1. Waves: Wave equation and solution
Electrical Analogs
Waves: Wave equation and solution
Wave Velocity

2. Coupled Oscillations Problem: Damped forced coupled oscillations
A force F0 Cos ωt is applied on one
of the masses. Write down the
equations of motion try to solve the
system qualitatively using your
knowledge of forced oscillations and
coupled oscillations.

3. We can write the equation of motion:
Rearranging:

4. Decouple the equations by identifying the normal modes
The solution:
where
Find

5. Find amplitudes of original masses
It can be shown that:
Considering damping to be very small

6. Amplitude resonances for normal modesω
= 5 ω
= 7 o 1 β
= 0.1

7. Amplitude resonances for normal modes
Amplitudes of mass 2
Mechanical Band-pass filter

8. Electrical Analogs

9. Simple harmonic oscillation
Forced oscillation w/o damping
Forced oscillation with damping

10. LC Oscillator
© SB/SPK

11. LCR Resonance

12. Using the ‘complex’ method of analysis2β

13. Complex impedance of the circuit??
Therefore,
Complex impedance of the circuit

14. (Inductively coupled resistance free LC circuits )
Coupled Oscillator
(Inductively coupled resistance free LC circuits )
Mutual inductance
From Pain
If the two circuits are coupled by flux linkage and allowed to oscillate at some frequency , then the voltage equations are:
Assuming that the primary current varies with eiωt.

15. General Characteristics Mechanical properties
Electrical Analog of Mechanical properties
General Characteristics
Mechanical properties
Electrical property
Indep. variable
Dep. Variable
Inertia
Resistance
Stiffness
Time (t)
Position (x)
Mass (m)
Drag coeff.
Stiffness (k)
Time (t)
Charge (q)
Inductance (L)
Resistance
(R=2β L)
(r=2β m)
(Capacitance)-1

16. Resonant frequency
Period
Figure of merit 2β

17. WAVES

19. Webster's dictionary defines a wave as
"a disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric or magnetic intensity, electric potential, or temperature." 19

20. Longitudinal Wave
In Longitudinal waves, the particles in a medium oscillate back and forth about their equilibrium positions but it is the disturbance which travels, not the individual particles in the medium.
                                                                                                                                                                                                                                                                                              20

21. Transverse wave
In a transverse wave the particle displacement is perpendicular to the direction of wave propagation. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by.
                                                                                                                                                                                                                                                                                              21

22. Is the concept very new or it has been known for many centuries?
Actually, most of current understanding of wave motion has come from the study of acoustics.
Ancient Greek philosophers, many of whom were interested in music, hypothesized that there was a connection between waves and sound, and that vibrations, or disturbances, must be responsible for sounds. Indian musicians had also understood the relationship between sound and vibrations.
In 550 BC, Pythogoras, after closely observing that vibrating strings produce sound, worked and gave mathematical relationships between the lengths of strings that made harmonious tones.
In 17th Century AD, Galileo ( ) published his pioneering work detailing the connection between vibrating bodies and the sounds they produce.
Boyle, in a classic experiment from 1660, proved that sound can not travel through a vacuum. 22

23. Put simply, “a wave is a traveling disturbance”.
Some examples....
Huge waves – “tsunamis”.
Cause of tsunamis - A major earthquake, itself consisting of waves traveling through the earth, which triggers an underwater landslide that creates the tsunamis.
Radio stations - transmitting electromagnetic radio waves.
.....many-many more examples can be listed.
Therefore, waves of one form or another can be found in an amazingly diverse range of physical applications, from the oceans to the science of sound.
Put simply, “a wave is a traveling disturbance”. 23

24. Lets try to find a link from what we have learnt till now so as to understand the phenomena of WAVES.
Coupled oscillations forms the natural link between simple harmonic motion (of a single particle) and wave motion (of a continuous infinity of particles as in a medium) 24

25. y T x Oscillation of a single particle : one frequency ω0;
Oscillation of a two coupled particles :
two normal frequency ω0, ω1
Oscillation of n coupled particles :
n normal frequencies ω0, ω1….. ωn-1 y x T
Oscillation of infinite no. of coupled particles (lattice/medium):

26. Two-Mass, Three-Spring System
Longitudinal Motion (along x-axis):
System equations
Normal frequencies
Multiple mass system
Each additional mass adds another natural mode of vibration per axis of motion.

27. Lets us extend the concept of oscillations of a coupled spring mass system to a linear chain of springs and masses.
Also, let there be long linear chain of identical springs of stiffness k connecting the identical blocks of mass m.
Suppose we consider the limit when the number of springs and masses tends continuously to infinity.
This kind of limiting case can be envisaged in an elastic rod (which is a continuous distribution of elasticity and inertia).
If longitudinal disturbances are created on such a rod, they propogate much as in a linear chain of springs and masses, with the discrete system replaced by a continuous system.

28. Elastic Wave A L
Rod made of elastic substance
©SB/SPK

29. Disturbance in the rod
ξ: Xi
©SB/SPK

30. Elasticity : Spring constant
What happens to an elastic solid when it is compressed or stretched?
Stress is the internal force (per unit area) associated with a strain.
Strain is the relative change in shape or size of an object due to externally applied forces. ’

31. Elasticity : Spring constant

33. i
i-1
i+1
©SB/SPK

34. Displacement of ith mass satisfies differential equation
July 31, 2018
Waves_2

35. Displacement of ith mass satisfies differential equation
is a function of two continuous variable x and t

36. Notation of partial derivatives
: variation of with t while x is kept constant
: variation of with x while t is kept constant

37. Let
a: separation between the masses
a  where

38. In the Continuum limit i
i-1
i+1

39. Taylor series expansion
A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

40. and

42. Longitudinal wave in elastic rod
Y: Young’s modulus
A: Cross sectional area
r=mass density
We have:
Wave equation
cs: wave velocity

43. Wave equation
Speed of the wave

44. For disturbance propagating in all directions
(Laplacian operator)

45. Transverse vibrations in strings

46. Loaded string case 2 1 m
yr-yr-1
yr-yr+1 yr
yr-1
yr+1 a 46 46

47. In-phase / Pendulum mode
Anti-phase / Breathing mode

48. The string is fixed at both ends; it has a length (n+1)a and
Loaded string case
a light string supporting n equal masses m spaced at equal distance a along its length
The string is fixed at both ends; it has a length (n+1)a and
a constant tension T exists at all points and all times in the string.
Small SHO of the masses are allowed in only one plane and the problem is to find the frequencies of the normal modes and the displacement of each mass in a particular normal mode. 48

49. 2 1 m
yr-yr-1
yr-yr+1 yr
yr-1
yr+1 a
The equation of motion of this mass may be written by considering the components of the tension directed towards the equilibrium position. The rth mass is pulled downwards towards the equilibrium position by a force Tsin1, due to the tension on its left and a force Tsin2 due to the tension on its right. 49

50. Hence, eq. of motion is given by:50

51. We know,
In a given mode all masses oscillate with the same mode frequency , so all yr’s have the same time dependence.
However, the transverse displacement yr also depends upon the value of r i.e., the position of the rth mass on the string.
This means that yr is a function of two independent variables, the time t and the location of r on the string.
If we use the separation a ≈ x and let x  0, the masses become closer and we can consider positions along the string in terms of a continuous variable x and any transverse displacement as y(x,t), a function of both x and t. In this case, partial derivative notations can be used. 51

52. If we now locate the transverse displacement yr at a position x = xr along the string then, we get:
where y is evaluated at x = xr, and now, as a = x  0, (in the Continuum limit) we may write
xr = x,
xr+1 = x + x and
xr-1 = x - x
with
yr(t)  y(x,t),
yr+1 (t)  y(x + x, t) and
yr-1 (t)  y(x - x, t) 52

53. Taylor series expansion53

54. So, the eq. of motion becomes after the substitution:
yr(t)  y(x,t),
yr+1 (t)  y(x + x, t)
yr-1 (t)  y(x - x, t) 54

55. This is the WAVE EQUATION.
If we now write m = x
where  is the linear density (mass per unit length) of the string,
the masses must 0 as x0 to avoid infinite mass density. Thus, we have:
This is the WAVE EQUATION.
T/ has the dimension of the square of the velocity, the velocity with which the waves; i.e., the phase of oscillation, is propagated. The solution for y at any particular point along the string is always that of a harmonic oscillator. 55

56. For disturbance propagating in all directions
(Laplacian operator) 56

57. Transverse wave on a string
(String Wave)

59. Length of the curved element

60. Perpendicular force on the element
Since  is very small, sin  = tan 
The difference between the two terms defines the differential coefficient of the partial derivative ξ/x times the space derivative dx:

61. Equation of motion of small element
Wave equation

62. Wave equation and solution

63. Wave equation
Solution
General Solution

64. Therefore, (nothing but wave equation)
‘Chain rule’
Therefore, (nothing but wave equation)

65. Physical significance of x = f(ct-x)
Disturbance: ξ = f(x,t)
If we photograph the wave at t=0: S x’ ct x S
After a time t, the pulse has moved a distance ct
Introduce S’ which travels with the pulse S’
ξ = f(x’)
x’ = x – ct
For someone at rest in S: ξ = f(x-ct)
So, x = f(x-ct) denotes a wave moving to right

66. Physical significance of x = f(ct+x)
wave moving to the left

67. Wave velocity

68. The bracket in x = f(ct-x) has dimension of length
If ξ is SHO of an oscillator at position x and time t, we can express it as:
The bracket in x = f(ct-x) has dimension of length
For this function to be a sine or cosine its argument must have dimensions of radians.
So, we can write the solution as :
Where

69. For a sinusoidal plane wave
x is a point along the x-axis. y and z are not part of the equation because the wave's magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are.
n: frequency of oscillation in time

70. The number of waves that exist over a specified distance

71. c: wave velocity

72. t: period of oscillation

73. Different forms of solution of wave equations

74. Wave velocity
Particle velocity

75. Particle velocity :

76. Dispersion Relation A link between spatial and temporal oscillations
For the wave equation
Plane wave solution
Oscillation frequency

77. For monochromatic wave in a non-dispersive medium
Plot dispersion relation w
w = ck
More
later k
Slope (c)  phase velocity of the wave

78. Boundary conditions
Standing waves on a Stretched string l

79. -A

80. (n-1) nodes between boundaries

81. -A
kl = nπ

82. For an arbitrary plucking, what
are the modes excited? F P

83. F P
All those modes having a
node at P will be absent.

84. WAVES
Phase velocity
Group velocity

85. Sinusoidal waves
They are Progressive Wave -A
This is standing wave

86. T t

87. x

88. New position of at

89. Phase velocity =The speed with which the
constant phase moves

90. So far: Monochromatic waves
In reality: Common to have group of component frequencies (White light)

91. Group velocity
Superposition of two waves of almost equal frequencies:

92. Phase velocity
Group velocity

94. vp < vg
t = 0

95. vp < vg
t = 1

96. vp < vg
t = 2
Wave Velocity

97. vp < vg
t = 3

98. vp < vg
t = 4

99. vp < vg
t = 5

100. vp < vg
t = 6

101. vp < vg
t = 7

102. vp < vg
t = 8

103. vp < vg
t = 9

104. vp < vg
t = 10

105. vp = vg
t = 0

106. vp = vg
t = 1

107. vp = vg
t = 2

108. vp = vg
t = 3

109. vp = vg
t = 4

110. vp = vg
t = 5

111. vp = vg
t = 6

112. vp = vg
t = 7

113. vp = vg
t = 8

114. vp = vg
t = 9

115. vp = vg
t = 10

116. vp > vg
t = 0

117. vp > vg
t = 1

118. vp > vg
t = 2

119. vp > vg
t = 3

120. vp > vg
t = 4

121. vp > vg
t = 5

122. vp > vg
t = 6

123. vp > vg
t = 7

124. vp > vg
t = 8

125. vp > vg
t = 9

126. vp > vg
t = 10

127. The red dot moves with the phase velocity, and the green dots propagate with the group velocity.

128. SUMMARY
Phase velocity: Velocity of the phase of the wave.
Group Velocity: Velocity of the maximum amplitude of the group.

129. Later we came across cases where vp < vg and vp > vg.
We have initially considered superposition of amplitude and phase of two waves whose phase velocities were equal.
Later we came across cases where vp < vg and vp > vg.
This happens when
Then,
- The group velocity is different from the individual velocities.
- The superposition is no longer constant and the group profile will change with time.