Oscillations & Waves Class 11 Notes Physics Chapter 13

 Page 1


Physics Class XI
246
10.1 Periodic Motion
A motion, which repeat itself over and over again after a regular interval of 
time is called a periodic motion and the fixed interval of time after which 
the motion is repeated is called period of the motion. Examples : Revolution 
of earth around the sun (period one year).
10.2 Oscillatory or Vibratory Motion.
The motion in which a body moves to and fro or back and forth repeatedly 
about a fixed point in a definite interval of time. Oscillatory motion is also 
called as harmonic motion. Example : The motion of the pendulum of a wall 
clock.
10.3 Harmonic and Non-harmonic Oscillation.
Harmonic oscillation is that oscillation which can be expressed in terms of 
single harmonic function (i.e. sine or cosine function). Example : y = a sin 
?t or y = a cos ?t.
Non-harmonic oscillation is that oscillation which can not be expressed in 
terms of single harmonic function. Example : y = a sin ?t + b sin 2 ?t.
10.4 Some Important Definitions.
(1) Time period : It is the least interval of time after which the periodic 
motion of a body repeats itself. S.l. units of time period is second.
(2) Frequency : It is defined as the number of periodic motions executed 
by body per second. S.l unit of frequency is hertz (Hz).
(3) Angular Frequency :
(4) Displacement: Its deviation from the mean position.
2pn
Page 2


Physics Class XI
246
10.1 Periodic Motion
A motion, which repeat itself over and over again after a regular interval of 
time is called a periodic motion and the fixed interval of time after which 
the motion is repeated is called period of the motion. Examples : Revolution 
of earth around the sun (period one year).
10.2 Oscillatory or Vibratory Motion.
The motion in which a body moves to and fro or back and forth repeatedly 
about a fixed point in a definite interval of time. Oscillatory motion is also 
called as harmonic motion. Example : The motion of the pendulum of a wall 
clock.
10.3 Harmonic and Non-harmonic Oscillation.
Harmonic oscillation is that oscillation which can be expressed in terms of 
single harmonic function (i.e. sine or cosine function). Example : y = a sin 
?t or y = a cos ?t.
Non-harmonic oscillation is that oscillation which can not be expressed in 
terms of single harmonic function. Example : y = a sin ?t + b sin 2 ?t.
10.4 Some Important Definitions.
(1) Time period : It is the least interval of time after which the periodic 
motion of a body repeats itself. S.l. units of time period is second.
(2) Frequency : It is defined as the number of periodic motions executed 
by body per second. S.l unit of frequency is hertz (Hz).
(3) Angular Frequency :
(4) Displacement: Its deviation from the mean position.
2pn
(5)  Phase : It is a physical quantity, which completely express the position 
and direction of motion, of the particle at that instant with respect to its 
mean position.
Y = a sin ? = a sin (?t + f
0
) here ? = ?t + f
0
  = phase of vibrating 
particle.
(i) Initial phase or epoch : It is the phase of a vibrating particle  
at t= 0.
(ii) Same phase: Two vibrating particle are said to be in same phase, 
if the phase difference between them is an even multiple of n or 
path difference is an even multiple of (?/2) or time interval is an 
even multiple of (T/2).
(iii) Opposite phase : Opposite phase means the phase difference 
between the particle is an odd multiple of  or the path difference 
is an odd multiple of ? or the time interval is an odd multiple of 
(T/2).
(iv) Phase difference : If two particles performs S.H.M and their 
equation are y
1
 = a sin (?t + f
1
) and y
2
 = a sin (?t + f
2
) then phase 
difference ?f = (?t + f
2
) –  (?t + f
1
) = f
2
 – f
1
10.5 Simple Harmonic Motion.
Simple harmonic motion is a special type of periodic motion, in which 
Restoring force ? Displacement of the particle from mean position.
 F = – kx
Where k is known as force constant. Its S.l. unit is Newton/meter and 
dimension is [MT
–2
].
10.6 Displacement in S.H.M.
Simple harmonic motion is defined as the projection of uniform circular 
motion on any diameter of circle of reference
(i) y = a sin ?t when at t = 0 the vibrating particle is at mean position.
(ii) y = a cos ?t when at t = 0 the vibrating particle is at extreme position.
(iii)  y = a sin (?t ± f) when the vibrating particle is f phase leading or 
lagging from the mean position.
Page 3


Physics Class XI
246
10.1 Periodic Motion
A motion, which repeat itself over and over again after a regular interval of 
time is called a periodic motion and the fixed interval of time after which 
the motion is repeated is called period of the motion. Examples : Revolution 
of earth around the sun (period one year).
10.2 Oscillatory or Vibratory Motion.
The motion in which a body moves to and fro or back and forth repeatedly 
about a fixed point in a definite interval of time. Oscillatory motion is also 
called as harmonic motion. Example : The motion of the pendulum of a wall 
clock.
10.3 Harmonic and Non-harmonic Oscillation.
Harmonic oscillation is that oscillation which can be expressed in terms of 
single harmonic function (i.e. sine or cosine function). Example : y = a sin 
?t or y = a cos ?t.
Non-harmonic oscillation is that oscillation which can not be expressed in 
terms of single harmonic function. Example : y = a sin ?t + b sin 2 ?t.
10.4 Some Important Definitions.
(1) Time period : It is the least interval of time after which the periodic 
motion of a body repeats itself. S.l. units of time period is second.
(2) Frequency : It is defined as the number of periodic motions executed 
by body per second. S.l unit of frequency is hertz (Hz).
(3) Angular Frequency :
(4) Displacement: Its deviation from the mean position.
2pn
(5)  Phase : It is a physical quantity, which completely express the position 
and direction of motion, of the particle at that instant with respect to its 
mean position.
Y = a sin ? = a sin (?t + f
0
) here ? = ?t + f
0
  = phase of vibrating 
particle.
(i) Initial phase or epoch : It is the phase of a vibrating particle  
at t= 0.
(ii) Same phase: Two vibrating particle are said to be in same phase, 
if the phase difference between them is an even multiple of n or 
path difference is an even multiple of (?/2) or time interval is an 
even multiple of (T/2).
(iii) Opposite phase : Opposite phase means the phase difference 
between the particle is an odd multiple of  or the path difference 
is an odd multiple of ? or the time interval is an odd multiple of 
(T/2).
(iv) Phase difference : If two particles performs S.H.M and their 
equation are y
1
 = a sin (?t + f
1
) and y
2
 = a sin (?t + f
2
) then phase 
difference ?f = (?t + f
2
) –  (?t + f
1
) = f
2
 – f
1
10.5 Simple Harmonic Motion.
Simple harmonic motion is a special type of periodic motion, in which 
Restoring force ? Displacement of the particle from mean position.
 F = – kx
Where k is known as force constant. Its S.l. unit is Newton/meter and 
dimension is [MT
–2
].
10.6 Displacement in S.H.M.
Simple harmonic motion is defined as the projection of uniform circular 
motion on any diameter of circle of reference
(i) y = a sin ?t when at t = 0 the vibrating particle is at mean position.
(ii) y = a cos ?t when at t = 0 the vibrating particle is at extreme position.
(iii)  y = a sin (?t ± f) when the vibrating particle is f phase leading or 
lagging from the mean position.
10.7 Comparative Study of Displacement, Velocity and Ac-
celeration.
Displacement y  = a sin ?t
Velocity v  = a? cos ?t
?t  = 
Acceleration A  = – a?
2
 sin ?t
?t  = a?
2
 sin (?t + p)
(i) All the three quantities displacement, velocity and acceleration show 
harmonic variation with time having same period.
(ii) The velocity amplitude is ? times the displacement amplitude
(iii) The acceleration amplitude is ?
2
 times the displacement amplitude
(iv) In S.H.M. the velocity is ahead of displacement by a phase angle p/2.
(v) In S.H.M. the acceleration is ahead of velocity by a phase angle p/2.
(vi)  The acceleration is ahead of displacement by a phase angle of p.
(vii)  Various physical quantities in S.H.M. at different position :
Page 4


Physics Class XI
246
10.1 Periodic Motion
A motion, which repeat itself over and over again after a regular interval of 
time is called a periodic motion and the fixed interval of time after which 
the motion is repeated is called period of the motion. Examples : Revolution 
of earth around the sun (period one year).
10.2 Oscillatory or Vibratory Motion.
The motion in which a body moves to and fro or back and forth repeatedly 
about a fixed point in a definite interval of time. Oscillatory motion is also 
called as harmonic motion. Example : The motion of the pendulum of a wall 
clock.
10.3 Harmonic and Non-harmonic Oscillation.
Harmonic oscillation is that oscillation which can be expressed in terms of 
single harmonic function (i.e. sine or cosine function). Example : y = a sin 
?t or y = a cos ?t.
Non-harmonic oscillation is that oscillation which can not be expressed in 
terms of single harmonic function. Example : y = a sin ?t + b sin 2 ?t.
10.4 Some Important Definitions.
(1) Time period : It is the least interval of time after which the periodic 
motion of a body repeats itself. S.l. units of time period is second.
(2) Frequency : It is defined as the number of periodic motions executed 
by body per second. S.l unit of frequency is hertz (Hz).
(3) Angular Frequency :
(4) Displacement: Its deviation from the mean position.
2pn
(5)  Phase : It is a physical quantity, which completely express the position 
and direction of motion, of the particle at that instant with respect to its 
mean position.
Y = a sin ? = a sin (?t + f
0
) here ? = ?t + f
0
  = phase of vibrating 
particle.
(i) Initial phase or epoch : It is the phase of a vibrating particle  
at t= 0.
(ii) Same phase: Two vibrating particle are said to be in same phase, 
if the phase difference between them is an even multiple of n or 
path difference is an even multiple of (?/2) or time interval is an 
even multiple of (T/2).
(iii) Opposite phase : Opposite phase means the phase difference 
between the particle is an odd multiple of  or the path difference 
is an odd multiple of ? or the time interval is an odd multiple of 
(T/2).
(iv) Phase difference : If two particles performs S.H.M and their 
equation are y
1
 = a sin (?t + f
1
) and y
2
 = a sin (?t + f
2
) then phase 
difference ?f = (?t + f
2
) –  (?t + f
1
) = f
2
 – f
1
10.5 Simple Harmonic Motion.
Simple harmonic motion is a special type of periodic motion, in which 
Restoring force ? Displacement of the particle from mean position.
 F = – kx
Where k is known as force constant. Its S.l. unit is Newton/meter and 
dimension is [MT
–2
].
10.6 Displacement in S.H.M.
Simple harmonic motion is defined as the projection of uniform circular 
motion on any diameter of circle of reference
(i) y = a sin ?t when at t = 0 the vibrating particle is at mean position.
(ii) y = a cos ?t when at t = 0 the vibrating particle is at extreme position.
(iii)  y = a sin (?t ± f) when the vibrating particle is f phase leading or 
lagging from the mean position.
10.7 Comparative Study of Displacement, Velocity and Ac-
celeration.
Displacement y  = a sin ?t
Velocity v  = a? cos ?t
?t  = 
Acceleration A  = – a?
2
 sin ?t
?t  = a?
2
 sin (?t + p)
(i) All the three quantities displacement, velocity and acceleration show 
harmonic variation with time having same period.
(ii) The velocity amplitude is ? times the displacement amplitude
(iii) The acceleration amplitude is ?
2
 times the displacement amplitude
(iv) In S.H.M. the velocity is ahead of displacement by a phase angle p/2.
(v) In S.H.M. the acceleration is ahead of velocity by a phase angle p/2.
(vi)  The acceleration is ahead of displacement by a phase angle of p.
(vii)  Various physical quantities in S.H.M. at different position :
249
Physical quantities Equilibrium position (y = 0) Extreme Position (y =  ± a)
Displacement y = a sin ?t Minimum (Zero) Maximum (a)
Velocity Maximum (a?) Minimum (Zero)
Acceleration A = – ?
2
y Minimum (Zero) Maximum (?
2
a)
10.8 Energy in S.H.M.
A particle executing S.H.M. possesses two types of energy : Potential energy 
and Kinetic energy
(1)  Potential energy : 
 (i)    when y = ± a; ?t = p/2; t = T/4
 (ii)     when y = 0; ?t = 0; t = 0
(2)  Kinetic energy :
 or  
 (i)       when y = 0; t = 0; ?t = 0
 (ii)       when y = a; t = T/4, ?t = p/2
(3) Total energy : Total mechanical energy 
 =  Kinetic energy + Potential energy
Total energy is not a position function i.e. it always remains constant.
(4)  Energy position graph :
Page 5


Physics Class XI
246
10.1 Periodic Motion
A motion, which repeat itself over and over again after a regular interval of 
time is called a periodic motion and the fixed interval of time after which 
the motion is repeated is called period of the motion. Examples : Revolution 
of earth around the sun (period one year).
10.2 Oscillatory or Vibratory Motion.
The motion in which a body moves to and fro or back and forth repeatedly 
about a fixed point in a definite interval of time. Oscillatory motion is also 
called as harmonic motion. Example : The motion of the pendulum of a wall 
clock.
10.3 Harmonic and Non-harmonic Oscillation.
Harmonic oscillation is that oscillation which can be expressed in terms of 
single harmonic function (i.e. sine or cosine function). Example : y = a sin 
?t or y = a cos ?t.
Non-harmonic oscillation is that oscillation which can not be expressed in 
terms of single harmonic function. Example : y = a sin ?t + b sin 2 ?t.
10.4 Some Important Definitions.
(1) Time period : It is the least interval of time after which the periodic 
motion of a body repeats itself. S.l. units of time period is second.
(2) Frequency : It is defined as the number of periodic motions executed 
by body per second. S.l unit of frequency is hertz (Hz).
(3) Angular Frequency :
(4) Displacement: Its deviation from the mean position.
2pn
(5)  Phase : It is a physical quantity, which completely express the position 
and direction of motion, of the particle at that instant with respect to its 
mean position.
Y = a sin ? = a sin (?t + f
0
) here ? = ?t + f
0
  = phase of vibrating 
particle.
(i) Initial phase or epoch : It is the phase of a vibrating particle  
at t= 0.
(ii) Same phase: Two vibrating particle are said to be in same phase, 
if the phase difference between them is an even multiple of n or 
path difference is an even multiple of (?/2) or time interval is an 
even multiple of (T/2).
(iii) Opposite phase : Opposite phase means the phase difference 
between the particle is an odd multiple of  or the path difference 
is an odd multiple of ? or the time interval is an odd multiple of 
(T/2).
(iv) Phase difference : If two particles performs S.H.M and their 
equation are y
1
 = a sin (?t + f
1
) and y
2
 = a sin (?t + f
2
) then phase 
difference ?f = (?t + f
2
) –  (?t + f
1
) = f
2
 – f
1
10.5 Simple Harmonic Motion.
Simple harmonic motion is a special type of periodic motion, in which 
Restoring force ? Displacement of the particle from mean position.
 F = – kx
Where k is known as force constant. Its S.l. unit is Newton/meter and 
dimension is [MT
–2
].
10.6 Displacement in S.H.M.
Simple harmonic motion is defined as the projection of uniform circular 
motion on any diameter of circle of reference
(i) y = a sin ?t when at t = 0 the vibrating particle is at mean position.
(ii) y = a cos ?t when at t = 0 the vibrating particle is at extreme position.
(iii)  y = a sin (?t ± f) when the vibrating particle is f phase leading or 
lagging from the mean position.
10.7 Comparative Study of Displacement, Velocity and Ac-
celeration.
Displacement y  = a sin ?t
Velocity v  = a? cos ?t
?t  = 
Acceleration A  = – a?
2
 sin ?t
?t  = a?
2
 sin (?t + p)
(i) All the three quantities displacement, velocity and acceleration show 
harmonic variation with time having same period.
(ii) The velocity amplitude is ? times the displacement amplitude
(iii) The acceleration amplitude is ?
2
 times the displacement amplitude
(iv) In S.H.M. the velocity is ahead of displacement by a phase angle p/2.
(v) In S.H.M. the acceleration is ahead of velocity by a phase angle p/2.
(vi)  The acceleration is ahead of displacement by a phase angle of p.
(vii)  Various physical quantities in S.H.M. at different position :
249
Physical quantities Equilibrium position (y = 0) Extreme Position (y =  ± a)
Displacement y = a sin ?t Minimum (Zero) Maximum (a)
Velocity Maximum (a?) Minimum (Zero)
Acceleration A = – ?
2
y Minimum (Zero) Maximum (?
2
a)
10.8 Energy in S.H.M.
A particle executing S.H.M. possesses two types of energy : Potential energy 
and Kinetic energy
(1)  Potential energy : 
 (i)    when y = ± a; ?t = p/2; t = T/4
 (ii)     when y = 0; ?t = 0; t = 0
(2)  Kinetic energy :
 or  
 (i)       when y = 0; t = 0; ?t = 0
 (ii)       when y = a; t = T/4, ?t = p/2
(3) Total energy : Total mechanical energy 
 =  Kinetic energy + Potential energy
Total energy is not a position function i.e. it always remains constant.
(4)  Energy position graph :
250
(5)  Kinetic energy and potential energy vary periodically double the 
frequency of S.H.M.
10.9 Time Period and Frequency of S.H.M.
  Time period (T) =   as   = 
Frequency (n) =  = 
In general m is called inertia factor and k is called spring factor.
Thus                T = 2p
10.10 Differential Equation of S.H.M.
For S.H.M. (linear)      [As 
For angular S.H.M. 
10.11 Simple Pendulum
Mass of the bob = m
Effective length of simple pendulum = l ; T = 2p 
(i) Time period of simple pendulum is independent of amplitude as long 
as its motion is simple harmonic.
(ii) Time period of simple pendulum is also independent of mass of the 
bob. 
(iii) If the length of the pendulum is comparable to the radius of earth 
then 
If  l >> R (?8) 1/l< 1/R   so  84.6 minutes
(iv) The time period of simple pendulum whose point of suspension moving  
2p
2p