CH APTER

WAVE MOTION & DOPPLER'S EFFECT

A wave is a disturbance that propagates in space, transports energy and momentum from one point to another
without the transport of matter.
CLASSIFICATION OF WAVES
Mechanical (Elastic) waves Progressive waves
Medium Propagation
Necessity of energy
Non-mechanical waves (EM waves) Stationary (standing) waves

1-D (Waves on strings)

Transverse waves
Vibration of Dimension 2-D (Surface waves or ripples on water)
medium particle Longitudinal waves 3-D (Sound or light waves)
• A mechanical wave will be transverse or longitudinal depending on the nature of medium and mode of excitation.
• In strings, mechanical waves are always transverse.
• In gases and liquids, mechanical waves are always longitudinal because fluids cannot sustain shear.
• Partially transverse waves are possible on a liquid surface because surface tension provide some rigidity on
a liquid surface. These waves are called as ripples as they are combination of transverse & longitudinal.
• In solids mechanical waves (may be sound) can be either transverse or longitudinal depending on the mode
of excitation.
• In longitudinal wave motion, oscillatory motion of the medium particles produce regions of compression
(high pressure) and rarefaction (low pressure).

PLANE PROGRESSIVE WAVES

• General equation of wave y = A sin ( wt ± kx + f )

If both wt & kx have same sign then wave moving towards –ve direction
2p
where k = = wave propagation constant
l
¶2 y 1 ¶2 y
• Differential equation : =
¶x 2 v 2 ¶t 2

dx w dx w
Wave velocity (phase velocity) v= = Q wt - kx= constant Þ =
dt k dt k

dy æ dy ö
• Particle velocity v p = = Aw cos ( wt - kx ) v p = - v ´ slope = - v çè ÷ø
dt dx

¶2 y
• Particle acceleration : a p = = -w 2 A sin ( wt - kx ) = -w 2 y
¶t 2 y
For particle 1 : vp ¯ and ap ¯ 1
For particle 2 : vp ­ and ap ¯ 2
t
For particle 3 : vp ­ and ap ­ 3
4

For particle 4 : vp ¯ and ap ­

• Relation between phase difference, path difference & time difference

p 2p
0 Df Dl DT
T/2 T = =
2p l T
l
 ENERGY IN WAVE MOTION WAVE FRONT
• Spherical wave front (source ®point source)
KE 1 æ Dm ö 2
• = ç ÷ vp
volume 2 è volume ø
1 1
= rv 2p = rw2 A 2 cos2 ( wt - kx )
2 2

2
• Cylindrical wave front (source ®linear source)
PE 1 æ dy ö 1
• = rv 2 ç ÷ = rw 2 A 2 cos 2 ( wt - kx )
volume 2 è dx ø 2

TE
• = rw 2 A2 cos2 ( wt - kx ) • Plane wave front (source ® point / linear
volume
source at very large distance)

1 2 2
• Energy density u = rw A
2

[i.e. Average total energy / volume]

• Power : P = (energy density) (volume/ time) INTENSITY OF WAVE
1
æ 1 2 2ö • Due to point source I µ
P = çè rw A ÷ø ( Sv ) r2
2 A r r
y ( r,t ) = sin ( wt - k × r )
[where S = Area of cross-section] (V = wave velocity) r
1
• Due to cylindrical source I µ
Power 1 r
• Intensity : I = = rw 2 A 2 v
area of cross-section 2 A r r
y ( r, t ) = sin ( wt - k × r )
Speed of transverse wave on string : r
• Due to plane source I = constant
r r
T y ( r,t ) = A sin ( wt - k × r )
v= where m = mass/length and
m
INTERFERENCE OF WAVES
T = tension in the string.
y1 = A1sin(wt–kx), y2 = A2sin(wt–kx+f0)
y = y1 + y2 = A sin (wt – kx + f)
KEY POINTS
A2
• A wave can be represent ed by fu ncti on A
y=f(kx ± wt) because it satisfy the differential
f0
¶2 y 1 æ ¶2 y ö w f
equation = 2 ç 2 ÷ where v = .
¶x 2
v è ¶t ø k A1

• A pulse whose wave function is given by where A= A12 + A 22 + 2A1 A 2 cos 0

y=4 / [(2x +5t)2+2] propagates in –x direction as
this wave function is of the form y=f (kx + wt) A 2 sin f0
and tan f =
which represent a wave travelling in –x direction. A1 + A 2 cos f 0
Wave speed = coeff. of t/coeff. of x As I µ A2
• Longitudinal waves can be produced in solids,
So I = I1 + I2 + 2 I1 I2 cos f0
liquids and gases because bulk modulus of elasticity
is present in all three.
• For constructive interference [Maximum intensity] • Rarer Denser

f0 = 2np or path difference = nl where n = 0, 1, yi=Aisin(w t-k1x) yt=A tsin(wt-k2x)

( )
2
2, 3, .... Imax = I1 + I2

• For destructive interference [Minimum Intensity]

l yr=-A rsin(wt+k1x)
f0 = (2n+1)p or path difference = (2n+1)
2 • Denser Rarer

( )
2
where n = 0, 1, 2, 3, .... Imin = I1 - I2 yi=Aisin(wt-k 1x) yt=A tsin(wt-k 2x)

I max - I min
• Degree of hearing= I + I ´ 100
max min

yr=A rsin(wt+k 1x)

REFLECTION AND REFRACTION
BEATS
(TRANSMISSION) OF WAVES
When two sound waves of nearly equal (but not
exactly equal) frequencies travel in same direction,
v1 v2 at a given point due to their super position, intensity
Incident wave alternatively increases and decreases periodically. This
periodic waxing and waning of sound at a given
yi=Aisin(wt-k1x) position is called beats.
Ai Beat frequency = difference of frequencies of two
Medium-1

Medium-2

transmitted wave interfering waves

Ði=Ðr

i
r t
STATIONARY WAVES OR STANDING WAVES
Ar
At When two waves of same frequency and amplitude travel
in opposite direction at same speed, their superstition
gives rise to a new type of wave, called stationary waves
Reflected wave or standing waves. Formation of standing wave is possible
only in bounded medium.
• Let two waves are y1=Asin(wt–kx); y2=Asin(wt+kx)
• The frequency of the wave remain unchanged. by principle of superposition y=y1+y2 =2Acoskxsinwt
¬ Equation of stationary wave
æ v 2 - v1 ö
• Amplitude of reflected wave® A r = ç v + v ÷ A i • As this equation satisfies the wave equation
è 1 2ø
¶2 y 1 ¶2 y
= , it represent a wave.
æ 2v 2 ö ¶x 2 v 2 ¶t 2
• Amplitude of transmitted wave ® A t = ç v + v ÷ A i • Its amplitude is not constant but varies periodically
è 1 2ø
with position.
• If v2 > v1 i.e. medium-2 is rarer • Nodes®amplitude is minimum :
Ar > 0 Þ no phase change in reflected wave l 3l 5l
cos kx = 0 Þ x =
, , ,......
• If v2 < v1 i.e. medium-1 is rarer 4 4 4
Ar < 0 Þ There is a phase change of p in reflected • Antinodes ® amplitude is maximum :
wave l 3l
cos kx = 1 Þ x = 0, , l, ,......
• As At is always positive whatever be v1 & v2 the phase 2 2
of transmitted wave always remains unchanged. • The nodes divide the medium into segments (loops).
All the particles in a segment vibrate in same phase
• In case of reflection from a denser medium or rigid
but in opposite phase with the particles in the adjacent
support or fixed end, there is inversion of reflected
segment.
wave i.e. phase difference of p between reflected and • As nodes are permanently at rest, so no energy can
incident wave. be transmitted across them, i.e. energy of one region
• The transmitted wave is never inverted. (segment) is confined in that region.
Transverse stationary waves in stretched string Sound Waves
Velocity of sound in a medium of elasticity E and
[Fixed at both ends] [fixed end ®Node &
density r is
free end®Antinode]
v= E
Fundamental or v r
f=
first harmonic 2l
Solids Fluids
l= l (Young's Modulus) (Bulk Modulus)
2
v= Y v= B
r r
second harmonic 2v
f= • Newton's formula : Sound propagation is
first overtone 2l
l=l
P
isothermal B = P Þ v =
third harmonic 3v r
f=
second overtone 2l • Laplace correction : Sound propagation is
3l
l=
2 gP
adiabatic B = gP Þ v =
r
fourth harmonic 4v
f= KEY POINTS
third overtone 2l
l=2l • With rise in temperature, velocity of sound
gRT
Fixed at one end in a gas increases as v =
MW
v • With rise in humidity velocity of sound
Fundamental f=
4l increases due to presence of water in air.
l
l= • Pressure has no effect on velocity of sound
4
in a gas as long as temperature remains
third harmonic 3v constant.
f= Displacement and pressure wave
first overtone 4l
3l
l= A sound wave can be described either in terms
4
of the longitudinal displacement suffered by the
particles of the medium (called displacement
fifth harmonic 5v wave) or in terms of the excess pressure generated
f=
second overtone 4l due to compression and rarefaction (called
5l
l= pressure wave).
4
Displacement wave y=Asin(wt–kx)
Pressure wave p = p0cos(wt–kx)
seventh harmonic 7v
f= where p0 = ABk = rAvw
third overtone 4l
7l Note : As sound-sensors (e.g., ear or mike) detect
l=
4 pressure changes, description of sound as
pressure wave is preferred over displacement
wave.
Sonometer
KEYPOINTS
N A
• The pressure wave is 90° out of phase w.r.t.
\\\\\\\\\\\\\\\\\\

N
fixed displacement wave, i.e. displacement will be
\\ \\
\\\\\\\\\\\\\\\\\\\\\\\\\

p T maximum when pressure is minimum and

\ \\\

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
fn =
\

plucking 2l m vice-versa.
(free end) • Intensity in terms of pressure amplitude
[p : number of loops] p20
I=
2rv
 Vibrations of organ pipes Intensity of sound in decibels
æ I ö
Stationary longitudinal waves closed end ® displacement Loudness L = 10log10 ç ÷ dB (decibel)
node, open end® displacement antinode è I0 ø
Where I0 = threshold of human ear = 10–12 W/m2
• Closed end organ pipe
Characteristics of sound
l v
l= Þf= • Loudness ® Sensation received by the ear due to
4 4l
intensity of sound.
3l 3v
l= Þf= • Pitch ® Sensation received by the ear due to
4 4l frequency of sound.
5l 5v • Quality (or Timbre)® Sensation received by the ear
l= Þf=
4 4l due to waveform of sound.
• Only odd harmonics are present Doppler's effect in sound :
• Maximum possible wavelength = 4l
v A stationary source emits wave fronts that propagate
• Frequency of mth overtone = (2m + 1) with constant velocity with constant separation
4l between them and a stationary observer encounters
• Open end organ pipe
them at regular constant intervals at which they were
l v emitted by the source.
l= Þf=
2 2l A moving observer will encounter more or lesser
2v number of wavefronts depending on whether he is
l= lÞf=
2l approaching or receding the source.
3l 3v A source in motion will emit different wave front
l= Þf= at different places and therefore alter wavelength
2 2l
i.e. separation between the wavefronts.
• All harmonics are present
• Maximum possible wavelength is 2l. The apparent change in frequency or pitch due to
v relative motion of source and observer along the
• Frequency of mth overtone = ( m + 1) line of sight is called Doppler Effect.
2l
Source vS v0 observer
• End correction : n Sound Wave
Due to finite momentum of air molecules in organ
Observed frequency
pipes reflection takes place not exactly at open end
but some what above it, so antinode is not formed speed of sound wave w.r.t. observer
exactly at free end but slightly above it. n¢ =
observed wavelength
v v + v0 æ v + v0 ö
In closed organ pipe f1 = n¢ = = n
4 (l + e ) æ v - v s ö çè v - v s ÷ø
where e = 0.6 R (R=radius of pipe) çè ÷
n ø
v æ v0 + vs ö
In open organ pipe f1 = If v0, vs <<<v then n ¢ » çè 1 + ÷n
2 ( l + 2e ) v ø
• Resonance Tube speed of source
• Mach Number=
speed of sound
a a
B
b b Doppler's effect in light :
A A
Case I : Observer Light Source
l1 l/4
O Sv
c æ 1+ vö ü
S N l2
N
3l/4 Frequency n¢ = ç c n » æ1 + v ö nï
v÷ çè ÷
è 1- cø cø ïï
T
ý Violet Shift
æ 1 - vc ö æ vö ï
Wavelength l ¢ = ç v ÷ l » çè 1 - c ÷ø l ï
N è 1 + cø ïþ
P Light Source
Case II : Observer
O v S
æ 1 - vc ö æ vö ü
Frequency n¢ = ç ÷÷ n » ç1 - ÷ n ï
ç v
è 1+ c ø è cø ïï
Wavelength l =2(l2- l1) l2 -3l1 ý Red Shift
End correction e= æ 1 + vc ö
2 æ vö ï
Wavelength l¢ = ç ÷ l » ç1 + ÷ l ï
ç 1- v ÷ è c ø
è c ø þï