to 
their ear. Instead, you create a disturbance in the air around your lips. This involves 
transferring energy to nearby air particles, either pushing them forward or pulling them 
back. This action temporarily changes the density of the air in that region. The disturbed 
air particles then affect the next layer of air, passing on the disturbance. This chain 
reaction continues until the air near your friend's ear is disturbed, enabling them to hear 
your greeting. 
 {Note: - In the above example air itself does not move.} 
A wave is a disturbance that propagates in space and transports energy and momentum 
from one point to another without the transport of matter. 
A few examples of waves: 
The ripples on a pond (water waves), the sound we hear, visible light, radio and TV 
signals, etc. 
Wave is a phenomenon of energy & momentum transfer from one part of the space to 
another part of space without the transfer of the bulk of the medium. 
? Stationary waves & progressive waves 
? One dimensional (e.g. String wave) 
? 2-dimensional (e.g. Ripple wave) 
? 3-dimensional (e.g. Light and sound waves) 
 
Classification of waves: 
 
 
WAVES 
Mechanical waves 
? Medium is required 
(e.g.) waves on string sound wave ? 
Transverse wave 
 
In this type of wave, the elements of the material move back and forth sideways, at right 
angles to the direction the wave travels. An example of this is the waves that move along a 
string. (e.g.) Waves on string 
Non-mechanical waves 
? Medium is not required 
(e.g.) Electromagnetic wave 
Light wave 
Page 3


WAVE ON STRING  
WAVE AND ITS CHARACTERISTICS 
Introduction of Waves 
What is wave motion? 
? When a particle moves through space, it carries KE with itself. Wherever the 
particle goes, the energy goes with it. (One way of transporting energy from one 
place to another place) 
? There is another way (wave motion) to transport energy from one part of space to 
another without any bulk motion of material together with it. Sound is 
transmitted in the air in this manner. 
Ex. You (Haryana) want to communicate with your friend (Delhi) 
 
1
st 
 option involves the concept of particle & the second choice involves the concept of 
wave. 
 
When you say "Namaste" to your friend, no physical particles are sent from your lips to 
their ear. Instead, you create a disturbance in the air around your lips. This involves 
transferring energy to nearby air particles, either pushing them forward or pulling them 
back. This action temporarily changes the density of the air in that region. The disturbed 
air particles then affect the next layer of air, passing on the disturbance. This chain 
reaction continues until the air near your friend's ear is disturbed, enabling them to hear 
your greeting. 
 {Note: - In the above example air itself does not move.} 
A wave is a disturbance that propagates in space and transports energy and momentum 
from one point to another without the transport of matter. 
A few examples of waves: 
The ripples on a pond (water waves), the sound we hear, visible light, radio and TV 
signals, etc. 
Wave is a phenomenon of energy & momentum transfer from one part of the space to 
another part of space without the transfer of the bulk of the medium. 
? Stationary waves & progressive waves 
? One dimensional (e.g. String wave) 
? 2-dimensional (e.g. Ripple wave) 
? 3-dimensional (e.g. Light and sound waves) 
 
Classification of waves: 
 
 
WAVES 
Mechanical waves 
? Medium is required 
(e.g.) waves on string sound wave ? 
Transverse wave 
 
In this type of wave, the elements of the material move back and forth sideways, at right 
angles to the direction the wave travels. An example of this is the waves that move along a 
string. (e.g.) Waves on string 
Non-mechanical waves 
? Medium is not required 
(e.g.) Electromagnetic wave 
Light wave 
Longitudinal wave 
? In this wave constituents of the medium oscillate along the direction of wave 
propagation. 
(e.g.) Sound wave 
(i) Based on medium necessity: - Waves can be categorized based on whether they 
need a medium to travel through or not. Waves that can travel without a medium are 
called non-mechanical waves, such as light, heat (infrared), and radio waves. On the 
other hand, waves that need a medium to move through are called mechanical waves. In 
the case of mechanical waves, the elasticity and density of the medium are crucial 
factors, which is why they are also known as elastic waves. 
Example: Sound waves in water, seismic waves in earth's crust. 
(ii) Based on energy propagation: - Waves can be divided into two parts based on 
energy propagation  
(i) Progressive wave  
(ii)  Stationary waves. 
 The progressive wave propagates with constant velocity in a medium. In 
stationary waves particles of the medium vibrate with different amplitude but 
energy does not propagate. 
(iii) Based on the direction of propagation: - Waves can exist in one, two, or three 
dimensions depending on how they carry energy. For example, waves traveling along 
strings are one-dimensional. Ripples on the surface of the water are two-dimensional. 
Sound or light waves from a point source spread out in three dimensions. 
(iv) Based on the motion of particles of the medium: 
 
Waves are of two types based on the motion of particles of the medium. 
(i) Longitudinal waves 
(ii) Transverse waves 
In a transverse wave, the disturbance (like the motion of particles in the medium) happens 
at a right angle to the direction the wave moves. However, in a longitudinal wave, the 
disturbance occurs in the same direction as the wave's movement. 
Page 4


WAVE ON STRING  
WAVE AND ITS CHARACTERISTICS 
Introduction of Waves 
What is wave motion? 
? When a particle moves through space, it carries KE with itself. Wherever the 
particle goes, the energy goes with it. (One way of transporting energy from one 
place to another place) 
? There is another way (wave motion) to transport energy from one part of space to 
another without any bulk motion of material together with it. Sound is 
transmitted in the air in this manner. 
Ex. You (Haryana) want to communicate with your friend (Delhi) 
 
1
st 
 option involves the concept of particle & the second choice involves the concept of 
wave. 
 
When you say "Namaste" to your friend, no physical particles are sent from your lips to 
their ear. Instead, you create a disturbance in the air around your lips. This involves 
transferring energy to nearby air particles, either pushing them forward or pulling them 
back. This action temporarily changes the density of the air in that region. The disturbed 
air particles then affect the next layer of air, passing on the disturbance. This chain 
reaction continues until the air near your friend's ear is disturbed, enabling them to hear 
your greeting. 
 {Note: - In the above example air itself does not move.} 
A wave is a disturbance that propagates in space and transports energy and momentum 
from one point to another without the transport of matter. 
A few examples of waves: 
The ripples on a pond (water waves), the sound we hear, visible light, radio and TV 
signals, etc. 
Wave is a phenomenon of energy & momentum transfer from one part of the space to 
another part of space without the transfer of the bulk of the medium. 
? Stationary waves & progressive waves 
? One dimensional (e.g. String wave) 
? 2-dimensional (e.g. Ripple wave) 
? 3-dimensional (e.g. Light and sound waves) 
 
Classification of waves: 
 
 
WAVES 
Mechanical waves 
? Medium is required 
(e.g.) waves on string sound wave ? 
Transverse wave 
 
In this type of wave, the elements of the material move back and forth sideways, at right 
angles to the direction the wave travels. An example of this is the waves that move along a 
string. (e.g.) Waves on string 
Non-mechanical waves 
? Medium is not required 
(e.g.) Electromagnetic wave 
Light wave 
Longitudinal wave 
? In this wave constituents of the medium oscillate along the direction of wave 
propagation. 
(e.g.) Sound wave 
(i) Based on medium necessity: - Waves can be categorized based on whether they 
need a medium to travel through or not. Waves that can travel without a medium are 
called non-mechanical waves, such as light, heat (infrared), and radio waves. On the 
other hand, waves that need a medium to move through are called mechanical waves. In 
the case of mechanical waves, the elasticity and density of the medium are crucial 
factors, which is why they are also known as elastic waves. 
Example: Sound waves in water, seismic waves in earth's crust. 
(ii) Based on energy propagation: - Waves can be divided into two parts based on 
energy propagation  
(i) Progressive wave  
(ii)  Stationary waves. 
 The progressive wave propagates with constant velocity in a medium. In 
stationary waves particles of the medium vibrate with different amplitude but 
energy does not propagate. 
(iii) Based on the direction of propagation: - Waves can exist in one, two, or three 
dimensions depending on how they carry energy. For example, waves traveling along 
strings are one-dimensional. Ripples on the surface of the water are two-dimensional. 
Sound or light waves from a point source spread out in three dimensions. 
(iv) Based on the motion of particles of the medium: 
 
Waves are of two types based on the motion of particles of the medium. 
(i) Longitudinal waves 
(ii) Transverse waves 
In a transverse wave, the disturbance (like the motion of particles in the medium) happens 
at a right angle to the direction the wave moves. However, in a longitudinal wave, the 
disturbance occurs in the same direction as the wave's movement. 
Wave Function OR Wave Equation: 
The wave function of a wave tells about the displacement of particles of the medium at a 
given time. 
(i) ?? =?? (?? ,?? )  [wave equation] 
(ii) It ' ?? ' is kept fixed (?? =?? 0
) in the wave equation then the resultant equation 
?? (?? ,?? =?? 0
)=?? (?? ) described the shape of the wave at given time ' ?? 
0
 
0
. 
(iii) If ' ?? ' is kept fixed ( ?? =?? 0
) in the wave equation then the resultant equation 
?? (?? =?? 0
,?? )=?? (?? ) represents the equation of motion of particle present at ?? =?? 0
. 
Finding wave equation if the equation of its shape at the same time is given: 
 
Consider the shape of the wave given by ?? =?? (?? ) at ?? =0 as shown in figure (1). Wave is 
moving with constant velocity (v) in +ve ?? -direction due to propagation of wave the new 
shape fig. (2) will be shifted by amount vt from its previous location. 
?  Wave equation at time ' t ' will be given by y=f(x-vt) 
?? (?? )??? (?? -???? )
?? (?? )??? (?? -?? /?? )
 
Example. ?? =?? (?? ) at ?? =?? 0
 
v?
+
 
Solution: ?? =?? (?? -?? (?? -?? 0
) wave equation 
Example. ?? =?? 2
+2?? at ?? =0 
3 m/s
-ve
 
Solution: ?? =(?? +3?? )
2
+2(?? +3?? ) is wave equation 
Example. ?? =4?? at ?? =1 
Page 5


WAVE ON STRING  
WAVE AND ITS CHARACTERISTICS 
Introduction of Waves 
What is wave motion? 
? When a particle moves through space, it carries KE with itself. Wherever the 
particle goes, the energy goes with it. (One way of transporting energy from one 
place to another place) 
? There is another way (wave motion) to transport energy from one part of space to 
another without any bulk motion of material together with it. Sound is 
transmitted in the air in this manner. 
Ex. You (Haryana) want to communicate with your friend (Delhi) 
 
1
st 
 option involves the concept of particle & the second choice involves the concept of 
wave. 
 
When you say "Namaste" to your friend, no physical particles are sent from your lips to 
their ear. Instead, you create a disturbance in the air around your lips. This involves 
transferring energy to nearby air particles, either pushing them forward or pulling them 
back. This action temporarily changes the density of the air in that region. The disturbed 
air particles then affect the next layer of air, passing on the disturbance. This chain 
reaction continues until the air near your friend's ear is disturbed, enabling them to hear 
your greeting. 
 {Note: - In the above example air itself does not move.} 
A wave is a disturbance that propagates in space and transports energy and momentum 
from one point to another without the transport of matter. 
A few examples of waves: 
The ripples on a pond (water waves), the sound we hear, visible light, radio and TV 
signals, etc. 
Wave is a phenomenon of energy & momentum transfer from one part of the space to 
another part of space without the transfer of the bulk of the medium. 
? Stationary waves & progressive waves 
? One dimensional (e.g. String wave) 
? 2-dimensional (e.g. Ripple wave) 
? 3-dimensional (e.g. Light and sound waves) 
 
Classification of waves: 
 
 
WAVES 
Mechanical waves 
? Medium is required 
(e.g.) waves on string sound wave ? 
Transverse wave 
 
In this type of wave, the elements of the material move back and forth sideways, at right 
angles to the direction the wave travels. An example of this is the waves that move along a 
string. (e.g.) Waves on string 
Non-mechanical waves 
? Medium is not required 
(e.g.) Electromagnetic wave 
Light wave 
Longitudinal wave 
? In this wave constituents of the medium oscillate along the direction of wave 
propagation. 
(e.g.) Sound wave 
(i) Based on medium necessity: - Waves can be categorized based on whether they 
need a medium to travel through or not. Waves that can travel without a medium are 
called non-mechanical waves, such as light, heat (infrared), and radio waves. On the 
other hand, waves that need a medium to move through are called mechanical waves. In 
the case of mechanical waves, the elasticity and density of the medium are crucial 
factors, which is why they are also known as elastic waves. 
Example: Sound waves in water, seismic waves in earth's crust. 
(ii) Based on energy propagation: - Waves can be divided into two parts based on 
energy propagation  
(i) Progressive wave  
(ii)  Stationary waves. 
 The progressive wave propagates with constant velocity in a medium. In 
stationary waves particles of the medium vibrate with different amplitude but 
energy does not propagate. 
(iii) Based on the direction of propagation: - Waves can exist in one, two, or three 
dimensions depending on how they carry energy. For example, waves traveling along 
strings are one-dimensional. Ripples on the surface of the water are two-dimensional. 
Sound or light waves from a point source spread out in three dimensions. 
(iv) Based on the motion of particles of the medium: 
 
Waves are of two types based on the motion of particles of the medium. 
(i) Longitudinal waves 
(ii) Transverse waves 
In a transverse wave, the disturbance (like the motion of particles in the medium) happens 
at a right angle to the direction the wave moves. However, in a longitudinal wave, the 
disturbance occurs in the same direction as the wave's movement. 
Wave Function OR Wave Equation: 
The wave function of a wave tells about the displacement of particles of the medium at a 
given time. 
(i) ?? =?? (?? ,?? )  [wave equation] 
(ii) It ' ?? ' is kept fixed (?? =?? 0
) in the wave equation then the resultant equation 
?? (?? ,?? =?? 0
)=?? (?? ) described the shape of the wave at given time ' ?? 
0
 
0
. 
(iii) If ' ?? ' is kept fixed ( ?? =?? 0
) in the wave equation then the resultant equation 
?? (?? =?? 0
,?? )=?? (?? ) represents the equation of motion of particle present at ?? =?? 0
. 
Finding wave equation if the equation of its shape at the same time is given: 
 
Consider the shape of the wave given by ?? =?? (?? ) at ?? =0 as shown in figure (1). Wave is 
moving with constant velocity (v) in +ve ?? -direction due to propagation of wave the new 
shape fig. (2) will be shifted by amount vt from its previous location. 
?  Wave equation at time ' t ' will be given by y=f(x-vt) 
?? (?? )??? (?? -???? )
?? (?? )??? (?? -?? /?? )
 
Example. ?? =?? (?? ) at ?? =?? 0
 
v?
+
 
Solution: ?? =?? (?? -?? (?? -?? 0
) wave equation 
Example. ?? =?? 2
+2?? at ?? =0 
3 m/s
-ve
 
Solution: ?? =(?? +3?? )
2
+2(?? +3?? ) is wave equation 
Example. ?? =4?? at ?? =1 
3 m/s
-ve
 
Solution: ?? =4[?? +3(?? -1)] 
Finding wave equation if the equation of motion of the same particle is given : 
?? =(?? ,?? )=?? (?? -
?? ?? ) ?? =0
?? =?? 0
 ?? =?? (?? )
?? =?? (?? -
(?? -?? 0
)
?? )
 
Example. ·y=e
2t
 for x=2 ?
3 m/s
 
Solution: ?? =?? 2(?? -
(?? -2)
3
)
 
?? =?? 2(?? +
(?? -2)
3
)
 
KEY PONT 
To represent a traveling wave the quantities ' ?? ' & ' ?? ' must appear in linear combination 
(keeping function defined). 
Example. Which of the following are valid wave functions?  
(1) ?? =?? sin
2
 (?? (?? -
?? ?? )) 
(2) ?? =?? -3?? +23
 
(3) ?? =sin (?? 2
-?? 2
?? 2
) 
(4) ?? =
1
?? -3?? 
?? (? Function is not always defined ) 
(E.g. function is not defined at =3&?? =1 ) 
 Velocity of wave =-
 Coefficient of ?? Coefficient of ?? 
Example. ?? =5sin (?? -3?? ) 
?? =-
-3
1
?+3