K.V. Lumding; K.V. Karimganj; K.V. Langjing
Oscillations and Waves
Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ωt [ω = 2π/T]
If the argument of this function ωt is increased by an integral multiple of 2π radians,
the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T=
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ωt + B cos ωt
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
187
,K.V. Lumding; K.V. Karimganj; K.V. Langjing
f(t) = D sin (ωt + ø)
D=√ and ø =
Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves
to and fro about a mean position under the action of a restoring force which is
directly proportional to its displacement from mean position and is always
directed towards mean position.
Restoring Force Displacement
F
Where ‘k’ is force constant.
Amplitude: Maximum displacement of oscillating particle from its mean position.
xMax =
Time Period: Time taken to complete one oscillation.
Frequency: . Unit of frequency is Hertz (Hz).
1 Hz = 1
Angular Frequency: = 2πν
S.I unit ω = rad
Phase:
1. The Phase of Vibrating particle at any instant gives the state of the particle
as regards its position and the direction of motion at that instant.
It is denoted by ø.
2. Initial phase or epoch: The phase of particle corresponding to time t = 0.
It is denoted by ø.
Displacement in SHM :
( ø0)
Where, = Displacement,
A = Amplitude
ωt = Angular Frequency
ø0 = Initial Phase.
188
, K.V. Lumding; K.V. Karimganj; K.V. Langjing
Case 1: When Particle is at mean position x = 0
v= √ =
vmax = =
Case 2: When Particle is at extreme position x =
v= √ =0
Acceleration
Case 3: When particle is at mean position x = 0,
acceleration = ( ) = 0.
Case 4: When particle is at extreme position then
acceleration =
Formula Used :
1. ( ø0)
2. v = √ , vmax = ωA.
3. ω2 ( ø0)
amax = ω2A
4. Restoring force F = = mω2
Where = force constant & ω2 =
5. Angular freq. ω = 2 = ⁄
6. Time Period T = 2π√ = 2π√
7. Time Period T = 2π√ = 2π√
8. P.E at displacement ‘y’ from mean position
EP = ky2= mω2y2= mω2A2 sin2ωt
189
Oscillations and Waves
Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ωt [ω = 2π/T]
If the argument of this function ωt is increased by an integral multiple of 2π radians,
the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T=
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ωt + B cos ωt
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
187
,K.V. Lumding; K.V. Karimganj; K.V. Langjing
f(t) = D sin (ωt + ø)
D=√ and ø =
Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves
to and fro about a mean position under the action of a restoring force which is
directly proportional to its displacement from mean position and is always
directed towards mean position.
Restoring Force Displacement
F
Where ‘k’ is force constant.
Amplitude: Maximum displacement of oscillating particle from its mean position.
xMax =
Time Period: Time taken to complete one oscillation.
Frequency: . Unit of frequency is Hertz (Hz).
1 Hz = 1
Angular Frequency: = 2πν
S.I unit ω = rad
Phase:
1. The Phase of Vibrating particle at any instant gives the state of the particle
as regards its position and the direction of motion at that instant.
It is denoted by ø.
2. Initial phase or epoch: The phase of particle corresponding to time t = 0.
It is denoted by ø.
Displacement in SHM :
( ø0)
Where, = Displacement,
A = Amplitude
ωt = Angular Frequency
ø0 = Initial Phase.
188
, K.V. Lumding; K.V. Karimganj; K.V. Langjing
Case 1: When Particle is at mean position x = 0
v= √ =
vmax = =
Case 2: When Particle is at extreme position x =
v= √ =0
Acceleration
Case 3: When particle is at mean position x = 0,
acceleration = ( ) = 0.
Case 4: When particle is at extreme position then
acceleration =
Formula Used :
1. ( ø0)
2. v = √ , vmax = ωA.
3. ω2 ( ø0)
amax = ω2A
4. Restoring force F = = mω2
Where = force constant & ω2 =
5. Angular freq. ω = 2 = ⁄
6. Time Period T = 2π√ = 2π√
7. Time Period T = 2π√ = 2π√
8. P.E at displacement ‘y’ from mean position
EP = ky2= mω2y2= mω2A2 sin2ωt
189
