− +
( )
F
m b
m
d d
0
2 2 2 2 1 2
( )
/
w w w
and tanf w= −vxd00
where w is the natural angular frequency of the oscillator, x0 and v0 are the displacement and velocity of the oscillator at time t = 0, when the periodic force is applied.
Special cases :
Case I. For small damping driven frequency is far from natural frequency
wd2b2 << m(w2 – wd2)2
\ A F
m d
= (w2−w2)
Case II. When driven frequency is close to natural frequency
m(w2 – wd2) ≈ 0
\ A F
db
= 2pw
SELFCHECK
11. A simple harmonic oscillator of angular frequency 2 rad s–1 is acted upon by an external force F = sint N.
If the oscillator is at rest in its equilibrium position at t = 0, its position at later times is proportional to (a) sint+ 1sin t
2 2 (b) sint+ 1cos t
2 2
(c) cost− 1sin t
2 2 (d) sint− 1sin t
2 2
(JEE Main 2015) 12. A pendulum with time period of 1 s is losing energy
due to damping. At certain time its energy is 45 J.
If after completing 15 oscillations, its energy has become 15 J, its damping constant (in s–1) is (a) 1
30ln 3 (b) 1 15ln3
(c) 2 (d) 1
2 (JEE Main 2015) Resonance : It is the phenomenon in which a system is made to oscillate by external force whose frequency is equal to the natural frequency of the system. At resonance, the amplitude of the system is maximum. It is a special case of forced oscillation.
Condition for resonance is ω = ωd. WaVes
A wave is defined as a disturbance which propagates energy and momentum from one place to the other without the transport of matter.
Waves can be one, two or three dimensional according to the number of dimensions in which they propagate energy. Wave moving along strings are one dimensional, surface waves or ripples on water are two dimensional, while sound or light waves travelling radially out from a point source are three dimensional waves. Waves are of two types :
Mechanical waves : The waves which require some material medium for their propagation are called mechanical waves. Sound waves, seismic waves, waves in strings and springs are examples of mechanical waves.
Elasticity and inertia of medium play an important role in propagation of mechanical waves.
Non-mechanical waves : The waves which do not require any material medium for their propagation are called non-mechanical waves. All electromagnetic
waves such as g-rays, X-rays, radio waves, light etc. are non-mechanical.
Any wave whether mechanical or non-mechanical, can be divided into two groups :
Longitudinal waves : Particles of the medium oscillate in the direction of wave motion. They are propagated as compression and rarefaction and are also known as pressure waves. Waves in springs and sound waves in air are example of longitudinal waves.
Transverse waves : In this case the oscillations are at right angles to the direction of wave motion or energy propagation. Waves in strings are transverse. These are propagated as crests and troughs.
A transverse wave travels through a medium in the form of crests and troughs.
A longitudinal wave travels through a medium in the form of compressions and rarefactions. At places of compression the density and pressure are maximum, while at places of rarefaction those are minimum.
Transverse waves can be polarised whereas longitudinal waves cannot be polarised. Hence, transverse or longitudinal nature of a wave can be decided on the basis of polarisation.
KEYPOINT
Transverse waves can propagate only in medium
• with shear modulus of elasticity such as solids and strings, but not in fluids.
Longitudinal waves need bulk modulus of elasticity
• and therefore possible in all media, solids, liquids and gases.
Ultrasonic waves in air produced by a vibrating quartz crystal are longitudinal waves.
Waves produced in a cylinder containing a liquid by moving its piston back and forth are longitudinal waves.
In case of a vibrating tuning fork the waves in the prongs are transverse while in the stem are longitudinal.
Some waves in nature are neither transverse nor longitudinal but a combination of the two. e.g. waves produced by a motorboat sailing in water is a combination of both longitudinal and transverse waves.
Various terms related to Wave motion
Amplitude : It is defined as the maximum displacement of an oscillating particle of the medium from the mean position. It is denoted by symbol A.
Wavelength : It is defined as the distance travelled by the wave during the time, the particle of the medium completes one oscillation about its mean position.
It may also be defined as the distance between two consecutive points in the same phase of wave motion. It is denoted by symbol l.
In case of transverse wave
l = distance between two consecutive crests and troughs.
In case of longitudinal waves
l = distance between two consecutive compressions and rarefactions.
Time period : It is defined as the time taken by a particle to complete one oscillation about its mean position. It is denoted by symbol T.
Frequency : It is defined as the number of oscillations made by the particle in one second. It is denoted by symbol u.
u = 1T
Wave speed or speed of a wave : It is defined as the distance travelled by the wave in one second. It is denoted by symbol v and is given by
v = ul ....(i)
As the speed of a wave is related to its wavelength and frequency by the equation (i) but it is determined by the properties of the medium.
Intensity of a wave : It is defined as the amount of energy flow per unit area per unit time in a direction perpendicular to the propagation of wave. It is denoted by the symbol I and is given by
I = 2p2u2A2rv
where u is the frequency, A is the amplitude, v is the velocity of the wave, r is the density of the medium.
The SI unit of intensity is W m–2.
Dimensional formula of intensity of a wave is [ML0T–3].
Energy density : It is defined as amount of energy flow per unit volume. It is denoted by symbol u and is given by u = 2p2A2u2r
where u is the frequency, A is the amplitude and r is the density of the medium.
The SI unit of energy density is joule/m3.
Dimensional formula of energy density is [ML–1T–2].
equation of Plane Progressive Wave
Equation of plane progressive wave travelling along the positive direction of x-axis is given by
y (x, t) = Asin(wt – kx + f) where y = displacement of a particle at time t A = amplitude of the wave,
w = angular frequency = 2pu = 2 1pT
k = propagation constant or angular wave number =2p
f = phase constant or initial phase.l
Phase of the wave is the argument (wt – kx + f) of the oscillatory term sin(wt – kx + f).
Wave velocity, v k= w .
It depends only on the nature of the medium in which the wave propagates.
Slope of the wave, dydx= −kAcos(wt kx− +f) Particle velocity,
vparticle = dy
dt =wAcos(kx−w ft+ ) = −
w k dy
or vparticle = –wave velocity × slope of the wavedx
Particle acceleration, a d y
dt y
= 22 = −w2
Equation of plane progressive wave travelling along negative direction of x-axis is given by
y = Asin(wt + kx + f)
The differential equation of one dimensional progressive wave is given by
∂
∂ = ∂
∂
2
2 2 2
2
y
t v y
x
relationship Between Phase difference, Path difference and time difference
Phase difference=2p×path difference l
Phase difference=2p×time difference
A path difference of l corresponds to a phase difference T of 2p radian.
To calculate phase difference between two waves, the equation of both waves must be in sine form or in cosine form.
displacement and Pressure Waves
A longitudinal sound wave can be expressed either in terms of the longitudinal displacement of the particles of the medium or in terms of excess pressures produced due to compression or rarefection. The first type is called the displacement wave and the second type the pressure wave.
Equation of displacement wave
y = Asin(wt – kx) Equation of pressure wave P = Pmcos(wt – kx)
Here Pm = amplitude of pressure wave.
speed of transverse Wave
Speed of a transverse waves on a stretched string is given by
v T
where T is the tension in the string, m is the mass per = m unit length of the string called linear density.
Speed of a transverse wave in a solid is given by v = h
where h is the modulus of rigidity, r is the density of a r solid.
speed of Longitudinal Wave
Speed of a longitudinal wave in a medium is given by
v E
= r
where E is the modulus of elasticity and r is the density of the medium.
Speed of a longitudinal wave in a metallic bar is given by
v Y
= r
where Y is the Young’s modulus and r is the density of material of a metallic bar.
Speed of a longitudinal wave in a fluid is given by
v B
= r
where B is the bulk modulus and r is density of a fluid.
Newton’s formula : Newton assumed that propagation of sound wave in gas is an isothermal process. Therefore, according to Newton, speed of sound in gas is given by
v P
= r
where P is the pressure of the gas and r is the density of the gas.
According to the Newton’s formula, the speed of sound in air at N.T.P. is 280 m s–1. But the experimental value of the speed of sound in air at N.T.P. is 332 m s–1. Newton could not explain this large difference.
Newton’s formula was corrected by Laplace.
Laplace’s correction : Laplace assumed that propagation of sound wave in gas is an adiabatic process. Therefore,
according to Laplace, speed of sound in a gas is given by
v P
= gr
According to Laplace’s correction the speed of sound in air at N.T.P. is 331.3 m s–1. This value agrees fairly well with the experimental values of the speed of sound in air at N.T.P.
Speed of sound in a gas, v= g3vrms. Principle of superposition of Waves
When two or more waves travel in a medium in such a way that each wave represents its separate motion individually, then the resultant displacement of particle of the medium at any time is equal to the vector sum of the individual displacements. This phenomenon is known as principle of superposition of waves.
If y y y1, 2, 3, ....,yn are the displacements at a point due to the n waves, then the resultant displacement at that point is given by
y y y= + + + +1 2 y3 .... yn
The superposition of waves give rise to following three phenomena :
Interference
• Stationary waves
• Beats
•
reflection of Waves
The reflection of waves at a boundary or interface between two media occurs as follows :
A travelling wave, at a rigid boundary or a closed end is reflected with a phase reversal of p but the reflection at an open boundary takes place without any phase change.
Let the incident wave be represented by yi = Asin(wt – kx)
For reflection at a rigid boundary, the reflected wave is represented by
yr = Asin (wt + kx + p) = –Asin(wt + kx) For reflection at an open boundary, the reflected wave is represented by
yr = A sin(wt + kx)
An echo can be cited as an example of reflection of sound from a distant object such as hill or cliff. If there is a sound reflector at a distance d from the source, the time interval between original sound and its echo at the site of source will be
t dv d
v d
= + = 2 .v
Now as persistence of ear is (1/10) s, echo of a sharp or momentary sound (such as clap) will be heard if
t d
v i e d v
> 1 > >
10 2 1
10 20
or , . .,
If a person standing between two parallel hills fires a gun and hears the first echo after t1 s, the second echo after t2 s, and v is the velocity of sound, then the distance between the two hills is given by
s1 + s2 = (vt1/2) + (vt2/2) = [v(t1 + t2)/2]
KEYPOINT
The concept of rarer and denser media for a wave
• is through the velocity of propagation and not density. Lesser the velocity, denser is the medium and vice versa.
Interference of waves : When two waves of same frequency or wavelength having constant phase difference travelling with same speed in the same direction superpose on each other, they give rise to an effect called interference of waves.
Condition for constructive interference Phase difference φ = 2nπ where n = 0, 1, 2, ...
Path difference δ = l
p f l
2 × = n where n = 0, 1, 2,..
Condition for destructive interference
Phase difference φ = (2n + 1)π where n = 0, 1, 2, ...
Path difference δ l
p f l
= × = +
2 1
n 2 . where n = 0, 1, 2 ....
KEYPOINT
The phenomenon of interference is based on
• conservation of energy.
stationary Waves
When two waves of same frequency, wavelength and amplitude travel in opposite directions at same speed, their superposition gives rise to a new type of waves known as stationary waves or standing waves. Energy does not propagate in this type of wave hence, it is named as stationary wave.
Stationary waves are of two types Longitudinal stationary waves
• Transverse stationary waves
•
Longitudinal stationary waves : It is produced in organ pipe and resonance tube.
Transverse stationary waves : It is produced in stretched string and sonometer.
Equation of a stationary wave is given by y = (2Asinkx)coswt
Stationary waves are characterised by nodes and antinodes.
Nodes are the points for which the amplitude is minimum whereas antinodes are the points for which the amplitude is maximum.
In a stationary wave nodes and antinodes are formed alternately and distance between them is l/4.
At antinodes, displacement and velocity is maximum.
At nodes, displacement and velocity is zero.
Distance between two consecutive nodes or antinodes is l/2. Distance between a node and adjoining antinode is l/4.
Vibrations in a stretched string of Length l fixed at Both ends
Speed of waves in a stretched string is given by
v T
= m
where T is the tension of the string, m is the mass per unit length of the string.
Fundamental mode or first mode, l1 = 2L
Fundamental frequency
u1 l1 2 1 m
= v = v =2
L L T
This frequency is called first harmonic.
Second mode, l2 = L
Frequency u l2 u
2 2 1
= v = =v
L
This frequency is called second harmonic or first overtone.
Third mode, l3 2
= L3 Frequency
u3 l u
3 3 1
2 3
= v = v =
L
This frequency is called third harmonic or second overtone.
For the nth mode, ln L
= 2n Frequency of nth mode
un lvn nv u m
L n n
L T
= =2 = 1=2 where n = 1, 2, 3, ....
This frequency is called nth harmonic or (n – 1)th overtone.
Note : In general, up p m L T
= 2 ,
where p = number of loops.
Laws of vibrating stretched string :
The fundamental frequency of a stretched string is given by u= 12L Tm
. Law of length,
• u ∝ 1L when T and m are constants.
Law
• of tension, u ∝ T when L and µ are constants
Law of mass,
• u
∝ 1 when L and T are constantsm Note : If r is the density of the material of the string and D is the diameter of string, then mass per unit length,
m p r= D42 .
\ u
p r pr
= 1 =
2 4 1
L T2
D LD T .
Laws of vibration of stretched string can be verified experimentally by using a sonometer.
melde’s experiment
In longitudinal mode, the prongs of the tuning fork vibrate in a direction parallel to the length of string.
In the longitudinal mode when fork completes one vibration, the string completes only half the vibration.
So, the frequency of the string is one half of that of the fork.
uL p m L T
=
In transverse mode, the prongs of the tuning fork vibrate in a direction perpendicular to the length of the string.
In the transverse mode of vibration when fork completes one vibration, the string also completes one vibration.
So, the frequency of the string is equal to the frequency of the fork.
u m u
T p L
L T
=2 = 2 .
The number of loops in the transverse mode is twice that in the longitudinal mode.
Ampere’s circuital law The line integral of magnetic field around any closed path in vacuum is equal to 0 times them total current passing through
that closed path.
Electromagnetic spectrum
The orderly distribution of e l e c t r o m a g n e t i c w av e s i n a c c o r d a n c e w i t h t h e i r