Wave Motion Theory1

7/18/2019 Wave Motion Theory1

http://slidepdf.com/reader/full/wave-motion-theory1 1/28

Wave Motion 1

genius PHYSICS

16.1 Wave.

A wave is a disturbance which propagates energy and momentum from one place to theother without the transport of matter

!"# Necessary properties of the medium for wave propagation :

!i# $lasticity % So that particles can return to their mean position& after having beendisturbed

!ii# Inertia % So that particles can store energy and overshoot their mean position

!iii# Minimum friction amongst the particles of the medium

!iv# 'niform density of the medium

!(# Characteristics of wave motion :

!i# It is a sort of disturbance which travels through a medium

!ii# Material medium is essential for the propagation of mechanical waves

!iii# When a wave motion passes through a medium& particles of the medium only vibratesimple harmonically about their mean position )hey do leave their position and move with the

disturbance!iv# )here is a continuous phase di*erence amongst successive particles of the medium i.e.&

particle ( starts vibrating slightly later than particle " and so on

!v# )he velocity of the particle during their vibration is di*erent at di*erent position

!vi# )he velocity of wave motion through a particular medium is constant It depends only onthe nature of medium not on the fre+uency& wavelength or intensity

!vii# $nergy is propagated along with the wave motion without any net transport of themedium

!,# Mechanical waves : )he waves which re+uire medium for their propagation are called

mechanical wavesExample % Waves on string and spring& waves on water surface& sound waves& seismic

waves

!-# Non-mechanical waves : )he waves which do not re+uire medium for theirpropagation are called non. mechanical or electromagnetic waves

Examples % /ight& heat !Infrared#& radio waves& γ . rays& X .rays etc

!0# Transverse waves : Particles of the medium e1ecute simple harmonic motion abouttheir mean position in a direction perpendicular to the direction of propagation of wave motion

!i# It travels in the form of crests and troughs

!ii# A crest is a portion of the medium which is raised temporarily above the normal positionof rest of the particles of the medium when a transverse wave passes through it



Normal LevelCrest

Trough Wave

Vi b r a t i on

Particle

B D

A C E

Trough

Crest

Transverse wave on water surface

T T

CCC

Transverse-wave in a ro Transverse wave in a string

Vibration of !article

C C C" " "

Wave Motion

genius PHYSICS

!iii# A trough is a portion of the medium which is depressed temporarily below the normalposition of rest of the particles of the medium& when transverse wave passes through it

!iv# $1amples of transverse wave motion % Movement of string of a sitar or violin& movementof the membrane of a )abla or 2hola3& movement of 3in3 on a rope& waves set.up on the surfaceof water

!v# )ransverse waves can be transmitted through solids& they can be setup on the surface of li+uids 4ut they can not be transmitted into li+uids and gases

!5# !ongitudinal waves : If the particles of a medium vibrate in the direction of wavemotion the wave is called longitudinal

!i# It travels in the form of compression and rarefaction!ii# A compression !C# is a region of the medium in which

particles are compressed

!iii# A rarefaction !R# is a region of the medium in which particlesare rare6ed

!iv# $1amples sound waves travel through air in the form of longitudinal waves& 7ibration of air column in organ pipes are longitudinal& 7ibration of air column above the surface of water inthe tube of resonance apparatus are longitudinal

!v# )hese waves can be transmitted through solids& li+uids and gases because for thesewaves propagation& volume elasticity is necessary

!8# "ne dimensional wave : $nergy is transferred in a single direction onlyExample % Wave propagating in a stretched string

!9# Two dimensional wave : $nergy is transferred in a plane in two mutually perpendiculardirectionsExample % Wave propagating on the surface of water

!:# Three dimensional wave : $nergy in transferred in space in all directionExample % /ight and sound waves propagating in space

16. #mportant Terms $egarding Wave Motion.

!"# Wavelength : !i# It is the length of one wave

!ii# Wavelength is e+ual to the distance travelled by the wave during the time in which any

one particle of the medium completes one vibration about its mean position



" C C C C" "

Wave Motion %

genius PHYSICS

!iii# Wavelength is the distance between any two nearest particles of the medium& vibratingin the same phase

!iv# 2istance travelled by the wave in one time period is 3nown as wavelength

!v# In transverse wave motion %

λ ; 2istance between the centres of two consecutive crests

λ ; 2istance between the centres of two consecutive troughs

λ ; 2istance in which one trough and one crest are contained

!vi# In longitudinal wave motion %

λ ; 2istance between the centres of two consecutive compression

λ ; 2istance between the centres of two consecutive rarefaction

λ ; 2istance in which one compression and one rarefaction contained

!(# &re'uency : !i# <re+uency of vibration of a particle is de6ned as the number of vibrations completed by particle in one second

!ii# It is the number of complete wavelengths traversed by the wave in one second

!iii# 'nits of fre+uency are hert= !Hz # and per second

!,# Time period : !i# )ime period of vibration of particle is de6ned as the time ta3en by theparticle to complete one vibration about its mean position

!ii# It is the time ta3en by the wave to travel a distance e+ual to one wavelength

!-# $elation (etween fre'uency and time period : )ime period ; "><re+uency ⇒ T ;

">n

!0# $elation (etween velocity) fre'uency and wavelength : v ; nλ

7elocity !v # of the wave in a given medium depends on the elastic and inertial property of the medium

<re+uency !n# is characterised by the source which produces disturbance 2i*erent sources may

produce vibration of di*erent fre+uencies Wavelength #!λ will di*er to 3eep nλ = v = constant

16.% *ound Waves.

)he energy to which the human ears are sensitive is 3nown as sound In general all types of waves are produced in an elastic material medium& Irrespective of whether these are heard ornot are 3nown as sound

According to their fre+uencies& waves are divided into three categories %!"# +udi(le or sound waves : ?ange (@ Hz to (@ KHz. )hese are generated by vibrating

bodies such as vocal cords& stretched strings or membrane

!(# #nfrasonic waves : <re+uency lie below (@ Hz.

Example % waves produced during earth +ua3e& ocean waves etc

!,# ,ltrasonic waves : <re+uency greater than (@ KHz. Human ear cannot detect thesewaves& certain creatures such as mos+uito& dog and bat show response to these As velocity of

sound in air is ,,( m>sec so the wavelength of ultrasonics λ "55 cm and for infrasonics λ B"55 m

ote % *upersonic speed : An obDect moving with a speed greater than the speedof sound is said to move with a supersonic speed



Wave Motion

genius PHYSICS

Mach num(er : It is the ratio of velocity of source to the velocity of sound

Mach umber ; soundof 7elocity

sourcof 7elocity

2i*erence between sound and light waves %

!i# <or propagation of sound wave material medium is re+uired but no materialmedium is re+uired for light waves

!ii# Sound waves are longitudinal but light waves are transverse

!iii# Wavelength of sound waves ranges from "50 cm to "50 meter and for light itranges from -@@@ E to (@@@ E

16. elocity of *ound /Wave motion0.

!"# *peed of transverse wave motion :

!i# Fn a stretched string % m

T v =

T ; )ension in the stringG m ; /inear density of string!mass per unit length#

!ii# In a solid body % ρ

η =v

η ; Modulus of rigidityG ρ ; 2ensity of the material!(# *peed of longitudinal wave motion :

!i# In a solid medium ρ

η ,

-+

=k

v

k ; 4ul3 modulusG η ; Modulus of rigidityG ρ ; 2ensity

When the solid is in the form of long bar ρ

Y

v = Y ; Youngs modulus of material of rod

!ii# In a li+uid medium ρ

k v =

!iii# In gases ρ

k v =

16. elocity of *ound in 2lastic Medium.

When a sound wave travels through a medium such as air& water or steel& it will set particlesof medium into vibration as it passes through it <or this to happen the medium must possessboth inertia i.e mass density !so that 3inetic energy may be stored# and elasticity !so that P$may be stored# )hese two properties of matter determine the velocity of sound

i.e. velocity of sound is the characteristic of the medium in which wave propagate

ρ

Ev =

!E ; $lasticity of the mediumG ρ ; 2ensity of the medium#

Important points

!"# As solids are most elastic while gases least i.e. GLS EEE >> So the velocity of sound isma1imum in solids and minimum in gases

v steel B v water B v air

0@@@ m>s B "0@@ m>s B ,,@ m>s



Wave Motion

genius PHYSICS

As for sound v water B v Air while for light v w v A

Water is rarer than air for sound and denser for light

)he concept of rarer and denser media for a wave is through the velocity of propagation!and not density# /esser the velocity& denser is said to be the medium and vice.versa

!(# Newton3s formula : He assumed that when sound propagates through air temperature

remains constant!i.e. the process is isothermal# v air ; ρ ρ

K =

As K ; Eθ = G Eθ ; IsothermalelasticityG ; Pressure

4y calculation v air ; (8: m>sec

However the e1perimental value of sound in air is ,,( m>sec which is greater than that given byewtons formula

!,# !aplace correction : He modi6ed ewtons formula assuming that propagation of sound in air as adiabatic process

v ; ρ k

; ρ

φ E

!As k ; E φ γρ = ; Adiabatic elasticity#

v ; #$%$ (8:; ,,", m>s ! -""Air =γ #

!-# 24ect of density : v ; ρ

γ

⇒ v ρ

"∝

!0# 24ect of pressure : !

T Rv

γ

ρ

γ ==

7elocity of sound is independent of the pressure

of gas provided the temperature remains constant ! ρ ∝

when T ; constant#

!5# 24ect of temperature : !

RT v

γ =

⇒ ( )K T v in∝

When the temperature change is small then v t ; v @ !" J α t #v " ; velocity of sound at @KC& v t ; velocity of sound at t KC & α ; temp.coeLcient of velocity

of sound

7alue of α ; @5@9 C

sm#

>

; @5" !Appro1#

Temperat$re c#e%cient #& vel#cit' of sound is de6ned as the change in the velocity of

sound& when temperature changes by "KC!8# 24ect of humidity : With increase in humidity& density of air decreases So with rise inhumidity velocity of sound increases

)his is why sound travels faster in humid air !rainy season# than in dry air !summer# at thesame temperature



A

&&oun 'v(

L

Win 'w(wcos

6 Wave Motion

genius PHYSICS

!9# 24ect of wind velocity : 4ecause wind drifts the medium !air# along its direction of motion therefore the velocity of sound in a particular direction is the algebraic sum of thevelocity of sound and the component of wind velocity in that direction ?esultant velocity of

sound along SL ; v J w cosθ

!:# Sound of any fre+uency or wavelength travels through a given medium with the samevelocity

!v ; constant# <or a given medium velocity remains constant All other factors li3e phase&

loudness pitch& +uality etc have practically no e*ect on sound velocity!"@# ?elation between velocity of sound and root mean s+uare velocity

v sound ; !

RT γ

and v rms ; !

RT ,

so s#$n(

rms

v

v

;γ

,

or v sound ; γ >,N">( v rms

!""# )here is no atmosphere on moon& therefore propagation of sound is not possible there )o do conversation on moon& the astronaut uses an instrument which can transmit and detectelectromagnetic waves

16.6 $e5ection and $efraction of Waves.



)ncient wave

i

r

eflecte wave

Transmitte wave

t

&ource

Detector

Wave Motion

genius PHYSICS

When sound waves are incident on a boundary between two media& a part of incident wavesreturns bac3 into the initial medium !reOection# while the remaining is partly absorbed and partlytransmitted into the second medium !refraction# In case of reOection and refraction of sound

!"# )he fre+uency of the wave remains unchanged that meansω i ; ω r ; ω t ; ω ; constant

!(# )he incident ray& reOected ray& normal and refracted ray all lie in the same plane

!,# <or reOection angle of incidence !i# ; Angle of reOection !r #

!-# <or refraction t

i

v

v

t

i=

sin

sin

!0# In reOection from a denser medium or rigid support& phase changes by "9@K and

direction reverses if incident wave is ' ; A" sin #! kx t −ω then reOected wave becomes ' ; Ar sin !

#π ω ++kx t ; Ar sin #! kx t +ω

!5# In reOection from a rarer medium or free end& phase does not change and direction

reverses if incident wave is ' ; AI sin ! #kx t −ω then reOected wave becomes ' ; Ar sin ! #kx t +ω

!8# $cho is an e1ample of reOection



7 Wave Motion

genius PHYSICS

If there is a sound reOector at a distance ( from the source then time interval between

original sound and its echo at the site of source will be v

(t

(=

16. $e5ection of Mechanical Waves

Medium !ongitudinalwave

Transversewave

Changein

direction

8hasechange

Timechange

8athchange

?eOection fromrigid end>densermedium

Compression asrarefaction andvice.versa

Crest as crestand )rough astrough

?eversed π

(

T

(

λ

?eOection fromfree end>rarermedium

Compression ascompressionand rarefactionas rarefaction

Crest as troughand trough ascrest

o change Qero Qero Qero

16.7 8rogressive Wave.

!"# )hese waves propagate in the forward direction of medium with a 6nite velocity

!(# $nergy and momentum are transmitted in the direction of propagation of waves withoutactual transmission of matter

!,# In progressive waves& e+ual changes in pressure and density occurs at all points of medium

!-# 7arious forms of progressive wave function

!i# ' ; A sin !ω t kx #

!ii# ' ; A sin !ω t x

λ π (

#

!iii# ' ; A sin (

−

λ π

x

T

t

!iv# ' ; A sin λ

π (

!vt x #

!v# ' ; A sin ω

−

v

x t

Important points!a# If the sign between t and x terms is negative the wave is propagating along positive X .

a1is and if the sign is positive then the wave moves in negative X .a1is direction

!b# )he coeLcient of sin or cos functions i.e. Argument of sin or cos function i.e. !ω t . kx # ;

Phase

!c# )he coeLcient of t gives angular fre+uency ω ; ( T n

π π

(=

; vk

!d# )he coeLcient of x gives propagation constant or wave number =k v

ω

λ

π =

(

!e# )he ratio of coeLcient of t to that of x gives wave or phase velocity i.e. v ; k

ω

where ' ; displacement A ; amplitudeω ; angular fre+uencyn ; fre+uencyk ; propagation constantT ; time periodλ ; wave lengthv ; wave velocityt ; instantaneous time

x ; position of particle from origin



Wave Motion 9

genius PHYSICS

!f# When a given wave passes from one medium to another its fre+uency does not change

!g# <rom v ; λ n ⇒ v λ ∝ n ; constant ⇒ (

"

(

"

λ

λ =

v

v

!0# *ome terms related to progressive waves

!i# Wave num(er /n0 : )he number of waves present in unit length is de6ned as the wave

number ! n # ; λ

"

'nit ; meter " G 2imension ; L"N

!ii# 8ropagation constant /k 0 : k ; x

φ

; thembetween2istance

particlebetweendi*erencePhase

velocityWave

velocityAngular==

v k ω

and k ;λ π

λ

π (

(=

!iii# Wave velocity /v 0 : )he velocity with which the crests and troughs or compression and

rarefaction travel in a medium& is de6ned as wave velocity v ; k

ω

; n T

λ

π

λ ω λ ==

( !iv# 8hase and phase di4erence : Phase of the wave is given by the argument of sine or

cosine in the e+uation of wave It is represented by#!

(#&! x vt t x −=

λ

π φ

!v# At a given position !for 61ed value of x # phase changes with time !t #

T

v

(t

( π

λ

π φ ((==

⇒ (φ ;(t

T

(π

⇒ Phase di*erence ;×

T

π (

)ime di*erence!vi# At a given time !for 61ed value of t # phase changes with position ! x #

(x ((x

(×=⇒=

λ

π φ

λ

π φ ((

⇒ Phase di*erence ;×

λ

π (

Path di*erence

⇒ )ime di*erence ; λ

T

Path di*erence

Sample problems based on Progressive wave

P roblem 1. )he speed of a wave in a certain medium is :5@ m>sec If ,5@@ waves pass over a certainpoint of the medium in " minute& the wavelength is

M8 8MT ;;;<

!a# ( meters !b# - meters !c# 9 meters !d# "5 meters

S#l$ti#n % !d# v ; :5@ m>s G n ;Hz

5@,5@@

So===

5@:5@

nv λ

"5 meters

P roblem . A simple harmonic progressive wave is represented by the e+uation ' ; 9 sin (π !@" x (t #where x and ' are in cm and t is in seconds At any instant the Phase di*erence between twoparticles separated by (@ cm in the x .direction is

M8 8MT ;;;<

!a# "9K !b# ,5K !c# 0-K !d# 8(K

S#l$ti#n % !d# ' ; 9 sin (

− t

x (

"@π

given by comparing with standard e+uation ' ; a sin (

−

λ π

x

T

t

cm"@=λ

So Phase 2i*erence ; λ

π (

path di*erence ; °×=× "9@0

(("@

(π

; 8(K



1; Wave Motion

genius PHYSICS

P roblem %. )he fre+uency of sound wave is n and its velocity is v if the fre+uency is increased to -n thevelocity of the wave will be

M8 82T ;;;<

!a# v !b# (v !c# -v !d# v >-S#l$ti#n % !a# Wave velocity does not depends on the fre+uency It depends upon the $lasticity and inertia

of the mediumP roblem . )he displacement of a particle is given by 1 ; , sin !0 #t π J - cos !0 t π # )he amplitude of

particle isM8 8MT 1999<

!a# , !b# - !c# 0 !d# 8

S#l$ti#n % !c# Standard e+uation % x ; a sin ω t J ) cos t ω

( )( )a)t )a x >tansin "(( −++= ω

Riven e+uation ( ) #0cos!-0sin, t t x π π +=

( ),>-tan0sin"5: "−++= t x π

##,>-!tan0sin!0 "−+= t x π

P roblem . )he e+uation of a transverse wave travelling on a rope is given by #@@(@"@!sin"@ t x ' −= π

where ' and x are in cm and t in seconds )he ma1imum transverse speed of a particle inthe rope is about M8 82T 1999<

!a# 5, cm>sec !b# 80 cm>s !c# "@@ cm>sec !d# "(" cm>sec

S#l$ti#n % !a# Standard e+ of travelling wave ' ; A sin #! t x k ω −

4y comparing with the given e+uation ' ; "@ sin !@@" t x π π (− # A ; "@ cm& ω ; ( π

Ma1imum particle velocity ; Aω ; (π "@ ; 5, cm>sec

P roblem 6. In a wave motion ' ; a sin ' t x k #&!

ω − can represents##T-=22 1999<

!a# $lectric <ield !b# magnetic 6eld !c# 2isplacement !d# PressureS#l$ti#n % !a&b&c&d#

P roblem . <ind the ratio of the speed of sound in nitrogen gas to that of helium gas& at ,@@ k is##T-=22 1999<

!a# (

"

!b# ,

(

!c# 0

,

!d# 0-

S#l$ti#n % !c# !

RT v

γ =

(9-

,>00>8

(

( ==*

He

He

*

He

*

!!

v v

γ γ

; 0

,

P roblem 7. )he displacement x !in metres# of a particle performing simple harmonic motion is related to

time t !in seconds# as x ; @@0 cos

+

--

π π t

)he fre+uency of the motion will beM8 8MT > 82T 1997<

!a# @0 Hz !b# "@ Hz !c# "0 Hz !d# (@ Hz

S#l$ti#n % !d# <rom the given e+uation& coeLcient of t ; π ω #=

∴ n ;

((

-

(==

π

π

π

ω

Hz



Wave Motion 11

genius PHYSICS

P roblem 9. A wave is represented by the e+uation Y ; 8 sin

+−

,@-@8 π π π x t

x is in meters and t is inseconds )he speed of the wave is

M8 82T 1996<

!a# "80 m+sec !b# -: π m+s !c# sm>

-:

π !d# @(9 sm *π

S#l$ti#n % !a# Standard e+uation ' ; A sin ( )@φ ω +− kx t

In a given e+uation π π ω @-@&8 == k

π

π ω

@-8

==k

v ; "80 m>sec

P roblem 1;. A wave is represented by the e+uation y ; @0 sin !"@ t J 1#m It is a travelling wavepropagating along the x direction with velocity

$oor?ee 199<

!a# "@ m>s !b# (@ m>s !c# 0 m>s !d# one of these

S#l$ti#n % !a# v ; smk >"@">"@> ==ω

P roblem 11. A transverse progressive wave on a stretched string has a velocity of "@ ms" and afre+uency of "@@ Hz )he phase di*erence between two particles of the string which are (0cm apart will be M8 8MT 199<

!a# 9>π !b# ->π !c# 9>,π !d# (>π

S#l$ti#n % !d#cmmnv "@"@

"@@

"@> ====λ

Phase di*erence ; λ

π (

path di*erence "@

(π =

(0 ; (

π

P roblem 1. In a stationary wave& all particles are M8 8MT 199<

!a# At rest at the same time twice in every period of oscillation

!b# At rest at the same time only once in every period of oscillation!c# ever at rest at the same time!d# ever at rest at all

S#l$ti#n % !a#

P roblem 1%. )he path di*erence between the two waves

−=

λ

π ω

x t a ' (

sin""

and

+−= φ

λ

π ω

x t a ' (

cos((

is M8 8MT 199<

!a#φ

π

λ

( !b#

+

((

π φ

π

λ

!c#

−

(

( π φ

λ

π

!d#( )φ

λ

π (

S#l$ti#n % !b#

−= λ

π ω

x t a '

(sin""

G

++−= (

(sin

((

π φ

λ

π ω

x t a '

Phase di*erence ;

−−

++−

λ

π ω

π φ

λ

π ω

x t

x t

(

(

(

+=

(

π φ

Path di*erence ; π

λ

( Phase di*erence π

λ

(=

+

(

π φ

P roblem 1. A plane wave is described by the e+uation

−−=

("@

-cos,

π t

x '

)he ma1imum velocityof the particles of the medium due to this wave is

M8 8MT 199<

!a# ,@ !b# , (>π !c# ,>- !d# -@S#l$ti#n % !a# Ma1imum velocity ; Aω ; , "@ ; ,@



1 Wave Motion

genius PHYSICS

P roblem 1. A wave represented by the given e+uation ' ; A sin !"@#

,"0

π π π ++ t x

where x is in meterand t is in second )he e1pression represents

##T-=22 199;<

!a# A wave travelling in the positive x .direction with a velocity of "0 m>sec

!b# A wave travelling in the negative x .direction with a velocity of "0 m>sec!c# A wave travelling in the negative x,direction with a wavelength of @( m

!d# A wave travelling in the positive x .direction with a wavelength of @( m

S#l$ti#n % !b& c# 4y comparing with standard e+uation Y ; A sin ( ),>π ω ++ t x k

K ; "@ π & π ω "0=

We 3now that % k v ω =

; "0 m>secG k

π λ

(=

; @( meter

P roblem 16. A transverse wave is described by the e+uation @ ' Y = sin (π

−

λ

x &t

)he ma1imumparticle velocity is four times the wave velocity if

##T 197) M8 8MT 199) M8 8MT>82T 1997<

!a# -

@ ' π λ =

!b# (

@ ' π λ =

!c# @ ' π λ = !d# @( ' π λ =

S#l$ti#n % !b# Ma1imum particle velocity ; - wave velocity

Aω ; - & λ

' @ ( λ π & & -=

(@ ' π

λ =

P roblem 1. )he e+uation of a wave travelling in a string can be written as ' ; , cosπ !"@@ t x # Itswavelength is

M8 8MT 1991) 9) 9@ MN$ 197<

!a# "@@ cm !b# ( cm !c# 0 cm !d# one of these

S#l$ti#n % !b# ' ; A cos ! −− ( xk t ω standard e+uation

' ; , cos !"@@ xt π π − # given e+uation

So K ; π and==

k

π λ

(

( cm

P roblem 17. A plane wave is represented by x ; "( sin !,"- t J "(05 y# where x and ' are distancesmeasured along in x and ' direction in meter and t is time in seconds )his wave has

M8 82T 1991<

!a# A wave length of @(0 m and travels m J ve x,direction

!b# A wavelength of @(0 m and travels in J ve ',direction!c# A wavelength of @0 m and travels in ve ',direction!d# A wavelength of @0 m and travels in ve x,direction

S#l$ti#n % !c# <rom given e+uation k ; "(05

k

π λ

(=

@0 m direction ; '

P roblem 19. A wave is reOected from a rigid support )he change in phase on reOection will beM8 8MT 199;<

!a# ->π !b# (>π !c# π !d# π (

S#l$ti#n % !c#



Wave Motion 1%

genius PHYSICS

P roblem ;. )he e+uation of displacement of two waves are given as ' " ; "@ sin

+

,,

π π t

G ' ( ; 0

N,cos,,Msin t t π π +

)hen what is the ratio of their amplitudes +##M* 199<

!a# " % ( !b# ( % " !c# " % " !d# one of these

S#l$ti#n % !c# ' ( ; 0 N,cos,,Msin t t π π + ; 0

++

,,sin," π π t

; "@ sin

+

,,

π π t

So& A" ; "@ and A( ; "@

P roblem 1. )he e+uation of a wave travelling on a string is y ; - sin

−

99

(

x t

π

if x and ' are in cm& thenvelocity of wave is M8 82T 199;<

!a# 5- cm>sec in x direction !b# ,( cm>sec in x direction!c# ,( cm>sec in J1 direction !d# 5- cm>sec in J x

direction

S#l$ti#n % !d# ' ; - sin

− x t

"5-

π π

"5>&- π π ω == k

sec>5-"5>

-cm

k v ===

π

π ω

in J x direction

P roblem . )he e+uation of wave is ' ; ( sin #(@@0@! t x −π where x and ' are e1pressed in cm and t insec )he wave velocity is

M8 8MT 1976<

!a# "@@ cm>sec !b# (@@ cm>sec !c# ,@@ cm>sec !d# -@@ cm>sec

S#l$ti#n % !d# π

π ω

0@(@@==

k v

; -@@ cm>sec

16.9 8rinciple of *uperposition.

)he displacement at any time due to any number of waves meeting simultaneously at apoint in a medium is the vector sum of the individual displacements due each one of the wavesat that point at the same time

If ,&(" & ' ' ' are the displacements at a particular time at a particular position& due to

individual waves& then the resultant displacement,(" +++= ' ' ' '

Examples



'B(

1 Wave Motion

genius PHYSICS

!i# ?adio waves from di*erent stations having di*erent fre+uencies cross the antenna 4ut our )7>?adio set can pic3 up any desired fre+uency

!ii# When two pulses of e+ual amplitude on a string approach each other 6g !A#N& then onmeeting& they superimpose to produce a resultant pulse of =ero amplitude 6g !4#N Aftercrossing& the two pulses travel independently as shown in 6g !C#N as if nothing hadhappened

Important applications of superposition principle %

!a# Interference of waves !b# Stationary waves !c# 4eats

16.1; #nterference of *ound Waves.When two waves of same fre+uency& same wavelength& same velocity !nearly e+ual

amplitude# moves in the same direction& )heir superimposition results in the interference 2ue tointerference the resultant intensity of sound at that point is di*erent from the sum of intensitiesdue to each wave separately )his modi6cation of intensity due to superposition of two or morewaves is called interference

/et at a given point two waves arrives with phase di*erence φ and the e+uation of thesewaves is given by

' " ; a" sin t ω & ' ( ; a( sin ! #φ ω +t then by the principle of superposition

(" ' ' ' += ⇒ ' ; A sin ( )θ ω +t where φ cos( ("((

(" aaaa A ++= and tan θ ; φ

φ

cos

sin

("(aa

a

+

and since Intensity( A∝

So - ; φ cos&( ("((

(" aaaa ++ ⇒ φ cos( ("(" ----- ++=

Important points!"# Constructive interference : Intensity will be ma1imum

when &-&(&@ π π φ = nπ ( G where n ; @&"&(

when x ; @& λ& (λ& nλ G where n ; @& "



)

Phase Diff% + , # . /

Path Diff% + *, *, , .*, 0

(("ma1 #! ---

("( --

("( --

(" --

(("min #T! ---

Wave Motion 1

genius PHYSICS

-max ; -" J -( J ("( -- ; ( ) ( )(("

(

(" A A-- +∝+

It means the intensity will be ma1imum at those points where path di*erence is an integral

multiple of wavelength λ )hese points are called points of constructive interference or

interference ma1ima!(# Aestructive interference : Intensity will be minimum

when 0&,& π π π φ = π #"(! −n G where n; "& (& ,

when x ; λ>(& ,λ>(& !(n."#λ /2G where n ; "& (& ,

-min ; -"J-( ( ("-- ⇒ Imin ;( )((" -- − ( ) (("T A A∝

!,# All ma1ima are e+ually spaced and e+ually loud Same is also true for minima Also

interference ma1ima and minima are alternate as for ma1imum (&&@ etc x λ λ =∆ and for minimum

etc x (

0&

(

,&

(

λ λ λ =∆

!-#

( )( )

( )

( ) ((

("

(

"

(("

(("

(

("

(

("

min

ma1 withTT A

A

-

-

A A

A A

--

--

-

-=

+=

+=

!0# If @(" --- == then -ma1 ; #-- and -min

; @

!5# In interference the intensity in

ma1imum ( )((" -- + e1ceeds the sum of individual intensities !-" J -(# by an amount

("( -- while in minima

( ) ( ) bylac3sT ("

(

(" ---- + the same amount ("( --

Hence in interference energy is neither created nor destroyed but is redistributed

16.11 *tanding Waves or *tationary Waves.

When two sets of progressive wave trains of same type !both longitudinal or both

transverse# having the same amplitude and same time period>fre+uency>wavelength travellingwith same speed along the same straight line in opposite directions superimpose& a new set of waves are formed )hese are called stationary waves or standing waves

Characteristics of standing waves %!"# )he disturbance con6ned to a particular region between the starting point and reOecting

point of the wave!(# )here is no forward motion of the disturbance from one particle to the adDoining particle

and so on& beyond this particular region!,# )he total energy associated with a stationary wave is twice the energy of each of

incident and reOected wave 4ut there is no Oow or transference of energy along the stationary

wave



0 1 + 0 1 L

2$

2,

N N

("λ

L

A

N NA A

N

L 1 ,

16 Wave Motion

genius PHYSICS

!-# )here are certain points in the medium in a standing wave& which are permanently at

rest )hese are called nodes )he distance between two consecutive nodes is (

λ

!0# Points of ma1imum amplitude is 3nown as antinodes )he distance between two

consecutive antinodes is also (>λ )he distance between a node and adDoining antinode is ->λ !5#)he medium splits up into a number of segments $ach segment is vibrating up and down

as a whole!8# All the particles in one particular segment vibrate in the same phase Particles in two

consecutive segments di*er in phase by "9@K!9# All the particles e1cept those at nodes& e1ecute simple harmonic motion about their

mean position with the same time period!:# )he amplitude of vibration of particles varies from =ero at nodes to ma1imum at

antinodes!"@# )wice during each vibration& all the particles of the medium pass simultaneously through

their mean position

!""# )he wavelength and time period of stationary waves are the same as for thecomponent waves

!"(# 7elocity of particles while crossing mean position varies from ma1imum at antinodes to=ero at nodes

!",# In standing waves& if amplitude of component waves are not e+ual ?esultant amplitudeat nodes will be minimum !but not =ero# )herefore& some energy will pass across nodes andwaves will be partially standing

16.1 *tanding Waves on a *tring.

When a string under tension is set into vibration& transverse harmonic waves propagatealong its length When the length of string is 61ed& reOected waves will also e1ist )he incident

and reOected waves will superimpose to produce transverse stationary waves in a string

Incident wave ' " ; a sin λ

π (

!vt J x #

?eOected wave( )[ ]π

λ

π +−= x vt a '

(sin(

( ) x vt a −−=

λ

π (sin

According to superposition principle % ' ; ' " J ' ( ; ( a cos λ

π

λ

π x vt (sin

(

Reneral formula for wavelength n

L(=λ

where n ; "&(&,& correspond to "st

& (nd

& ,rd

modesof vibration of the string

!"# <irst normal mode of vibration L

v v n

("" ==

λ ⇒ m

T

Ln

(

"" =

)his mode of vibration is called the fundamental mode andthe fre+uency is called fundamental fre+uency )he sound fromthe note so produced is called fundamental note or 6rstharmonic

!(# Second normal mode of vibration %

#!((

("

(( n

L

v

L

v v n ====

λ

)his is second harmonic or 6rst over tone



N

N

N

NA A A

N

("λ

L

A

N A A N

-

, (λ

L

N A

N

A

N

A

Wave Motion 1

genius PHYSICS

!,# )hird normal mode of vibration %"

,, ,

(

,n

L

v v n ===

λ

)his is third harmonic or second overtone

8osition of nodes :Ln

L

n

L

n

L x

,&

(&&@=

<or 6rst mode of vibration x ; @ & x ; L )wo nodesN

<or second mode of vibration x ; @& x ;L x

L=&

( )hree nodesN

<or third mode of vibration x ; @ & x ;L x

L x

L== &

,

(&

, <our nodesN

8osition of antinodes :

(

0&

(

,&

( n

L

n

L

n

L x =

( )

n

L

(

"( −η

<or 6rst mode of vibration x ; L>( Fne antinodeN

<or second mode of vibration 1 ; -

,&

-

LL

)wo antinodeN

16.1% *tanding Wave in a Closed "rgan 8ipe.

Frgan pipes are the musical instrument which are used for producing musical sound byblowing air into the pipe /ongitudinal stationary waves are formed on account of superimpositionof incident and reOected longitudinal waves

$+uation of standing wave y ; (a cos λ

π

λ

π x vt (sin

(

Reneral formula for wavelength ( )"(

-

−=

n

Lλ

!"# <irst normal mode of vibration % L

v n

-" =

)his is called fundamental fre+uency )he note so producedis called fundamental note or 6rst harmonic

!(# Second normal mode of vibration %"

(( ,

-

,n

L

v v n ===

λ

)his is called tir( arm#nic or /rst #vert#ne

!,# )hird normal mode of vibration %", 0

-

0n

L

v n ==

)his is called /&t arm#nic or sec#n( #vert#ne

8osition of nodes % x ; @& #"(!

(#"(!

5&#"(!

-&#"(!

(

−−−− n

nL

n

L

n

L

n

L

<or 6rst mode of vibration x ; @ Fne nodeN

<or second mode of vibration x ; @& x ; ,

(L

)wo nodesN



A A N

("λ

L

L 1 ,

N

A

N

A A

(

, ,λ L

NA

NA A

N

A

17 Wave Motion

genius PHYSICS

<or third mode of vibration x ; @& 0

-&

0

( LL x =

)hree nodesN

8osition of antinode : x ;L

n

L

n

L

n

L&

"(

0&

"(

,&

"( −−−

<or 6rst mode of vibration x ; L Fne antinodeN

<or second mode of vibration x ;L x

L=&

, )wo antinodeN

<or third mode of vibration x ;L

LL&

0

,&

0 )hree antinodeN

16.1 *tanding Waves in "pen "rgan 8ipes.

Reneral formula for wavelength

n

L(

=λ

where n ; "&(&,

!"# <irst normal mode of vibration %="n L

v v

("

=λ

)his is called fundamental fre+uency and the note so producedis called &$n(amental n#te or /rst arm#nic

!(# Second normal mode of vibration

"("

(

( (((

( nnnL

v

L

v v n =⇒=

===

λ

)his is called sec#n( arm#nic or /rst #vert#ne

!,# )hird normal mode of vibration L

v v n

(

,

,, ==

λ & ", ,nn =

)his is called tir( arm#nic or sec#n( #vert#ne

Important points!i# Comparison of closed and open organ pipes shows that fundamental note in open organ

pipe

=

L

v n

("

has double the fre+uency of the fundamental note in closed organ pipe

-"

=

L

v n

<urther in an open organ pipe all harmonics are present whereas in a closed organ pipe& only

alternate harmonics of fre+uencies &0&,& """ nnn etc are present )he harmonics of fre+uencies (n"&-n"& 5n" are missing

Hence musical sound produced by an open organ pipe is sweeter than that produced by aclosed organ pipe



+%/ r

A N N3i0e

en3i0e

en

c4ing

Wave Motion 19

genius PHYSICS

!ii# Harmonics are the notes>sounds of fre+uency e+ual to or an integral multiple of fundamental fre+uency !n# )hus the 6rst& second& third& harmonics have fre+uencies """ ,&(& nnn &

!iii# Fvertones are the notes>sounds of fre+uency twice>thrice> four times the fundamental

fre+uency !n# e0 ( nnn -&,& and so on

!iv# In organ pipe an antinode is not formed e1actly at the open end rather it is formed alittle distance away from the open end outside it )he distance of antinode from the open end of the pipe is 3nown as end correction

16.1 i(ration of a *tring.

<undamental fre+uency m

T

Ln

(

"=

Reneral formula m

T

L

pn p

(=

L ; /ength of string& T ; )ension in the string

m ; Mass per unit length !linear density#& p ; mode of vibration

Important points



; Wave Motion

genius PHYSICS

!"# As a string has many natural fre+uencies& so when it is e1cited with a tuning for3& thestring will be in resonance with the given body if any of its natural fre+uencies concides with thebody

!(# !i# Ln

"

∝ if T and m are constant !ii# T n∝ if L and m are constant !iii# mn

"∝

if T andL are constant

!,# If ! is the mass of the string of length L& L

!m=

So !L

T

L!

T

Lm

T

Ln

(

"

>(

"

(

"===

; πρ ρ π

T

Lr r

T

L (

"

(

"( =

where m ; π r ( ρ !r ; ?adius& ρ ;

2ensity#

16.16 Comparative *tudy of *tretched *trings) "pen "rgan 8ipe and Closed

"rgan 8ipe

*.No.

8arameter *tretchedstring

"pen organ pipe Closed organpipe

!"# <undamental fre+uencyor "st harmonic l

v n

(" = l

v n

(" =

l

v n

-" =

!(# <re+uency of "st overtone or (nd harmonic

"( (nn = "( (nn = Missing

!,# <re+uency of (nd overtone or ,rd harmonic

", ,nn = ", ,nn = ", ,nn =

!-# <re+uency ratio ofovertones

( % , % - ( % , % - , % 0 % 8

!0# <re+uency ratio ofharmonics

" % ( % , % - " % ( % , % - " % , % 0 % 8

!5# ature of waves )ransversestationary

/ongitudinalstationary

/ongitudinalstationary

16.1 Beats.

When two sound waves of slightly di*erent fre+uencies& travelling in a medium along the

same direction& superimpose on each other& the intensity of the resultant sound at a particularposition rises and falls regularly with time )his phenomenon of regular variation in intensity of sound with time at a particular position is called beats

Important points!"# "ne (eat : If the intensity of sound is ma1imum at time t ; @& one beat is said to be

formed when intensity becomes ma1imum again after becoming minimum once in between!(# Beat period : )he time interval between two successive beats !i.e. two successive

ma1ima of sound# is called beat period!,# Beat fre'uency : )he number of beats produced per second is called beat fre+uency!-# 8ersistence of hearing : )he impression of sound heard by our ears persist in our mind

for ">"@th of a second If another sound is heard before ">"@ second is over& the impression of the

two sound mi1 up and our mind cannot distinguish between the twoSo for the formation of distinct beats& fre+uencies of two sources of sound should be nearly

e+ual !di*erence of fre+uencies less than "@#



Wave Motion 1

genius PHYSICS

!0# 2'uation of (eats : If two waves of e+ual amplitudes a and slightly di*erentfre+uencies n" and n( travelling in a medium in the same direction are

' " ; a sin t nat "" (sin π ω = G t nat a ' ((( (sinsin π ω ==

4y the principle of super position % (" ' ' ' +=

' ; A sin t nn #! (" +π where A ; (a cos t nn #! (" −π ; Amplitude of resultant wave

!5# Beat fre'uency : n ; n" T n(

!8# Beat period : T ; ("T

"

fre+uency4eat

"

nn=

16.17 Aetermination of ,n?nown &re'uency.

/et n( is the un3nown fre+uency of tuning for3 1& and this tuning for3 1 produce x beats persecond with another tuning for3 of 3nown fre+uency n"

As number of beat>sec is e+ual to the di*erence in fre+uencies of two sources& therefore n(

; n" ± x )he positive>negative sign of x can be decided in the following two ways %

By loading By ling

#f B is loaded with waD so its fre'uencydecreases

#f B is led) its fre'uency increases

If number of beats decreases n( ; n" J x If number of beats decreases n( ; n" x

If number of beats Increases n( ; n" x If number of beats Increases n( ; n" J x

If number of beats remains unchanged n( ;n" J x

If number of beats remains unchanged n( ;n" x

If number of beats becomes =ero n( ; n" J x If number of beats becomes =ero n( ; n" x

#f A is loaded with waD its fre'uencydecreases

#f A is led) its fre'uency increases

If number of beats decreases n( ; n" x If number of beats decreases n( ; n" J x

If number of beats increases n( ; n"J x If number of beats Increases n( ; n" x

If number of beats remains unchanged n( ;n" x

If number of beats remains unchanged n( ;n" J x

If number of beats becomes =ero n( ; n" x If no of beats becomes =ero n( ; n" J x

Sample problems based on Superposition of waves

P roblem %. )he stationary wave produced on a string is represented by the e+uation ' ; 0 cos#-@!sin

,t

x π

π

where x and ' are in cm and t is in seconds )he distance betweenconsecutive nodes is M8 8MT ;;;<

!a# 0 cm !b# π cm !c# , cm !d# -@ cm

S#l$ti#n % !c# 4y comparing with standard e+uation of stationary wave

' ; a cos λ

π x (

sin λ

π vt (

We get λ

π x,

; -

xπ

⇒ λ ; 5G 2istance between two consecutive nodes ;cm,

(=

λ

P roblem . Fn sounding tuning for3 A with another tuning for3 4 of fre+uency ,9- Hz & 5 beats areproduced per second After loading the prongs of A with wa1 and then sounding it again with



5A 5B

6+

78

8#,./

78

6+

Wave Motion

genius PHYSICS

1& - 4eats are produced per second what is the fre+uency of the tuning for3 A M8 8MT ;;;<

!a# ,99 Hz !b# 9@ Hz !c# ,89 Hz !d# ,:@ Hz S#l$ti#n % !c#

Probable fre+uency of A is ,:@ Hz and ,89 Hz and After loading the beats are decreasingfrom 5 to - so the original fre+uency of A will be n( ; n " x ; ,89 Hz

P roblem . )wo sound waves of slightly di*erent fre+uencies propagating in the same directionproduces beats due to

M8 82T ;;;<

!a# Interference !b# 2i*raction !c# Polari=ation !d# ?efractionS#l$ti#n % !a#

P roblem 6. 4eats are produced with the help of two sound waves on amplitude , and 0 units )he ratioof ma1imum to minimum intensity in the beats is

M8 8MT 1999<

!a# ( % " !b# 0 % , !c# - % " !d# "5 % "

S#l$ti#n % !d#

(

("

("

min

ma1

−+

= A A

A A

-

-

;

(

,0

,0

−+

; "5%"

P roblem . )wo tuning for3s have fre+uencies ,9@ and ,9- hert= respectively When they are soundedtogether& they produce - beats After hearing the ma1imum sound& how long will it ta3e tohear the minimum sound

M8 8MT>82T 1997<

!a# ">( sec !b# ">- sec !c# ">9 sec !d# ">"5 sec

S#l$ti#n % !c# 4eats period ; )ime interval between two minima

T ;sec

-

""

("

=− nn

)ime interval between ma1imum sound and minimum sound ; T >( ; ">9 sec

P roblem 7. )wo tuning for3 A and 1 give - beats per second when sounded together )he fre+uency of A

is ,(@ Hz When some wa1 is added to 1 and it is sounded with A& - beats per second areagain heard )he fre+uency of 1 is

M8 8MT 199<

!a# ,"( Hz !b# ,"5 Hz !c# ,(- Hz !d# ,(9 Hz

S#l$ti#n % !c# Since there is no change in beats )herefore the original fre+uency of 1 is

n( ; n " J x ; ,(@ J - ; ,(-

P roblem 9. -" for3s are so arranged that each produces 0 beat>sec when sounded with its near for3 If

the fre+uency of last for3 is double the fre+uency of 6rst for3& then the fre+uencies of the6rst and last for3 respectively

M8 8MT 199<

!a# (@@& -@@ !b# (@0& -"@ !c# ":0& ,:@ !d# "@@& (@@

S#l$ti#n % !a# /et the fre+uency of 6rst tuning for3 ; n and that of last ; (n

n& n J 0& n J "@& n J "0 (n this forms AP

<ormula of AP l ; a J !* "# r where l ; /ast term& a ; <irst term& * ; umber of term& r ;Common di*erence



Wave Motion %

genius PHYSICS

(n ; n J !-" "# 0

(n ; n J (@@

n ; (@@ and (n ; -@@

P roblem %;. In stationary waves& antinodes are the points where there is M8 8MT 1996<

!a# Minimum displacement and minimum pressure change!b# Minimum displacement and ma1imum pressure change

!c# Ma1imum displacement and ma1imum pressure change

!d# Ma1imum displacement and minimum pressure change

S#l$ti#n % !d# At Antinodes displacement is ma1imum but pressure change is minimum

P roblem %1. )he e+uation ' ; @"0 sin 0 x cos ,@@ t & describes a stationary wave )he wavelength of the

stationary wave is M8 8MT 199<

!a# Qero meter !b# "(05 meter !c# (0"( meter !d# @5(9 meter

S#l$ti#n % !b# 4y comparing with standard e+uation ∴ x

x 0

(=

λ

π

⇒ π λ

×= 0

(

; "(05 meter

P roblem %. )he e+uation of a stationary wave is ' ; @9 cos

(@

x π

sin (@@ t π where x is in cm and t is in

sec )he separation between consecutive nodes will be

!a# (@ cm !b# "@ cm !c# -@ cm !d# ,@ cm

S#l$ti#n % !a# Standard e+uation ' ; A cos λ

π

λ

π vt x (sin

(

4y comparing this e+uation with given e+uation (@

( x x π

λ

π =

⇒ λ ; -@ cm

2istance 4etween two nodes ; (

λ

; (@ cm

P roblem %%. Which of the property ma3es di*erence between progressive and stationary wavesM8 8MT 197<

!a# Amplitude !b# <re+uency !c# Propagation of energy !d# Phase of the wave

S#l$ti#n % !c# In stationary waves there is no transfer of energy

P roblem %. If amplitude of waves at distance r from a point source is A& the amplitude at a distance(r will be

M8 8MT 197<

!a# ( A !b# A !c# A>( !d# A>-

S#l$ti#n % !c# - ∝ A( and - (

"

r ∝

so r A

"∝

G⇒=

"

(

(

"

A

A

r

r

A( ; A"

(

"

r

r

; A

(

"

; A>(

P roblem %. If two waves of same fre+uency and same amplitude respectively on superimpositionproduced a resultant disturbance of the same amplitude the wave di*er in phase by

M8 8MT 199;<

!a# π !b# ,>(π !c# (>π !d# =ero

S#l$ti#n % !b# φ cos( ("((

(" A A A A A ++=

A( ; A( J A( J ( A( cos φ A" ; A( ; A givenN

cos φ ; ">( ⇒ ,

("(@

π φ =°=



Wave Motion

genius PHYSICS

P roblem %6. )he superposition ta3es place between two waves of fre+uency f and amplitude a )he totalintensity is directly proportional to

M8 8MT 1976<

!a# a !b# (a !c# (a( !d# -a(

S#l$ti#n % !d# - ( )(

(" aa

+∝ As a" ; a( ; aN

- (-a∝

P roblem %. )he following e+uation represent progressive transverse waves M8 82T 199%<

z " ; A cos !ω t kx #

z ( ; A cos !ω t J kx #

z , ; A cos !ω t J k' #

z - ; A cos ! (ω t (k' #

A stationary wave will be formed by superposing

!a# z " and z ( !b# z " and z - !c# z ( and z , !d# z , and z -

S#l$ti#n % !a# )he direction of wave must be opposite and fre+uencies will be same then by superposition&standing wave formation ta3es place

P roblem %7. When two sound waves with a phase di*erence of (>π and each having amplitude A and

fre+uency ω are superimposed on each other& then the ma1imum amplitude and fre+uency of resultant wave is M8 8MT 1979<

!a#(>G

(ω

A

!b#ω G

(

A

!c# G( A (

ω

!d# ω G( A

S#l$ti#n % !d# ?esultant Amplitude ;φ cos( ("

((

(" aaaa ++

E (cos( ((( π

A A A ++E A(

and fre+uency remains same ; ω

P roblem %9. )here is a destructive interference between the two waves of wavelength λ coming from two

di*erent paths at a point )o get ma1imum sound or constructive interference at that point&the path of one wave is to be increased by

M8 82T 197<

!a# ->λ !b# (>λ !c# -

,λ

!d# λ

S#l$ti#n % !b# 2estructive interference means the path di*erence is( )

("( λ −n

If it is increased by (>λ

)hen new path di*erence !(n "# ((

λ λ +

; nλ

which is the condition of constructive interference

P roblem ;. )he tuning for3 and sonometer wire were sounded together and produce - beats>second

when the length of sonometer wire is :0 cm or "@@ cm )he fre+uency of tuning for3 is M8 8MT 199;<

!a# "05 Hz !b# "0( Hz !c# "-9 Hz !d# "5@ Hz

S#l$ti#n % !a# <re+uency n m

T

l

n

l (

"As

"=∴∝

If n is the fre+uency of tuning for3



& L

v& vL

v

Wave Motion

genius PHYSICS

n J - :0"

∝⇒ n - "@@

"∝

⇒ !n J - # :0 ; !n -# "@@ ⇒ n ; "05 Hz

P roblem 1. A tuning for3 2 " has a fre+uency of (05 Hz and it is observed to produce 5 beats>second with

another tuning for3 2 ( When 2 ( is loaded with wa1 It still produces 5 beats>second with 2 "

)he fre+uency of 2 ( before loading wasM8 82T 199;<

!a# (0, Hz !b# (5( Hz !c# (0@ Hz !d# (0: Hz

S#l$ti#n % !b# o of beats does not change even after loading then n( ; n" J x ; (05 J 5 ; (5( Hz.

16.19 Aoppler 24ect.

Whenever there is a relative motion between a source of sound and the listener& theapparent fre+uency of sound heard by the listener is di*erent from the actual fre+uency of soundemitted by the source

When the distance between the source and listener is decreasing the apparent fre+uency

increases It means the apparent fre+uency is more than the actual fre+uency of sound )hereverse is also true

Reneral e1pression for apparent fre+uency n ;

( )[ ]

( )[ ]Sm

Lm

v v v

nv v v

−+−+

Here n ; Actual fre+uencyG v L ; 7elocity of listenerG v S ; 7elocity of source

v m ; 7elocity of medium and v ; 7elocity of sound wave

Sign convention % All velocities

along the direction S to L are ta3en as positive and all velocities along the direction L to S

are ta3en as negative If the medium is stationary v m ; @ then n ;n

v v v v

S

L

−−

Special cases %

!"# Source is moving towards the listener& but the listener at restn

v v

v n

S

H−

=

!(# Source is moving away from the listener but the listener is at restn

v v

v n

S

H+

=

!,# Source is at rest and listener is moving away from the sourcen

v

v v n L−=H

!-# Source is at rest and listener is moving towards the sourcen

v

v v n L H

+=

!0# Source and listener are approaching each othern

v v

v v n

S

L

−+

=H

!5# Source and listener moving away from each othern

v v

v v n

S

L

+−

=H

!8# 4oth moves in the same direction with same velocity n ; n& i.e. there will be no 2opplere*ect because relative motion between source and listener is =ero

!9# Source and listener moves at right angle to the direction of wave propagation n ; nIt means there is no change in fre+uency of sound heard if there is a small displacement of

source and listener at right angle to the direction of wave propagation but for a large



BA

s vs

vs

vs cos

L 'Listener at rest(

Cvs cos

9

r &

9

r L

6 Wave Motion

genius PHYSICS

displacement the fre+uency decreases because the distance between source of sound andlistener increases

Important points!i# If the velocity of source and listener is e+ual to or greater than the sound velocity then

2oppler e*ect is not seen!ii# 2oppler e*ect gives information regarding the change in fre+uency only It does not saysabout intensity of sound

!iii# 2oppler e*ect in sound is asymmetric but in light it is symmetric

16.; *ome Typical &eatures of AopplerFs 24ect in *ound.

!"# When a source is moving in a direction ma?ing an angle w.r.t. the listener : )he apparent fre+uency heard by listener L at rest

When source is at point A is θ cossv v

nv n

−=′

As source moves along A1& value of θ increases& cosθ decreases&n′goes on decreasing

At point C&#:@=θ & @:@coscos == #θ & nn =′

At point 1& the apparent fre+uency of sound becomes θ cossv v

nv n

+=′′

!(# When a source of sound approaches a high wall or a hill with a constant velocity

sv & the reOected sound propagates in a direction opposite to that of direct sound We can assume

that the source and observer are approaching each other with same velocity i.e. Ls v v =

∴nv v

v v

ns

L

−+

=′

!,# When a listener moves (etween two distant sound sources : /et Lv be the

velocity of listener away from "S and towards (S Apparent fre+uency from "S is v

nv v n L #! −

=′

and apparent fre+uency heard from LS is v

nv v n L #! +

=′′

∴ 4eat fre+uency v

nv nn L(

=′−′′=

!-# When source is revolving in a circle and listener L is on one side

ω r v s = so sv v

nv n

−=ma1

and sv v

nv n

+=min

!0# When listener L is moving in a circle and the source is on one side

ω r v L = so v

nv v n L #!ma1

+=

and v

nv v n L #!min

−=

!5# )here will be no change in fre+uency of sound heard& if the source issituated at the centre of the circle along which listener is moving

!8# Conditions for no Aoppler e4ect : !i# When source !S# and listener !L# both are atrest



& L

vL 1 +vs':(

& L

vL 1 +vs':(

& L

vL 1 +vs':(

Wave Motion

genius PHYSICS

!ii# When medium alone is moving

!iii# When S and L move in such a way that distance between S and L remains constant

!iv# When source S and listener L& are moving in mutually perpendicular directions

Sample problems based on Doppler eect

P roblem . A source of sound of fre+uency :@ vi)rati#n>sec is approaching a stationary observer with a

speed e+ual to ">"@ the speed of sound What will be the fre+uency heard by the observerM8 8MT ;;;<

!a# 9@ vi)rati#n>sec !b# :@ vi)rati#n>sec !c# "@@ vi)rati#n>sec !d# "(@ vi)rati#n>sec

S#l$ti#n % !c#n

v v

v n

s

H−

=⇒

nv

v

v n

"@

H

−=

⇒ "@@

:

:@"@

:

"@H =

×== nn

vibration>sec

P roblem %. A source of sound of fre+uency 0@@ Hz is moving towards an observer with velocity ,@ m>s

)he speed of the sound is ,,@ m>s )he fre+uency heard by the observer will be

M8 82T ;;;<!a# 00@ Hz !b# -09, Hz !c# 0,@ Hz !d# 0-00 Hz

S#l$ti#n % !a#n

v v

v n

s

−

=⇒

0@@,@,,@

,,@H

−=n

⇒ n ; 00@ Hz

P roblem . A motor car blowing a horn of fre+uency "(- vi)rati#n>sec moves with a velocity 8( km>r

towards a tall wall )he fre+uency of the reOected sound heard by the driver will be !velocityof sound in air is ,,@ m>s#

M8 82T 199<

!a# "@: vi)rati#n>sec !b# ",( vi)rati#n>sec !c# "-@ vi)rati#n>sec !d# (-9 vi)rati#n>sec

S#l$ti#n % !c# In the given condition source and listener are at the same position i.e. !car# for givencondition

nv v

v v n

car

car H−+

= n(@,,@

(@,,@

−+

= ; "-@ vi)rati#n>sec

P roblem . )he driver of car travelling with a speed ,@ meter >sec towards a hill sounds a horn of

fre+uency 5@@ Hz If the velocity of sound in air is ,,@ m>s the fre+uency of reOected soundas heard by the driver is M8 8MT 1996<

!a# 8(@ Hz !b# 0000 Hz !c# 00@ Hz !d# 0@@ Hz

S#l$ti#n % !a# )his +uestion is same as that of previous one soHz n

v v

v v n

car

car 8(@H =−+

=

P roblem 6. )he source of sound s is moving with a velocity 0@ m>s towards a stationary observer )he

observer measures the fre+uency of the source as "@@@ Hz What will be the apparentfre+uency of the source when it is moving away from the observer after crossing him U )hevelocity of sound in the medium is ,0@ m>s M8 8MT 199<

!a# 80@ Hz !b# 908 Hz !c# ""-, Hz !d# ",,, Hz

S#l$ti#n % !a# When source is moving towards the stationary listener

"-9080@,0@

,0@"@@@H =⇒

−=⇒

−= nnn

v v

v n

s

When source is moving away from the stationary observer sv v

v n

+=HH

E 9080@,0@

,0@

×+ ; 80@Hz



7 Wave Motion

genius PHYSICS

P roblem . A source and listener are both moving towards each other with speed v >"@ where v is thespeed of sound If the fre+uency of the note emitted by the source is & & the fre+uency heardby the listener would be nearly

M8 8MT 199<

!a# """ & !b# "(( & !c# & !d# "(8 &

S#l$ti#n % !b#n

v v

v v n

s

L

−+

= ⇒

nv

v

v v n

−

+=

"@

"@H

⇒& n

:

""H=

; "(( &

P roblem 7. A man is watching two trains& one leaving and the other coming in with e+ual speed of -m>s If they sound their whistles& each of fre+uency (-@ Hz & the number of beats heard by theman !velocity of sound in air ; ,(@ m>s# will be e+ual to

M8 82T 1999@ C8MT 199@ NC2$T 197<

!a# 5 !b# , !c# @ !d# "(

S#l$ti#n % !a# App <re+uency due to train which is coming inn

v v

v n

s

" −=

App <re+uency due to train which is leavingn

v v

v n

s

( +=

So number of beats n" n( ;(-@,(@

,(-

"

,"5

"×

− ⇒ n" n( ; 5

P roblem 9. At what speed should a source of sound move so that observer 6nds the apparent fre+uencye+ual to half of the original fre+uency

$8MT 1996<

!a# v > ( !b# (v !c# v > - !d# v

S#l$ti#n % !d#

nv v

v n

s

H+

=

⇒

nv v

v n

s

( +=

⇒ v s ; v

Documents

Wave Motion Theory1