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Spectral Analysis of Wave Motion
Dr. Chih-Peng Yu
Dr. C. P. Yu 2
Elastic wave propagation
• Unbounded solids– P-wave, S-wave
• Half space– Surface (Rayleigh) wave
• Double bounded media– Lamb waves
• Slender member
Dr. C. P. Yu 3
2001 Fall
• Waves in Slender members– longitudinal wave– flexural wave– torsional wave
• Waves by different approximate theories– Elementary member– Deep member
Dr. C. P. Yu 4
2002 Spring
• General derivation of waves in solids– P-wave, S-wave, Surface (Rayleigh) wave
• Modification due to bounded media– Lamb waves
Dr. C. P. Yu 5
General Function of Space and Time
• At a specific point in space, the spectral relationship can be expressed as
• In general, at arbitrary position
tin
neAtFtrf 111 )(),(
tinn
nerftrf ),(ˆ),(
Dr. C. P. Yu 6
• Imply (discrete) Fourier Transform pairs
• Or, in a simpler form as
),(ˆ),( nn
FT
IFTrftrf
)(ˆ)( ftf
Dr. C. P. Yu 7
Spectral representation of time derivatives
• Assuming linear functions
• Or, for simplicity,
ti
nnti
nnn efief
tt
f ˆˆ
fifit
fnn
ˆor ˆ
discrete continuous
Dr. C. P. Yu 8
• Derivatives of general order
• It is clear to see the advantage of using spectral approach– time derivatives replaced by algebraic
expressions in Fourier coefficients => simpler
fifit
f mn
mnm
mˆor ˆ
Dr. C. P. Yu 9
Spectral representation of spatial derivatives
• Nothing special
• Or, for simplicity,
tinti
nnn e
x
fef
xx
f ˆ
ˆ
x
f
x
f
x
f n
ˆ
or ˆ
discrete continuous
Dr. C. P. Yu 10
• It is clear to see another advantage of using spectral approach– partial differential equation becomes ordinary differenti
al equation in Frequency domain => solution form is solvable or at least easier to be solved
– This is also true for using other transform integrals, such as Laplace Transform, Bessel-Laplace Transform
Dr. C. P. Yu 11
Spectral relation
• Consider a general, linear, homogeneous differential equation for u(r,t) – with all coefficients independent of time
– Assume one dimensional problem
02
12
2
2
2
211
tr
uc
t
ubu
ra
t
ub
r
uau
n
tinn
nexutxutru ),(ˆ),(),(
Dr. C. P. Yu 12
• The spectral representation of the general differential equation becomes
0
ˆˆ
ˆ
ˆˆ
ˆ
122
2
2
2
11
n
ti
nnnn
n
nnn
n
ne
x
uciubi
x
ua
ubix
ua
u
Dr. C. P. Yu 13
• eint is independent for all n. Thus the spectral representation results in n simultaneous equations as
• Or, in a general form as
• Aj depend on frequency and are complex.
nrfor
x
uciaubibi r
rrrr
,1
0ˆ
ˆ1 1122
1
0ˆ
),(ˆ
),(ˆ),(2
2
321
x
uxA
x
uxAuxA
Dr. C. P. Yu 14
• When Aj(x,) independent of position, the original partial differential equation has been transformed into n simultaneous ordinary linear differential equation.
• The solution form is et , the transformed ODE becomes then
• The equation in the ( ) is called characteristic equation, which can be solved to give values for
can be complex, so the solution form is in a form as
02321 xeAAA
xikCexu )()(ˆ with = + ik
Dr. C. P. Yu 15
is referred to as the attenuation factor of the wave motion. It represents the non-propagating and the attenuated components of the wave.
• k is the wave number. It represents the propagating parts of the wave.
• So, for a propagating component of the wave, the solution can be expressed as
xikCexu )()(ˆ with = ± ik
ikxCexu )(ˆ
± stands for the traveling direction (to the right or left)
Dr. C. P. Yu 16
Propagating speeds
• Consider the propagating component
• The time response is then in the form as
ikxCexu )(ˆ
j
tixikj
ti eeCexutxu j )(ˆ),(
j represents the number of characteristic constant
Dr. C. P. Yu 17
• The time response is then in the form as
• For each j, we can see the response corresponds to (infinite) sinusoids traveling with a speed of
• cj is called the phase speed corresponding to j
j
tk
xik
jj
txkij
jj
j eCeCtxu
),(
jj k
c
Dr. C. P. Yu 18
• So, for a specific j with only components traveling towards one direction (say -kx), we have the wave response expressed as
• Consider the interaction between two propagating wave components, the resultant response is thus
t
kxik
tkxi CeCetxu
),(
tdk
dxktxki
txkir
txkir
eC
eCeCtxu rrrr
*
*2*
1
cos2
),(
**
11
2 1* rr xxxwith
Dr. C. P. Yu 19
• The first sinusoid is the average response called carrier wave . It travels at the average speed of the two interacting wave components, c* = * / k*.
• The second term represents the modulated effect between the interacting components. This is called group wave traveling at a speed
tdk
dx
ktxkieCtxu*
*2* cos2 ),( **
dk
dckc
dk
d
dk
dcg
*
*
Dr. C. P. Yu 20
• It can be expected when there are many waves interacting together, the overall effect would be a carrier wave modulated by a group wave.
• In reality, the individual sinusoids is hard to be observed unless through an FFT scheme.
• The wave energy and varying amplitude of the wave envelope travel at group speed.
Dr. C. P. Yu 21
Transfer function
• Let’s exam again the displacement function
jjxik
j HAeCxu j ˆˆ)(ˆ
ctionansfer fun-called tr is the soH
ectrumplitude sp is the amAwhere
j
j
ˆ
ˆ
In a displacement – force relationship, transfer function is then the inverse of dynamic stiffness function.
Dr. C. P. Yu 22
Summary of wave terms
• Angular frequency (rad/s) = • Cyclic frequency (Hz) = f = / 2• Period (sec)= T = 1/f = 2 / • Wave number (1/length) = k = 2 / = / c• Wave length (length) = = 2 c / = 2 / k• Phase (rad) = = (kx - t) = • Phase velocity (length/s) = c = / k = / 2
• Group speed (length/s) = cg = d / dk
)(2
)( ctxctxc
Dr. C. P. Yu 23
Specific terms
• Spectrum relation : vs k• Dispersion relation : vs c• Non-dispersion : phase velocity is constant for all f
requency• Evanescent wave : the attenuated non-propagating
components of waves• Carrier wave : main zero-crossing sinusoid waves • Group wave : modulation of wave groups
Dr. C. P. Yu 24
Simple wave examples
• Wave equation of the 1-D axial member– Non-dispersion
• Flexural wave in a beam– dispersion
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