Upload
hadat
View
272
Download
1
Embed Size (px)
Citation preview
Definition of Waves
• A wave is a disturbance that moves through a medium without giving the medium, as a whole, any permanent displacement.
• The general name for these waves is progressive wave.
• If the disturbance takes place perpendicular to the direction of propagation of the wave, the wave is called transverse.
• If the disturbance is along the direction of propagation of the wave, it is called longitudinal.
Characteristics of Waves
• At any point, the disturbance is a function of time and at any instant, the disturbance is a function of the position of the point.
• In a sound wave, the disturbance is pressure-variation in a medium.
• In the transmission of light in a medium or vacuum, the disturbance is the variation of the strengths of the electric and magnetic fields.
• In a progressive wave motion, it is the disturbance that moves and not the particles of the medium.
• To demonstrate wave motion, take the loose end of a long rope which is fixed at the other end quickly up and down
• Crests and troughs of the waves move down the rope
• If the rope is infinity long such waves are called progressive waves
Progressive Waves
• If the rope is fixed at both ends, the progressive waves traveling on it are reflected and combined to form standing waves
Standing Waves
The first four harmonics of the standing waves allowed between the two
fixed ends of a string
Transverse vs Longitudinal Waves
• Transverse wave: the displacements or oscillations in the medium are transverse to the direction of propagation e.g. electromagnetic (EM) waves , waves on strings
• Longitudinal wave: the oscillations are parallel to the direction of wave propagatione.g. sound waves
Plane Waves
• Take a plane perpendicular to the direction of wave propagation and all oscillators lying within that plane have a common phase
• Over such a plane, all parameters describing the wave motion remain constant
• The crests and troughs are planes of maximum amplitude of oscillation, which are rad out of phase
• Crest = a plane of maximum positive amplitude
• Trough = a plane of maximum negative amplitude
The Wave Equation
2
2
22
21
t
y
cx
y
Tc
2
+d
T
T
(x +dx, y +dy)
(x , y )
• The wave equation of small element of string of linear density and constant tension T
where
c is the phase or wave velocity.
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
222
tansin small very
sin)sin(
ignored becan thusand small very
1
t
y
Tx
y
t
ydxdx
x
yT
t
ydx
x
y
x
yT
x
y
t
ydxTdT
dxdsdxds
x
y
x
y
x
y
dx
ds
x
y
dx
dx
dx
sd
dydxds
xdxx
xdxx
xdxx
x
y
x
ydx
x
y
x
y
x
y
dxx
y
2
2
2
21
Waves in One Dimension
• Suppose a wave moves along the x-axis with constant velocity c and without any change of shape (i.e. with no dispersion) and the disturbance takes place parallel to the y-axis, then
y (x, t) = f (ct – x) (1)
defines a one-dimensional wave along the positive direction of the x-axis (forward wave)
t
xc
Waves in One Dimension
• A wave which is the same in all respect but moving in the opposite direction (i.e. along the direction of x decreasing) is given by Eqn. (1) with the sign of v changed:
y (x, t) = f (ct + x) (2)
• This is known as backward wave.
Waves in One Dimension
• Eqns. (1) and (2) satisfy the second-order partial differential equation:
(3)
• Eqn. (3) is known as the non-dispersive wave equation.
2
2
22
21
t
y
cx
y
Solution of Wave equation
• A solution to the wave equation
, where is the oscillation
frequency and
• The wave is moving in the positive x direction.
)(2
sin)sin( xctatay
x2
2
2
c
The Wave Equation
• At position x = 0, wave equation
• Any oscillator to its right at some position x will be set in motion at some later time t.
• Have a phase lag with respect to the oscillator at x = 0.
• The wavelength is the separation in space between any two oscillators with a phase difference 2 rad.
tay sin
)(2
sin)sin( xctatay
The Wave Equation
• The period of oscillation
• An observer at any point would be passed by wavelengths per second.
• If the wave is moving to the left the sign is changed.
• Wave moving to right
• Wave moving to left
1
c
)(2
sin)sin( xctatay
)(2
sin)sin( xctatay
Equivalent Wave Expressions
where is called wave number.
• Cosine functions are equally valid.
• For both sine and cosine
)(2
sin xctay
)(2sin
x
tay
)(sinc
xtay
)sin( kxtay
ck
2
kxtiaey
The Wave Equation
2
2
22
2
2
2
2
2
2
2
2
2
2
2
1
)sin(),cos(
)sin(),cos(
)sin(
t
y
ct
yk
x
y
x
y
t
x
x
yc
x
y
kt
y
kxtakx
ykxtka
x
y
kxtat
ykxta
t
y
kxtay
Three Velocities in Wave Motion
1. Particle velocitySimple harmonic velocity of the oscillator about its equilibrium position
2. Wave or phase velocityThe velocity with which planes of equal phase, crests or troughs, progress through the medium
3. Group velocityA number of waves of different frequencies, wavelengths and velocities may be superposed to form a group. Motion of such a pulse would be described by its group velocity
• Locus of oscillator displacements in a continuous medium as
a wave passes over them travelling in the positive x-direction
• The wavelength is defined as the distance between any two
oscillators having a phase difference of 2 rad
Wave or Phase velocity
• The wave or phase velocity is
• It is the rate at which disturbance moves across the oscillators.
• The oscillator or particle velocity is a simple harmonic velocity
t
xc
t
y
)cos(
)sin(
kxtat
y
kxtay
Wave or Phase Velocity
Wave or Phase Velocity = the rate at which disturbance moves across the oscillators
Wave or Phase Velocity =t
x
Oscillator or Particle Velocity is a simple harmonic velocity
Oscillator or Particle Velocity = t
y
Characteristic Impedance of a String
• Any medium through which waves propagate will present an impedance to those waves
• If the medium is lossless, and possesses no resistive or dissipation mechanism, for a string the impedance is determined by inertia and elasticity
• The presence of a loss mechanism will introduce a complex term into the impedance
(the string as a forced oscillator)
• The transverse impedance is define as:
Characteristic Impedance of a String
• Characteristic impedance of the string:
(the string as a forced oscillator)
v
FZ
velocitytransverse
forcetransverse
cc
TZ 2
since cT
Characteristic Impedance of a String(the string as a forced oscillator)
The string as a forced oscillator with a vertical force F0eit driving it at one end
For small :
x
yTTTeF
titansin
0
Characteristic Impedance of a String(the string as a forced oscillator)
displacement of the progressive waves may be represented exponentially by:
amplitude A may
be complex
At the end of the string, where x = 0
)( kxtie
Ay
)0(
0
0
kti
x
tieikT
x
yTeF A
T
c
i
F
ikT
F00
A)(0 kxti
eT
c
i
F
y
Characteristic Impedance of a String(the string as a forced oscillator)
transverse velocity:
velocity amplitude:
transverse impedance:
Characteristic Impedance of the string
Since the velocity c is determined by the inertia and the elasticity,
the impedance is also governed by these properties
)(
0
kxtie
T
cF
yv
ZFv /0
cc
TZ 2
since cT
Z1 = 1c1
Z2 = 2c2
Reflection and Transmission
• Suppose a string consists of two sections smoothly joined at
a point x = 0 with a tension T
• Waves on a string of impedance Z1= 1c1 reflected and
transmitted at the boundary x = 0 where the string changes to
impedance Z2= 2c2
Reflection and Transmission
Incident wave:
Reflected wave:
Transmitted wave:
find the reflection and transmission amplitude coefficients
i.e. the relative values of B1 and A2 with respect to A1
)(
1
1xkti
ieAy
)(
1
1xkti
reBy
)(
2
2xkti
teAy
)(
1
1xkti
ieAy
)(
1
1xkti
reBy
)(
2
2xkti
teAy
find the reflection and
transmission amplitude
coefficients i.e. the relative values
of B1 and A2 with respect to A1
Boundary condition No. 1 at the impedance discontinuity at x = 0
Reflection and Transmission
1. A geometrical condition that the displacement is the
same immediately to the left and right of x = 0 for all
time, so that there is no discontinuity of displacement
triyyy
)(
2
)(
1
)(
1
211xktixktixkti
eAeBeA
0At x )1(Eq211
ABA
Boundary condition No. 2 at the impedance discontinuity at x = 0
Reflection and Transmission
2. A dynamical condition that there is a continuity of the
transverse force T(y/x) at x = 0, and therefore a continuous
slope
tri
yx
Tyyx
T
at x = 0 for all t
221111TAkTBkTAk
2
2
1
1
1
1
Ac
TB
c
TA
c
T
Reflection and Transmission
These coefficients are independent of
222
2
111
1
and Zcc
TZc
c
T
)2(Eq)(22111
AZBAZ
Reflection coefficient of amplitude:21
21
1
1
ZZ
ZZ
A
B
Transmission coefficient of amplitude:21
1
1
22
ZZ
Z
A
A
Solving Eqs. (1) and (2)
)(
11
)(
11
)(
1
)(
1
11
11
xktixkti
ri
xktixkti
ri
eBikeAikyyx
eBeAyy
tri
yx
Tyyx
T
)(
22
)(
2
2
2
xkti
t
xkti
t
eAikyx
eAy
1111
0,0At BikAikyyx
txri
220,0At Aiky
xtx
t
tri
yx
Tyyx
T
tx
0,0At
221111AikBikAik
2
2
1
1
1
1
Ac
TB
c
TA
c
T
221111AkBkAk
22111)( AZBAZ
222
2
111
1
Zcc
T
Zcc
T
• If Z2 = , B1/A1= 1
incident wave is completely reflected with a
phase change of
(conditions that necessary for standing waves to exist)
• If Z2 = 0 (x =0 is a free end of the string)
B1/A1= 1, A2/A1= 2
the flick at the end of a whip or free end string
Reflection and Transmission
21
21
1
1
ZZ
ZZ
A
B
21
1
1
22
ZZ
Z
A
A
• If Z2 = , B1/A1= 1
incident wave is completely
reflected with a phase change of
(conditions that necessary for
standing waves to exist)
• If Z2 = 0
(x =0 is a free end of the string)
B1/A1= 1, A2/A1= 2
the flick at the end of a whip or
free end string
Reflection and Transmission of Energy
What happens to the energy in a wave when it meets a
boundary between two media of different impedance values?
(the wave function of transferring energy throughout a medium)
Consider each unit length, mass , of the string as a simple
harmonic oscillator of maximum amplitude A
Total energy: = wave frequency
The rate at which energy is being carried along the string:
22
2
1AE
cA22
2
1velocity)(energy
Reflection and Transmission of Energy
The rate at which energy leaves the boundary, via the reflected
and transmitted waves:
the rate of energy arriving at the boundary x = 0 is the energy
arriving with the incident wave:
energy is conserved, and all energy arriving at the boundary in the
incident wave leaves the boundary in the reflected and transmitted waves
2
1
2
1
2
1
2
112
1
2
1AZAc
2
1
2
12
21
2
2
1
2
2112
1
2
2
2
2
2
2
1
2
1
2
2
2
22
2
1
2
11
2
1
)(
4)(
2
1
2
1
2
1
2
1
2
1
AZZZ
ZZZZZA
AZBZAcBc
Reflected and Transmitted Intensity Coefficients
If Z1 = Z2 no energy is reflected
and the impedances are said to be matched
2
21
21
2
1
1
2
11
2
11
EnergyIncident
Energy Reflected
ZZ
ZZ
A
B
AZ
BZ
2
21
21
2
11
2
224
EnergyIncident
Energy dTransmitte
ZZ
ZZ
AZ
AZ
Matching of Impedances
Why Important?
• Long distance cables carrying energy must be accurately matched at all joints to avoid wastage from energy reflection
Example:
• The power transfer from any generator is a maximum when the load matches the generator impedance
• A loudspeaker is matched to the impedance of the power output of an amplifier by choosing then correct turns ratio on the coupling transformer
Matching of Impedances
Insertion of a coupling element
between two mismatched impedances
Remark: when a smooth joint exists between two strings of different
impedances, energy will be reflected at the boundary
Goal: to eliminate energy reflection and match the impedances
Require to match the impedances Z1 = 1c1 and Z3 = 3c3
by the smooth insertion of a string of length l and
impedance Z2 = 2c2
Our problem is to find the values of l and Z2
Matching of Impedances
The impedances Z1 and Z3 of two strings are matched by the
insertion of a length l of a string of impedance Z2
Matching of Impedances
we seek to make the ratio
Boundary conditions:
y and T(y/x) are continuous across the junctions
x = 0 and x = l
1EnergyIncident
Energy dTransmitte
2
11
2
33
AZ
AZ
Matching of Impedances
Between Z1 and Z2 the continuity of y gives:
Continuity of T(y/x) gives
Dividing the above equation by and remember
At x = 0
)(
2
)(
2
)(
1
)(
1
2211xktixktixktixkti
eBeAeBeA
)0at(2211
xBABA
22221111
BikAikTBikAikT
ZcT/ckT /
222111
BAZBAZ
Matching of Impedances
At x = l
Continuity of T(y/x) gives:
Continuity of y gives:
From the four boundary equations, solve for the ratio A3/A1
Refer to the H.J. Pain, “The Physics of Vibrations and Waves”,6th Edition, pg 122-123 for detail derivation
322
22 AeBeAliklik
33222
22 AZeBeAZliklik
lkrrlkr
r
A
A
2
22
23122
22
13
2
13
2
1
3
sincos1
4
Matching of Impedances
2
1
2
3
13
2
11
2
331
EnergyIncident
Energy dTransmitte
A
A
rAZ
AZ
lkrrlkr
r
2
22
23122
22
13
13
sincos1
4
havewe1sinand0cos,4/chooseweif222
lklkl
23122
2312
13
2
11
2
33when1
4rr
rr
r
AZ
AZ
• A string of fixed length l with both ends rigidly clamped
• Consider wave with an amplitude a traveling in the positive x-direction and an amplitude b traveling in the negative x-direction
• The displacement on the string at any point is given by:
Standing Waves
with the boundary condition that y = 0 at x = 0 and x = l
)()( kxtikxtibeaey
Standing Waves
Boundary condition: y = 0 at x = 0
A wave in either direction meeting the infinite impedance at either end is completely reflected with a phase change in amplitude
a = b
An expression of y which satisfies the standing wave time dependent form of the wave equation:
kxaeieeaeytiikxikxti
sin2
02
2
2
yk
x
y
tieba
)(0
Standing Waves
Boundary condition: y = 0 at x = l
Limiting the value of allowed frequencies to:
22
n
n
n
nl
c
l
ncf
nc
l
c
lkl 0sinsin
l
cn
n
kliae
eeae
beae
ti
ikliklti
kltiklti
sin20
0
0)()(
Standing Waves
normal frequencies or modes of vibration:
Such allowed frequencies define the
length of the string as an exact
number of half wavelengths
(Fundamental mode)
The first four harmonics, n =1, 2, 3, 4
of the standing waves allowed
between the two fixed ends of a string
l
xn
c
xn
sinsin
2
nn
l
Standing Waves
• For n > 1, there will be a number of positions along the string where the displacement is always zero called nodes or nodal point
These points occur where
there are (n1) positions equally spaced along the string in the
nth harmonic where the displacement is always zero
• Standing waves arise when a single mode is excited and the incident and
reflected waves are superposed
• If the amplitudes of these progressive waves are equal and opposite
(resulting from complete reflection), nodal points will exist
0sinsin
l
xn
c
xn
),.....,3,2,1,0( nrrl
xn
0sin xkn
rxkn
Standing Waves
the complete expression for the displacement of the nth
harmonic is given by:
c
xtBtAy
n
nnnnn
sinsincos
c
xtitiay
n
nnn
sinsincos)(2
where the amplitude of the nth mode is given by aBAnn
22/1
22
we can express this in the form:
Standing Wave Ratio
• If a progressive wave system is partially reflected from a boundary, let the amplitude reflection coefficient B1/A1 = r, for r < 1
• The maximum amplitude at reinforcement is (A1 + B1), the minimum amplitude (A1 B1)
• The ratio of the maximum to minimum amplitudes is called standing wave ratio (SWR)
• Reflection coefficient:
r
r
BA
BA
1
1SWR
11
11
1SWR
1SWR
1
1
A
Br
Energy in Each Normal Mode of a Vibrating String
• A vibrating string possesses both kinetic and potential
energy
• Kinetic energy of an element of
length dx and linear density
• Total kinetic energy:
2
2
1ydx
dxyE2
1
02
1)kinetic(
Energy in Each Normal Mode of a Vibrating String
• Potential energy = the work done by thee tension T in
extending an element of length dx to a new length ds when
the string is vibrating
neglect higher powers of y/x
dxT
dxTdxdsTE
x
y
x
y
2
2
2
1
11)()potential(2
1
....112
2
12 2
1
x
y
x
y
...2
)1(1)1(
2
x
nnnxx
n
Energy in Each Normal Mode of a Vibrating String
• For standing waves:
c
xntBtAy
nnnnn
sinsincos
c
x
nnnnnnn
ntBtAy
sincossin
c
x
nnnnc
n nntBtA
x
y
cossincos
dxtBtAEc
xl
nnnnnn
n
2
0
22
2
1sincossin)kinetic(
dxtBtATEc
xl
nnnnc
n
nn
2
0
2
2
1cossincos)potential(
2
2
Energy in Each Normal Mode of a Vibrating String
where m is the mass of the string
= the square of the maximum displacement of the
mode
2cT
)(
)()potentialkinetic(
222
4
1
222
4
1
nnn
nnnn
BAm
BAlE
)(22
nnBA
a
axxaxdx
4
2sin
2sin
2
a
axxaxdx
4
2sin
2cos
2
2
sin0
)/(4
)/(2sin
2
2
0
ldx
l
c
xcx
c
xl
n
nn
2
cos0
)/(4
)/(2sin
2
2
0
ldx
l
c
xcx
c
xl
n
nn
dxtBtAEc
xl
nnnnnn
n
2
0
22
2
1sincossin)kinetic(
dxtBtATEc
xl
nnnnc
n
nn
2
0
2
2
1cossincos)potential(
2
2
2
22
2
1cossin)kinetic(
l
nnnnnntBtAE
22
4
122
4
1)kinetic(
nnnnnBmBlE At any time t:
2
2
2
1sincos)potential(
2
2
l
nnnnc
ntBtATE
n
At any time t: 22
4
122
4
1)potential(
nnnnnAmAlE
2cT
)(
)()potential()kinetic(
222
4
1
222
4
1
nnn
nnnnn
BAm
BAlEE
Wave Groups and Group Velocity
• Waves to occur as a mixture of a number or group of
component frequencies
e.g. white light is composed of visible wavelength spectrum
of 400 nm to 700 nm
• The behavior of such a group leads to the group velocity
dispersion causes the spatial separation
of a white light into components of
different wavelength (different colour)
Superposition of two waves of almost equal frequencies
• A group consists of two components of equal amplitude a but
frequencies 1 and 2 which differ by a small amount.
• Their displacements:
• Superposition of amplitude and phase:
)cos()cos(222111
xktayxktay
2
)(
2
)(cos
2
)(
2
)(cos2
21212121
21
xkktxkktayyy
a wave system with a frequency (1+ 2)/2 which is very close to
the frequency of either component but with a maximum amplitude
of 2a, modulated in space and time by a very slowly varying
envelope of frequency (1 2)/2 and wave number (k1 k2)/2
• The velocity of the new wave is
Superposition of two waves of almost equal frequencies
so that the component frequencies and their superposition, or
group will travel with the same velocity, the profile of their
combination in Figure 5.11 remaining constant
)/()(2121
kk
ckk 2211
//If the phase velocities , gives
ckk
kkc
kk
21
21
21
21)(
• For the two frequency components have different phase
velocities so that 1/k1 2/k2
Superposition of two waves of almost equal frequencies
The superposition of the two waves will no longer remain
constant and the group profile will change with time
Dispersive medium = medium in which the phase
velocity is frequency dependent
(i.e. /k not constant)
kkk
21
21velocityGroup
• If a group contain a number of components of frequencies
which are nearly equal the original, expression for the group
velocity is written:
Superposition of two waves of almost equal frequencies
Since = kv (v is the phase velocity)
group velocity:
gv
dk
d
k
dk
dvkvkv
dk
d
dk
dv
g
)(
d
dvvv
g
• A non-dispersive medium where /k is constant, so that
vg = v, for instance free space behaviour towards light waves
• A normal dispersion relation, vg < v
• An anomalous dispersion relation, vg > v
Characteristic of a Normal Mode
• all the masses move in SHM at the same frequency• normal modes are completely independent of each other • general motion of the system is a superposition of the
normal modes
• All of these properties of normal modes are shared by standing waves on a vibrating string
• all the particles of the string perform SHM with the same frequency
• the standing waves are the normal modes of the vibrating string
Standing Waves as Normal Modes
Superposition of Normal Modes
the expression for the n-th normal mode of a vibrating string of length L
the motion of the string will be a superposition of normal modes given by:
txkAtxynnnn
cossin),(
0sin xkn
Displacement zero (nodes) occur when sine term = 0
,....)2,1,0( nnxkn
Example: superposition of the 3rd normal mode with a relative amplitude of 1.0 and the 13th normal mode with a relative amplitude of 0.5
3rd harmonic y3(x, 0) of a string at t = 0
13th harmonic y13(x, 0) of a string at t = 0
The superposition of the two harmonics to give the resultant shape of the string at t = 0
(a)
(b)
(c)
• To excite the two normal modes in this way, we would somehow have to constrain the shape of the string as in (c) and then release it at time t = 0
• It is impractical to do this and in practice we pluck a string to cause it to vibrate
• Example the string is displaced a distance d at one quarter of its length
• Initially, the string has a triangular shape and this shape clearly does not match any of the shapes of the normal modes
it is possible to reproduce this triangular shape by adding together the normal modes of the string with appropriate amplitudes
The first three excited normal modes of the string: y1(x, 0), y2(x, 0) and y3(x, 0)
L
xAxy sin)0,(
11
L
xAxy
2sin)0,(
22
L
xAxy
3sin)0,(
33
• Even using just the first three normal modes we get a surprisingly good fit to the triangular shape
• By adding more normal modes, we would achieve even better agreement, especially with respect to the sharp corner
The superposition of the first three normal modes gives a good reproduction of the initial triangular shape of the string except for the sharp corner
When we pluck a string we excite many of its normal modes and the subsequent motion of the string is given by the superposition of these normal modes according to equation
Amplitudes of Normal Modes
The initial shape of the string f (x), i.e. at t = 0 is given by
The expansion of the above equation is known as a Fourier series and the amplitudes A1, A2, . . . as Fourier coefficients or Fourier amplitude
Any shape f (x) of the string with fixed end points [f (0) = f(L) = 0] can be written as a superposition of these sine functions with appropriate values for the coefficients A1, A2, . . . , i.e. in the form
Amplitudes of Normal Modes
Fourier series
Example
A string of length L is displaced at its mid-point by a
distance d and released at t = 0, as shown in figure below.
Find the first three normal modes that are excited and their
amplitudes in terms of the initial displacement d.
Let the shape of the string at time t = 0 by the function y = f (x)
Solution:
Inspection of figure
shows that:
To cope with the ‘kink’ in f (x) at x = L/2, we split the
integral in the Fourier amplitude equation (An in slide-9)
into two parts, so that
Substituting for f (x) over the appropriate ranges of x, the right-hand side
of this equation becomes:
Useful formula for the indefinite integrals
The final result is
Solution (continued…..):