Wave processes
Transfer of oscillations from a source to the medium (space)
vibrations in time,
disturbances in space,
moving disturbances in space-time associated
with the transfer/transformation of energy.
Wave: disturbance of any physical property
of a system around a reference value
travelling in space.
oSound
oLight
oRadiowaves
oSeismic waves
oOcean waves
oParticle waves
oGravitational waves
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L2-1
Mechanical waves
Mechanical wave:
The energy of a vibration is moving away from the source in the form
of a disturbance within the surrounding elastic medium!
The transmission medium is neither infinitely stiff nor infinitely pliable.
No mass transport!
Classification upon elastic properties :
Compressible media
– volumetric elasticity
Longitudinal waves: the displacement
of the medium is parallel to the propagation of the wave.
Compression and expansion wave.
In gas - pressure wave
- tensile elasticity– solid media
Transverse wave: the displacement of the medium
is perpendicular to the direction of propagation of the wave
crests (highs) and troughs (lows) combined waves
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L2-2
Waves
Wave’s dimension – number of coordinates necessary
to describe wave propagation
One dimensional wave
Two dimensional (planar)
3D wave
Wave front
Planar wave
Spherical wave
Wave ray
Single pulse
Travelling wave
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Harmonic wave
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L2-3
One dimensional travelling wave (1)
disturbance
 ( x, t )
Propagation to the right
 ( x  vt )
Wave function

t  0   ( x)
t0
vt
to the left
x
 ( x  vt )
 ( x , t )   ( x  vt )
Wave function describes change of the state at different points
at different times.
b2
eg .  ( x , t ) 
b  ( x  vt )2
;  ( x, t )  ( x
t  t1   ( x ) t1
Spacial profile at particular time
x  x1   ( t ) x1
Time evolution at the chosen point
1
 vt ) 2
Velocity of propagation of the certain state of disturbance
(a constant phase point) – phase velocity
   max  const  x  vt  const
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dx
v
dt
Phase velocity
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L2-4
One dimensional harmonic wave
x0
y
y   A sin(t )
Source oscillations
t
observer at a distance x records disturbance with a delay
y  A sin(
Oscillations at that point:
Angular frequence
of the wave:
wavelength:
2

x  t )
2

 2f
T
y
x
vT  
Wave number:
y( x , t )  A sin( kx  t )
2

Wave travelling to the left

k
  kv
phase : 
Dispersion relationship
y  A sin( kx  t )  A sin k ( x  vt )  f ( x  vt )
Wave travelling to the right
y  A sin( kx  t )
y  A sin( kx  t )

k

k
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t  2T
 ( k )  kv
 const
No dispersion
 f (v ,  )
dispersion
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L2-5
Wave equation
One dimensional travelling harmonic wave
y  A sin( kx  t   )
2 y
x
2

Periodical changes in time and in space

1 2 y
2
v t
2
k
v
One dimesional
wave equation
Solutions for planar one dimensional wave
2 y
x
2

1 2 y
y  A sin( kx  t   )
v 2 t 2
y  A sin(t  kx   ' )
equivalence
y  A cos(t  kx   ' ' )
For any wave function
 2
x
2

 2
y
2

 2
z
2

1  2
v 2 t 2
Wave equation
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L2-6
Dispersion of phase velocity
Phase velocity depends on the properties of the transmission medium
which may be dependent on the frequency of the deformation.

k

k
 v  const
v
Dispersion relationship
Dispersive medium
Dispersion curve
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v  f ( ) ; v  f ( )
Nondispersive medium
v  f ( ) ; v  f (  )
v ( )
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L2-7
Mechanical wave velocity
general:
Elastic property
v
transverse wave
in an elastic medium
Inertial property
Tensile elasticity
Longitudinal wave
in an elastic medium
v
Volumetric elasticity
v
B

Rigidity modulus

density
bulk modulus
density
Longitudinal wave in gases
v
F
Young modulus

density
v  f ( ) no dispersion

Dispersion of velocity
Low frequencies
-isothermal changes of volume
Higher frequencies
– adiabatic changes of volume
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E
v
Longitudinal wave
in a solid object:
Phase velocity in a string
(transverse wave)
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G
visot 
v ad
RT

 RT




molar mass
cp
cv
 1  vad  visot
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L2-8
Energy, power and intensity in wave motion
source
medium
energy
E k  max  y  0 E p  max  y  0
For a harmonic wave Ek i Ep are in phase
Average power transmitted
1T
P   Pdt
T0
I
P
S
 C  f 2 A2v
P  IS
2
Physics--2016

P f2
intensity: average power transmitted through
unit area normal to the direction of propagation
C  f 2 A2 v
I
S
IA
P  A2
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P  4 r12 I 1  4 r22 I 2
A
1
r
I1
r12

I2
r22
inverse square law
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L2-9
Superposition of waves
Interference:
superposition of waves of equal frequences and amplitudes
 1 (r , t )
 2 (r , t )
 1  A sin( kx  t   )
 2  A sin( kx  t )
 (r , t )   1 (r , t )   2 (r , t )


  2 A cos sin( kx  t  )
2
2
amplitude Aw  2 A cos
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
2
Resulting wave vs components:
•same frequency
•different amplitude
•shifted in phase
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L2-10
Interference (1)
Aw  2 A cos

2
1.   0    2n
Aw  2 A
constructive
2.
Aw  0
  ( 2n  1)
destructive
3.
  ( 2n  1)
   2 n
resultant wave is shifted in phase
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L2-11
Interference (3)
condition: coherency path difference
  const
  s  s2  s1
1  ks1  t
   2  1  k ( s2  s1 )
s1
 2  ks2  t

k
 s  n
1   2
and
 s
z1
k
2

s2
s  5
constructive
s  ( 2n  1)

2
z
s  3
destructive
O1
r1
s  
s  0
r2
O2
r2  r1  s  const  k
hyperbole
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L2-12
Standing wave
stationary wave
Result of interference between two waves travelling
in opposite directions.
eg due to the reflection at the interface
y1  y m sin( kx  t )
y 2  y m sin( kx  t )
y  y1  y 2  y m sin( kx  t )  sin( kx  t )
y  2 y m sin kx cos t
y  A( x ) cos t
nodes
W nW n1 

2
standing wave equation
A( x )  2 y m sin kx  0  kx  n
2

x  n  x  n

2
y
A( x )  2 ym  kx  ( 2n  1)
anti-nodes
S n S n  1 
2
 
2
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x  ( 2n  1)

2

t2
t1
2
 x  ( 2n  1)
x

4
S
W
S
W
S
W
S
W
S
W
S
W
t3
S
distance between conjugative nodes or antinodes
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L2-13
Standing waves – characteristic frequencies
Creation of the standing waves depends on the boundary conditions
eaxamples: transverse wave in a string
f 
v
1.
v

fI 
1.
 L    4L
4L
4
v
3
4
 3 fI
2.
 L    L f III  3
4L
4
3
f N  Nf I N  1,3,5.....
v

F


2
fundamental frequency
III harmonic
V harmonic
VII harmonic
IX harmonic
fI 
 L    2L
v
2L
v
f II   2 f I
L
f N  Nf I N  1,2,3,4....
2.   L
f min
; max
harmonics – overtones (aliquots)
f  f min
fundametal frequency
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fundamental frequency
II harmonic
III harmonic
IV harmonic
  max
V harmonic
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L2-14
Resonators
Characteristic frequencies of the standing waves in systems are equal to the
fundametal frequencies of their free oscillations (resonant frequencies)
Resonator:
Asborbs waves of resonant frequencies
Standing waves
in gas tubes
f N  Nf I
N  1,2,3,4....
f N  Nf I
N  1,3,5.....
Vibrating plates - Chladni figures
n=2
m=1
musical instruments
n=4
m=4
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n=2
m=2
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L2-15
Waves on water reservoirs
A seiche is a standing wave in an enclosed
or partially enclosed body of water
seiche
Windwaves
Tidal waves
A tsunami wave:
 earthquake – whole water depth (mass) shakes
small amplitude (wave height) offshore (0,2-1m)
low frequency (period :30min) (windwave: 10s)
very long wavelength (tens of km) (windwave: 150m)
high velocity at deep water of an open sea (>speed of sound)
(windwave: 15m/s)
v deep
travels huge distances (1000 km) without loosing energy
slows down on a shallow water
huge amplitude on the shore (12m)
accumulation of energy - onshore runup
v shallow  gd epth
intensity
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 g
d   v   height 
I  A2 v  const
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L2-16
Wave packet (1)
Wave packet: envelope or packet containing an arbitrary number of wave forms.
eg sinusoidal waves of slightly different frequencies
Combination of one dimensional waves
; 1   2  0
A1  A2
v
1

1   2
2
No dispersion
k1 k 2
y1  A cos(1t  k1 x )
y 2  A cos( 2 t  k 2 x )
superposition
y w  y1  y 2  A cos(1t  k1 x )  A cos( 2 t  k 2 x )
y w  2 A cos(
1   2
2
t
  2
k1  k 2
k  k2
x )  cos( 1
t 1
x)
2
2
2
packet
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Packet „content”
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L2-17
Wave packet (2)
y w  2 A cos(
1   2
2
t
  2
k1  k 2
k  k2
x )  cos( 1
t 1
x)
2
2
2
Travelling wave of frequency and wave number close
to the primary wave
amplitude:

k
y A  2 A cos(
t
x)
2
2
1   2
2
y

x
Depends on the coordinate and on time – modulated

A 
2
packet
frequency
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kA 
k
2
packet
wave number
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A  (
k

t
x)
2
2
packet
phase
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L2-18
Group velocity
packet’s velocity?
The group velocity of a wave is the velocity
with which the envelope of the wave propagates.
Group velocity
In a nondispersive medium
u  v phase
v phase  f (  )  f ( k )
group velocity of the packet = phase velocity
Energy is transmitted with a group velocity
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L2-19
Doppler’s effect (1)
change in frequency and wavelength of a wave as perceived
by an observer in result of observer’s and source relative motion.
f
v
vo
vz
source frequency
f'
perceived frequency
wave velocity in a medium
observer’s velocity
Christian Doppler
(1803-1853)
source velocity
assumptions:
- velocities < velocity of the wave v o
- unilinearity of source and observer
1842 r
, v z  v
observer and a source both in motion in the medium
inward motion
approaching each other:
outward motion
receding from each other:
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v  vo
f '
 f
v  vz
f '
v  vo
 f
v  vz
perceived frequency
increases
perceived frequency
decreases
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L2-20
Doppler’s effect for the electromagnetic waves
No material medium required
Moving observer case can not be distinguished from the moving source case.
Relative velocity creates the frequency shift
1. outward motion
receding from each other
f ' f
1  vw / c
 f
1  vw / c
„red shift”
2. inward motion
approaching each other
f ' f
1  vw / c
 f „blue shift”
1  vw / c
Astronomical observations
of distant planets
Doppler radar
The received frequency shift depends
on the radial velocity of target.
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L2-21
Sonic boom
vz
vz  v
v
Energy cumulation at the wave front.
Abrupt change of the gas pressure at the front – shock wave.
vv  v
Sonic boom
Conical shape of the front wave envelopes.
P2  P1  v z t
r  vt
vz t
vt
v
sin  

„Mach cone”
vzt vz
vz
 Ma
v
Mach number
Ernst Mach
(1838-1916)
Mach 1 is approximately 1,225 km/h (761 mph)
Light shock waves: "blue glow" of nuclear reactors
Cherenkov radiation. Electromagnetic radiation emitted
when a charged particle passes through an insulator at
a speed greater than the speed of light in that medium.
Physics--2016
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L2-22