Oscillations & Waves Class 11 Notes Physics Chapter 13

 Page 1


Oscillation s and Waves 
If a particle in periodic motion moves back and forth (or to and 
fro) over the same path, then its motion is called oscillatory or 
vibratory.  
Characteristics of a Harmonic Motion 
The basic quantities characterizing a periodic motion are 
the amplitude, period and frequency of vibrations. 
Amplitude (A) 
The amplitude of oscillations is the maximum displacement 
of a vibrating body from the position of equilibrium.  
Time Period (T) 
The time period of oscillations is defined as the time 
between two successive identical positions passed by  the 
body in the same direction.  
Frequency (f) 
The frequency of oscillations is the number of cycles of 
vibrations of a body completed in one second. The 
frequency is related to the time period as  
Page 2


Oscillation s and Waves 
If a particle in periodic motion moves back and forth (or to and 
fro) over the same path, then its motion is called oscillatory or 
vibratory.  
Characteristics of a Harmonic Motion 
The basic quantities characterizing a periodic motion are 
the amplitude, period and frequency of vibrations. 
Amplitude (A) 
The amplitude of oscillations is the maximum displacement 
of a vibrating body from the position of equilibrium.  
Time Period (T) 
The time period of oscillations is defined as the time 
between two successive identical positions passed by  the 
body in the same direction.  
Frequency (f) 
The frequency of oscillations is the number of cycles of 
vibrations of a body completed in one second. The 
frequency is related to the time period as  
    f = 
T
1
     
  The SI unit of frequency is s
-1
 or Hz (hertz)  
 
Simple Harmonic Motion  
Let us consider an oscillatory particle along a straight line 
whose potential energy function varies as 
  U(x) = 
2
2
1
kx   
  where k is a constant  
 
Simple Harmonic Motion. 
 x = A sin ( ?t + ?) is  the general equation of SHM. 
   
        
      
 Above equation is the standard differential equation of 
SHM. 
 
 
The Spring-Mass System  
O r
Page 3


Oscillation s and Waves 
If a particle in periodic motion moves back and forth (or to and 
fro) over the same path, then its motion is called oscillatory or 
vibratory.  
Characteristics of a Harmonic Motion 
The basic quantities characterizing a periodic motion are 
the amplitude, period and frequency of vibrations. 
Amplitude (A) 
The amplitude of oscillations is the maximum displacement 
of a vibrating body from the position of equilibrium.  
Time Period (T) 
The time period of oscillations is defined as the time 
between two successive identical positions passed by  the 
body in the same direction.  
Frequency (f) 
The frequency of oscillations is the number of cycles of 
vibrations of a body completed in one second. The 
frequency is related to the time period as  
    f = 
T
1
     
  The SI unit of frequency is s
-1
 or Hz (hertz)  
 
Simple Harmonic Motion  
Let us consider an oscillatory particle along a straight line 
whose potential energy function varies as 
  U(x) = 
2
2
1
kx   
  where k is a constant  
 
Simple Harmonic Motion. 
 x = A sin ( ?t + ?) is  the general equation of SHM. 
   
        
      
 Above equation is the standard differential equation of 
SHM. 
 
 
The Spring-Mass System  
O r
 Time period of a spring-mass is given by  
    T = 2 ?
k
m
     
Series and Parallel Combinations of springs  
  For Series Combinations of springs , the equivalent  
stiffness of the combination is given by  
  
12
1 2 1 2
1 1 1 k k
k
kk
? ? ? ?
? k k k
    
 For parallel Combinations of springs, the equivalent  stiffness 
of the combination is given by  
   k = k
1
 + k
2
    
 
 
ENERGY CONSERVATION IN SHM 
  In a spring-mass system, the instantaneous potential energy 
and kinetic energy are expressed as  
   U = ? ? ? ? ? ? t kA kx
2 2 2
sin
2
1
2
1
    
  and K =  ? ? ? ? ? ? ? t A m mv
2 2 2 2
cos
2
1
2
1
    
  Since ?
2
 = 
m
k
, therefore,  
Page 4


Oscillation s and Waves 
If a particle in periodic motion moves back and forth (or to and 
fro) over the same path, then its motion is called oscillatory or 
vibratory.  
Characteristics of a Harmonic Motion 
The basic quantities characterizing a periodic motion are 
the amplitude, period and frequency of vibrations. 
Amplitude (A) 
The amplitude of oscillations is the maximum displacement 
of a vibrating body from the position of equilibrium.  
Time Period (T) 
The time period of oscillations is defined as the time 
between two successive identical positions passed by  the 
body in the same direction.  
Frequency (f) 
The frequency of oscillations is the number of cycles of 
vibrations of a body completed in one second. The 
frequency is related to the time period as  
    f = 
T
1
     
  The SI unit of frequency is s
-1
 or Hz (hertz)  
 
Simple Harmonic Motion  
Let us consider an oscillatory particle along a straight line 
whose potential energy function varies as 
  U(x) = 
2
2
1
kx   
  where k is a constant  
 
Simple Harmonic Motion. 
 x = A sin ( ?t + ?) is  the general equation of SHM. 
   
        
      
 Above equation is the standard differential equation of 
SHM. 
 
 
The Spring-Mass System  
O r
 Time period of a spring-mass is given by  
    T = 2 ?
k
m
     
Series and Parallel Combinations of springs  
  For Series Combinations of springs , the equivalent  
stiffness of the combination is given by  
  
12
1 2 1 2
1 1 1 k k
k
kk
? ? ? ?
? k k k
    
 For parallel Combinations of springs, the equivalent  stiffness 
of the combination is given by  
   k = k
1
 + k
2
    
 
 
ENERGY CONSERVATION IN SHM 
  In a spring-mass system, the instantaneous potential energy 
and kinetic energy are expressed as  
   U = ? ? ? ? ? ? t kA kx
2 2 2
sin
2
1
2
1
    
  and K =  ? ? ? ? ? ? ? t A m mv
2 2 2 2
cos
2
1
2
1
    
  Since ?
2
 = 
m
k
, therefore,  
   K = ? ? ? ? ? t kA
2 2
cos
2
1
 
  The total mechanical energy is given by 
   E = K + U  or E = ? ? ? ? ? ? ? ? ? ? ? ? ? t t kA
2 2 2
cos sin
2
1
 
  or E = 
2
2
1
kA = constant   
  Thus, the total energy of 
SHM is constant and 
proportional to the 
square of the amplitude. 
  The variation of K and 
U as function of x is 
shown in figure. When 
x = ?A, the kinetic 
energy is zero and the 
total energy is equal to 
the maximum potential 
energy.  
   E = U
max
= 
2
2
1
kA 
 
Energy 
E 
U(x) 
K(x) x 
+A -A 
The variation of the kinetic energy , potential 
energy, and total energy as a function of 
position.  
 
  There are extreme points or turning points of the SHM. 
  At  x = 0, U = 0 and the energy is purely kinetic,  
  i.e. E =K
max
 =  ? ?
2
2
1
A m ?
 
Page 5


Oscillation s and Waves 
If a particle in periodic motion moves back and forth (or to and 
fro) over the same path, then its motion is called oscillatory or 
vibratory.  
Characteristics of a Harmonic Motion 
The basic quantities characterizing a periodic motion are 
the amplitude, period and frequency of vibrations. 
Amplitude (A) 
The amplitude of oscillations is the maximum displacement 
of a vibrating body from the position of equilibrium.  
Time Period (T) 
The time period of oscillations is defined as the time 
between two successive identical positions passed by  the 
body in the same direction.  
Frequency (f) 
The frequency of oscillations is the number of cycles of 
vibrations of a body completed in one second. The 
frequency is related to the time period as  
    f = 
T
1
     
  The SI unit of frequency is s
-1
 or Hz (hertz)  
 
Simple Harmonic Motion  
Let us consider an oscillatory particle along a straight line 
whose potential energy function varies as 
  U(x) = 
2
2
1
kx   
  where k is a constant  
 
Simple Harmonic Motion. 
 x = A sin ( ?t + ?) is  the general equation of SHM. 
   
        
      
 Above equation is the standard differential equation of 
SHM. 
 
 
The Spring-Mass System  
O r
 Time period of a spring-mass is given by  
    T = 2 ?
k
m
     
Series and Parallel Combinations of springs  
  For Series Combinations of springs , the equivalent  
stiffness of the combination is given by  
  
12
1 2 1 2
1 1 1 k k
k
kk
? ? ? ?
? k k k
    
 For parallel Combinations of springs, the equivalent  stiffness 
of the combination is given by  
   k = k
1
 + k
2
    
 
 
ENERGY CONSERVATION IN SHM 
  In a spring-mass system, the instantaneous potential energy 
and kinetic energy are expressed as  
   U = ? ? ? ? ? ? t kA kx
2 2 2
sin
2
1
2
1
    
  and K =  ? ? ? ? ? ? ? t A m mv
2 2 2 2
cos
2
1
2
1
    
  Since ?
2
 = 
m
k
, therefore,  
   K = ? ? ? ? ? t kA
2 2
cos
2
1
 
  The total mechanical energy is given by 
   E = K + U  or E = ? ? ? ? ? ? ? ? ? ? ? ? ? t t kA
2 2 2
cos sin
2
1
 
  or E = 
2
2
1
kA = constant   
  Thus, the total energy of 
SHM is constant and 
proportional to the 
square of the amplitude. 
  The variation of K and 
U as function of x is 
shown in figure. When 
x = ?A, the kinetic 
energy is zero and the 
total energy is equal to 
the maximum potential 
energy.  
   E = U
max
= 
2
2
1
kA 
 
Energy 
E 
U(x) 
K(x) x 
+A -A 
The variation of the kinetic energy , potential 
energy, and total energy as a function of 
position.  
 
  There are extreme points or turning points of the SHM. 
  At  x = 0, U = 0 and the energy is purely kinetic,  
  i.e. E =K
max
 =  ? ?
2
2
1
A m ?
 
WAVE 
  The wave function  
   y = A sin [k(x ?vt)] 
   y = A sin (kx ? t)    
  The negative sign is used when the wave travels along the 
positive x – axis, and vice-versa.  
   
Some Important Points 
  k = 
?
? 2
   
  
         
  where  is called the angular frequency (measured in 
rad/s)and T is the time period and f is the frequency.  
Time Period (T) 
                           T = 
1
f
 
 Frequency ( f ) 
  The number of complete vibrations of a point on the string 
that occur in one second or, the number of wavelengths that 
pass a given point in one second.  
k	is	called	the	wave	number,	and	?	is	called	the	wavelength.