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

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.

We can write the equation of motion: Rearranging:

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

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

Amplitude resonances for normal modes ω = 5 ω = 7 o 1 β = 0.1

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

Electrical Analogs

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

LC Oscillator © SB/SPK

LCR Resonance

Using the ‘complex’ method of analysis 2β

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

(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.

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

Resonant frequency Period Figure of merit 2β

WAVES

http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

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

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.                                                                                                                                                                                                                                                                                               http://www.kettering.edu/~drussell/Demos/waves-intro/waves-intro.html 20

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.                                                                                                                                                                                                                                                                                               http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html 21

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 (1564-1642) 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. http://www.visionlearning.com/library/module_viewer.php?mid=102 22

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

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

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):

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.

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.

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

Disturbance in the rod ξ: Xi ©SB/SPK

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. ’

Elasticity : Spring constant

i i-1 i+1 ©SB/SPK

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

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

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

Let a: separation between the masses a  where 0

In the Continuum limit i i-1 i+1

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.

and

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

Wave equation Speed of the wave

For disturbance propagating in all directions (Laplacian operator)

Transverse vibrations in strings

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

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

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

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

Hence, eq. of motion is given by: 50

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

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

Taylor series expansion 53

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

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

For disturbance propagating in all directions (Laplacian operator) 56

Transverse wave on a string (String Wave)

Length of the curved element

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:

Equation of motion of small element Wave equation

Wave equation and solution

Wave equation Solution General Solution

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

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

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

Wave velocity

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

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

The number of waves that exist over a specified distance

c: wave velocity

t: period of oscillation

Different forms of solution of wave equations

Wave velocity Particle velocity

Particle velocity :

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

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

Boundary conditions Standing waves on a Stretched string l

-A

(n-1) nodes between boundaries

-A kl = nπ

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

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

WAVES Phase velocity Group velocity

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

T t

x

New position of at

Phase velocity =The speed with which the constant phase moves

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

Group velocity Superposition of two waves of almost equal frequencies:

Phase velocity Group velocity

vp < vg t = 0

vp < vg t = 1

vp < vg t = 2 Wave Velocity

vp < vg t = 3

vp < vg t = 4

vp < vg t = 5

vp < vg t = 6

vp < vg t = 7

vp < vg t = 8

vp < vg t = 9

vp < vg t = 10

vp = vg t = 0

vp = vg t = 1

vp = vg t = 2

vp = vg t = 3

vp = vg t = 4

vp = vg t = 5

vp = vg t = 6

vp = vg t = 7

vp = vg t = 8

vp = vg t = 9

vp = vg t = 10

vp > vg t = 0

vp > vg t = 1

vp > vg t = 2

vp > vg t = 3

vp > vg t = 4

vp > vg t = 5

vp > vg t = 6

vp > vg t = 7

vp > vg t = 8

vp > vg t = 9

vp > vg t = 10

The red dot moves with the phase velocity, and the green dots propagate with the group velocity. http://en.wikipedia.org/wiki/Phase_velocity

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

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.