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

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2001 Fall

• Waves in Slender members– longitudinal wave– flexural wave– torsional wave

• Waves by different approximate theories– Elementary member– Deep member

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2002 Spring

• General derivation of waves in solids– P-wave, S-wave, Surface (Rayleigh) wave

• Modification due to bounded media– Lamb waves

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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),(

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• Imply (discrete) Fourier Transform pairs

• Or, in a simpler form as

),(ˆ),( nn

FT

IFTrftrf

)(ˆ)( ftf

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Spectral representation of time derivatives

• Assuming linear functions

• Or, for simplicity,

ti

nnti

nnn efief

tt

f ˆˆ

fifit

fnn

ˆor ˆ

discrete continuous

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• 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

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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

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• 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

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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),(),(

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• 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

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• 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

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• 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

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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)

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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

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• 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

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• 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

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• 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

*

*

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• 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.

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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.

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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

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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

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Simple wave examples

• Wave equation of the 1-D axial member– Non-dispersion

• Flexural wave in a beam– dispersion