Classification of waves
Waves
Necessity of medium
The medium should have elasticity,
inertia and uniform density
Characteristics of Wave
Energy propagation
=
> The particles of the medium are executing
simple harmonic motion.
Progressive
waves
One Dimensional
Transverse
waves
Non-mechanical
waves(EMW)
Stationary
waves
Two Dimensional
Longitudinal
waves
=
> The phase of vibration of the particle keep
on changing.
=
> Wave carries energy and momentum.
A
[
A Amplitude
Oscillating term
phase
wave length
ω Angular frequency
Initial phase
2π or 2π ν
ω =_
Direction
T
Propagation
_
1 μ Vω2A2
dK = _
dU= _
_
1 μ Vω2A2
dt dt
Rate of an power Transmission
Case 2
massive
string
x
l
V= ρπr2
K Wave Constant
2π
K= _
M-Mass of block
m-mass of string
ρ-density of material
of string
Case 3
Time taken to reach the pulse at top
Case 1
l
v
[
[
[
ρ
_
Mg 1V= _
μ
ρ-density of the liquid
-density of the material
μ-Linear
mass density of
the string
M
_
_ρ
C
C
C
amax=w2A
P
I=4_
π r2
If vmax of the particle = n x vof wave
R
R
T
T
Transverse-wave in a rod
2
Vmax=wA
C
Energy
2 2 2
_
I = time
+ Area = 2π ν a ρ ν
a=w y
V=w A2-y2
I a ν2 a2
Longitudinal wave in a rod
Medium should posses the property of
rigidity
Medium should posses the property of
elasticity
Transverse waves can be polarised
Longitudinal waves can not be polarised
wA=n
n= 2πA
x l
PHYSICS
WALLAH
l
2πA= n x l
l= 2πA
n
Sound waves travel through air,vibration
of air column in organ pipes vibration of
air column above the surface of water in
the tube of resonance apparatus.
A= nl
2π
Factors affecting velocity of sound
→ Pressure
mass of string m
velocity at bottom
Mg
V1 = μ
M
velocity at Top
→ Density
• Velocity of sound in air
is independent of pressure
V2
2
→ Temp
Vα 1
V=
V1
V1
V2
V1 _
_
M
= 1 = _
ρ2
ρ1
=
V2
m
=
T1
T2
M+m
g
V2= (M+m)
μ
Velocity of Longitudual Wave
massless
string
[
C
These waves can be transmitted through
solids,liquids and gases because for
these waves propagation,volume
elasticity is necessary.
Intensity of wave
Acceleration
of particle
velocity of particle
Temp Cofficient(α)
Increase in velocity of sound for 1oC
or 1K rise in temperature of gas
r2
_
r1
ρ
T=Mg 1- _
C
C R C R CR
Minimum
Transverse waves can be transmitted
through solids,they can be setup on the
surface of liquids.But they can not be
transmitted through liquids and gases.
Velocity at any Point
Velocity, v= T
μ
Time taken, t=
l
T
μ
E
V= _
ρ
(E=Elasticity of the medium; ρ=density of the medium)
(1) As solids are most elastic while gases least i.e. ES>EL>EG. So
the velocity of sound is maximum in solids and minimum in gases
+
V1 _
T
_
= T1
V
2
μ= ρA
Time period
T
l
T=2 _
g
M
_
T
Frequency
Time taken to reach the pulse at Top
l
T=Mg
m
μ= _
Maximum
pressure and
density
V= gx
Velocity of Transmission wave in a string.
T
V= _
μ
C
Case 4
2
4
1
dx
Velocity of particle=-Velocity wave+ Slope of the graph
It travels in the form of compression
(C) and rarefaction (R).
Movement of string of a sitar or violin,
movement of the membrane of a tabla or
Dholak,movement of kink on a rope,waves
set-up on the surface of water.
WAVES
Rate of Energy Transmission
T
k
S
C
I
S
Y
H
P
H
A
L
L
WA
Important Terms
ν
dt
Longitudinal wave in a fluid
It travels in the form of crests (C) and
troughs (T).
y(x,t)=a sin(ωt +
- kx +
- 0/0)
(
x
(v) y=A sin ω (t- _
ν
v
T
Angular frquency
Particles of the medium.
V
C
[
t -_
x
(iii) y=A sin 2π _
T
2π (ν t-x)
(iv) y=A sin _
Amplitude
2
2
d
1 d
_y = _
_y
2
2
dx ν dt2
Particles of the medium vibrates in the
direction of wave motion.
Transverse wave on a string
The general equation of a plane progressive wave with initial phase is
Displacement
Particles of the medium vibrates in a
direction perpendicular to the direction
of propagation of wave.
dy _
dy
_
=- ω + _
Equation of progressive wave
2π x)
(ii) y=A sin(ωt- _
Longitudinal waves
Three Dimensional
=
> The velocity of the particle is not equal to
velocity of wave.
(i) y=A sin(ωt-kx)
Transverse waves
Vibration of particles
Dimension
Mechanical
waves
can travel in
vacuum
All travelling wave satisfy a differential
equation called wave equation.
Classification of waves based on vibration of particles
According to
wave is a disturbance which propagates
energy and momentum from one place to
another without the transport of medium.
μ
= l
T
T= Tension in the string
μ= Linear mass density
Velocity Sound (air)
P _ Newton
V= _
ρ
P γ _ Laplace
V= _
ρ
Value of α =0.608
=0.61
m/s
C
o
Humidity
Humidity ↑
Speed of sound
↑
Sound travels faster in moist air
than in dry air
C
γ= P
CV
Mono atomic
γ= 5/3
diatomic
γ= 7/5
Relation between △X and
△Φ
△Φ= 2π △X
l
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Wave combination of string
Reflection of Mechanical waves
1) From rarer to denser medium
Echo
2) From rarer to denser medium
Source
Incident wave
Incident wave
Reflection from rigid end/denser medium→ Phase change by π
-x
Rigid boundary
Transverse
wave
Rarer
+x
Free boundary
-x
+x
Observer
d
Source at distance “d” from screen
Denser
t= dv + dv = 2d
v
Reflection from free end/rarer medium→ No phase change
Reflected wave
Transmitted wave
Persistance of hearing for
human ear is o.1 sec
S
C
I
S
Y
H
P
H
A
L
L
A
W
Conditions for echo:
if t > 0.1 ⇒ 2d > 0.1 ⇒ d >
Transmitted wave
v
v
20
Reflected wave
Principle of superposition
The displacement at any time due to number of waves meeting simulatoneously at a point in a medium is the
vector sum of individual displacements due each one of the waves at that point of same time
Superposition
Incident wave y1= a1 sin(ωt-k1x)
Incident wave yi= aisin(ωt-k1x)
Reflected wave yr= ar sin(ωt-k1(-x)+π)
Reflected wave yr= ar sin(ωt-k1(-x)+0)
= -ar sin(ωt+k1x)
= ar sin(ωt+k1x)
Transmitted wave yt= atsin(ωt-k2x)
Beat
• Constructive
• Destructive
Standing wave
Condition:•Two waves of same frequency, same wavelength, same velocity
•Resultant intensities will be different from the sum of intensities
of each wave seperately
y1= a1 sin ωt, y2= a2 sin(ωt + ϕ)
ϕ-Phase difference between two waves
y=y1+y2 ⇒ y= A sin(ωt + θ)
A= a12+a22+2a1a2cosϕ
tanθ=
Intensity
a2sinϕ
a1+a2cosϕ
A2
I = I1 + I2 + 2 I1
I1
I2
=
a1
a2
2
Imax
Imin
=
(a1 + a2)2
(a1 - a2)2
=
( I1 + I 2)2
( I1 - I 2)2
sound waves travelling in same medium with slightly different frequencies
superimpose on each other.
The intensity of resultant sound at particular position rises and falls regularly
with time.
The phenomenon of variation of intensity of sound with time at a particular
position is called beats.
Point to remember:-
i) For Constructive interference:-
ϕ = O, 2
, 4
, ---, 2
1) One beat:-
Maximum intensity
(at t=0)
n when n = 0, 1 , 2, ---
x=O, λ, 2λ, --- , nλ , when n= 0, 1, ---
Imax = I1 + I2 +2 I1I2
Becomes minimum
intensity
2
2
= ( I1 + I2) = (A1+A2)
ii) For Destructive interference:-
when ϕ=π, 3π, 5π, --- (2n-1)π ; where n=1,2,3,---
3
x= _ , _
,
2
2
Imin=I1+I2-2 I1I2
=
> Imin=( I1- I2)2
Again acheives
maximum
_
(2n-1) 2 , where,n=1,2,3,-----
(A1-A2)2
Beat period:-
One beat is formed
Time interval between two sucessive beats (ie.two sucessive maximum of sound)
is called beat period.
2
ϕ
I2 cos ϕ
A2 = a12 + a22 + 2a1a2 cos
r4π2
Beats:-
WAVES
Interference of sound wave
•This is due to the interference of waves
PHYSICS
WALLAH
• In a string
• In an open pipe
• In a closed pipe
Interference
Transmitted wave yt= atsin(ωt-k2x)
Beat frequency:No.of beats produced per second
Beat frequency:-
n= n1- n2
Beat period:- T=
1
= n 1n
Beat frequency
1 - 2
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End correction:1 l - 3l
e= _
1
2
2
(
Let n2 is the unknown frequency of tuning fork B, and this tuning fork B produce x beats per
second with another tuning fork of known frequency n1.
As number of beat/sec is equal to the difference in frequencies of two sources, therefore n2 = n1 ± x
By filing
If B is filed, its frequency increases
?) A Source of unknown frequency produces 4 beat/s when sounded with a
source of Known frequency 250 Hz. The second harmonic of the source
of unknown frequency gives 5 beat/s when Sounded with a source of
frequency 513 Hz. The unknown frequency is?
b) 246 Hz
c) 240 Hz
508
d) 260 Hz
250
246
492
5 be
at
21
t
bea
Standing Waves:
1:2:3
T
l=
n
x
(n+1) → node
l
2
= n
2l
l=
n
x
1
+ (n-1)
(n+1) → antinode
n → node
= n
2l
l=
n
x
l
4
S . NO
Note
Distance between two adjacent node & antinode is
4
WAVES
3
Relation between loudness and intensity
Resonance: The phenomenon of making a body vibrate with it‛s natural frequency under the
influence of another vibrating body with the same frequency is called resonance.
n=1,3,5....
Distance between two adjacent node & antinode is
7 λ
_
l4+e =
Comparative Study of Stretched Strings, Open Organ Pipe and Closed Organ Pipe
y=2a sin(kx) cos(ωt)
4
Unison: If the two frequencies are equal then vibrating bodies are said
to be in unison.
1:2:3
Closed pipe
5
_
l3+e =
(iii) Similarly if n2 = 2nn1, it means n2 is n-octave higher n1 is n octave lower.
n → antinode
l
2
4
1
(ii) If n2 = 23n1 it means n1 is 3-octave higher or n1 is 3-octave lower.
y=2a cos(kx) sin(ωt)
Open pipe
=
2
(i) If n2 = 2n1 it means n2 is an octave higher than n1 or n1 is an octave lower than n2.
y=2a sin(kx) cos(ωt)
2l
th
n
3
Octave: The tone whose frequency is double the fundamental frequency is defined as Octave.
•When two progressive waves (both longitudinal and transverse) having same amplitude, time
period,frequency moving along a straight line in opposite direction a superpose a new wave is
formed. It is called stationary Or standing wave.
= n
PHYSICS
WALLAH
513
Hence unknown frequency is 254 Hz
String
3
_
S
C
I
S
Y
H
P
H
A
L
L
A
W
n..................4
Solution:
254
Phase difference between 2 particle at both sides of node is 180
or
Strain and pressure is maximum at node and minimum at antinode
v
v
Fundamental frequency or
1st harmonic
2
Frequency of
or 2nd harmonic
n2 = 2n1
n2 = 2n1
1 overtone
1 overtone
3
Frequency of
or 3rd harmonic
2nd overtone
n1 =
2l
st
n3 = 3n1
Frequency ratio of
overtones
2:3:4.....
5
Frequency ratio of
harmonics
1:2:3:4.....
6
Nature of waves
Transverse
stationary
4
o
Stretched string Open organ pipe
1
2
4
Parameter
n1 =
2l
st
n3 = 3n1
2nd overtone
2:3:4.....
1:2:3:4.....
Longitudinal
stationary
Closed organ pipe
n1 =
L
I0=10-12W/m2
log10 Intensity
unit W/m2
unit(dB)
I0=Threshold intensity
I
I0
dB=10+ log10
Δ L = change in loudness
Δ I = change in intensity
v
4l
Missing
n3 = 3n1
1st overtone
3:5:7.....
1:3:5:7.....
Longitudinal
stationary
L1=10+ log10
I1
I0
L2=10+ log10
I2
I0
[
( II
L2-L1=10 log
Δ L =10 log
2
0
( II
2
1
( II
-log
(
a) 254 Hz
(
If B is loaded with wax so its frequency decreases
l2+e =
_
l1 +e=
4
l
1
0
I1
L1
I2
L2
[(
By loading
(
Resonance tube experiment
Frequency Increasing
Determination of Unknown Frequency
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Doppler Effect
Case 4
(source is stationary, listener is moving
away from the source)
Whenever there is a relative motion between a source
of sound and the listener, the apparent frequency of sound
heard by the listener is different from the actual
frequency of sound emitted by the source.
VS
S
C
I
S
Y
H
P
H
A
L
L
A
W
VS=0
General equation
(when both source & listener are moving)
(
Case 9
(source is moving away from stationary wall)
Case 5
(source is moving away from the listener,
listener is stationary)
(
Case 2
(The source is stationary & listener is approaching the source)
Case 10
Case 6
(source and listener moving in same direction)
VL=0
l= V+VL
l= _
V VS
x
)θ
VS COS θ
1
1=
V+VL
_
y
z
V+Vs
(
)
θ2
VS
VL
Case 3
(source & listener are approaching each other)
(
Note:
B
VL
l
A
(
V-V
= _L
V
l=
x
)θ
VS COS θ
1
y
)
_L
s
θ1
COS
( V+V
V-V COS
θ2
L
_
s
COS θ1=
VL
COS θ2=
θ1
x
_
x2+y2
y
_
x2+y2
(
A
( V-V
V+V
l
B
(
V+V
= _L
V
(
PHYSICS
WALLAH
l=
(
V+V
l= _L V-Vs
VS
θ2
VS
s
VL
z
VL
VS
θ2
L
_
Case 11
Case 7
(source and listener moving in opposite direction)
VL
COS
( V-V
V+V COS
(
(
S
V
l= _
V+ VS
(
VS
V
l= _
V- VS
(
( V+V
V-V
l= _S VL=0
(
(
Sound
VL=0
(
V-Vs
V-V
l= _L V
VS
Case 1
(listener is stationary & source is approaching the listener)
(
VL
(
+
V+
- VS
V- V
l= _L
V+V
l= _s (
(
VL
Case 8
(source approaching a stationary wall)
Beat(Δ) =lB-lA
=
[V+V -V+V ]
V
L
L
2VL
Δ= _
V
WAVES
4
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