The Hankel transform of first- and second-order tensor fields: definition and use for modeling circularly symmetric leaky waveguides.

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Eric DUCASSE, Slah YAACOUBI - The Hankel transform of first- and second-order tensor fields:

definition and use for modeling circularly symmetric leaky waveguides. - Physics Procedia - Vol.

3, n°1, p.505-514 - 2010

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The Hankel transform of rst- and se ond-order

tensor elds: denition and use for modeling

ir ularly symmetri leaky waveguides

Éri Du asse 1,2,3 , Slah Yaa oubi 4,2,3 . 1

ArtsetMétiersParisTe h,EsplanadedesArtsetMétiers,F-33405Talen e,e.du assei2m.u-bordeaux1.fr,Fran e.

2

Université deBordeaux, UMR 5469,351 ours delaLibération, F-33405Talen e,Fran e.

3

CNRS,UMR 5469,351 ours delaLibération, F-33405Talen e,Fran e.

4

Laboratoire Central des Ponts et Chaussées, Division Re onnaissan e et Mé anique des Sols, Route de Porni ,

BP4129, F-44341BouguenaisCedex, Fran e.

Abstra t

A tensor Hankel transform (THT) is dened for ve tor elds, su h as displa ement, and se ond-order tensor

elds, su h as stress or strain. The THT establishes a bije tion between the real spa e and the wave-ve tor

domain, and, remarkably, annotberedu ed to a s alartransform appliedseparately toea h omponent.

Oneof the advantagesof this approa his thatsome standard elasti ity problems an be on iselyrewritten

by applying this tensor integral transform oupled with an azimuthal Fourier series expansion. A simple and

ompa t formulationof the boundary onditions is alsoa hieved.

Thanks to the THT, we obtain for ea h azimuthal wavenumber and ea h azimuthal dire tion exa tly the

samewaveequationasforastandard2Dmodelofelasti wavepropagation. Thus,wavessimilartothestandard

plane P, SV and SH waves are naturally found.

Lastly,theTHT isusedto al ulatethe ultrasoni eldinanisotropi ylindri alleakywaveguide, thewalls

of whi hradiatingintoa surroundingelasti medium,by using astandard s attering approa h.

Keywords

Cir ular symmetri waveguide, Cylindri al oordinates, Elasti Wave S attering, P, SV and SH waves, Tensor

Hankeltransform, 2D Fourier transform.

1 Introdu tion

The modeling of elasti wave propagation in ir ularly symmetri media an be done using spa e integral

transform in ylindri al oordinates and does not ne essitate a potential formulation. In this ontext, The

Hankel transform of a s alar eld has been well known for a long time. It appears when the 2D Fourier

transform of a s alar eld is rewritten in polar oordinates (e.g.,[1℄, [2℄). Similarly,a tensor Hankel transform

(THT) an be naturally dened for ve tor elds and matrix elds. Even if we believe that su h a tensor

integral transformwas probablypublished between 1920 and 1960,we did not nd inthe literature neitherthe

elaboration of the on ept of THT nor the Hankel transform of a se ond-order tensor. In our knowledge, the

Hankel transform of a ve tor eld was introdu ed in ele tromagnetism only thirty years ago ([3℄, followed by

[4℄, [5℄). We onlyfound itin a re ent book [6℄ inanimpli it formulation forelasti ity appli ations. In the rst

se tion, denitions and properties of the THT are given for s alar,ve tor, and matrix elds.

Inthe se ondse tion,theTHT isappliedtoelastodynami s. Byusing it,somestandard elasti ityproblems

anbe on iselyrewritten. Adire tlinkisnotablyestablishedbetweena ylinderintheradial-wavenumber(

k

)/ axial-position(

z

)/frequen y(

ω

) domain and the standard 2D problem. Thus, the plane P, SV and SH waves an be transposed for a ir ularly symmetri geometry, for ea h azimuthal wavenumber and ea h azimuthal

The THT is also onvenient for easily a hieving the ultrasoni eld generated in an elasti half-spa e by a

transdu er in onta t with it. A dire t appli ation is the modeling of propagation in an isotropi ylindri al

leaky waveguide, the walls of whi h radiating into a surrounding elasti medium. Indeed, the in ident wave

generated at the end of the ylinder is rstly al ulated in the

(k, z, ω)

domain and then rewritten in the

(r, k

z

, ω)

domain,

k

z

denoting the axial wavenumber. Lastly, the ree ted (guided) wave in the ylinder and the transmitted wave radiated into the surrounding media are dedu ed. Only prin iples are des ribed here.

Detailed al ulation, approximations and numeri alresults are given in arelated paper[7℄.

2 Denitions

2.1 Hankel transform of a s alar eld

Thestandard2DFouriertransformofas alareld

φ

isanintegraltransformwhi hestablishesa orresponden e between

φ

in the real spa e and

φ

ˆ

in the wave-ve tor domain,both inCartesian oordinates (see Fig.1):

ˆ

φ(k) =

1

2π

Z

R

2

φ(x)

e i

k · x

d

x

|

{z

}

m

z

}|

{

φ(x) =

1

2π

Z

R

2

ˆ

φ(k)

e

−

i

k · x

d

k ,

(1)

where

x

,

k

and the dot denote the position

x

y



, the wave-ve tor

k

x

k

y



, and the standard s alar produ t,

respe tively. (a) (b) PSfragrepla ements

x

y

r

θ

O

n

r

n

θ

r

cos(θ)

r

sin(θ)

PSfragrepla ements

k

x

k

y

k

α

O

n

k

n

α

k

sin(α)

−k cos(α)

Figure1: Cartesian andpolar oordinates (a)intherealspa e and (b)inthewave-ve tor domain.

Inpolar oordinates (see Fig.1), the azimuthalFourier series expansionof the s alar eld

φ

leads to:

φ(x) = φ

0

(r) +

+∞

X

n=1

cos (n θ) φ

⊢

n

(r) + sin (n θ) φ

⊥

n

(r) ,

(2.a) with

φ

0

(r) =

1

2π

Z

π

−π

φ(x)

d

θ ,

(2.b)

φ

n

⊢

(r) =

1

π

Z

π

−π

cos (n θ) φ(x)

d

θ ,

(2. )

φ

⊥

n

(r) =

1

π

Z

π

−π

sin (n θ) φ(x)

d

θ .

(2.d)

The 2D Fouriertransform (1)be omes:

ˆ

φ(k) =

Z

+∞

0

 1

2π

Z

π

−π

φ(x)

e i

k r

sin(α−θ)

d

θ



r

d

r

(3)

and, by usingthe integral representation (e.g., [8,Ÿ17.23℄) of the Besselfun tion of the rst kind:

J

n

(a) =

1

2π

Z

π

−π

e i

( a sin(β)−n β )

d

β

(4)

and the property

J

−n

(a) = (−1)

n

_J

n

(a)

,we obtainaftersome algebrathe azimuthalFourier seriesexpansion of

ˆ

φ

:

ˆ

φ(k) = Φ

0

(k) +

+∞

X

n=2

even n

cos(n α) Φ

⊢

n

(k) + sin(n α) Φ

⊥

n

(k)

+

i

+∞

X

n=1

odd n

− cos(n α) Φ

⊥

n

(k) + sin(n α) Φ

⊢

n

(k) ,

(5)

the oe ientsof whi h being the

n

th

-order Hankel transform (e.g.,[1℄, [2℄) ofthe oe ients in(2):

Φ

⊥,⊢

n

(k) =

Z

+∞

0

φ

⊥,⊢

_n

(r)

J

n

(k r) r

d

r

|

{z

}

m

z

}|

{

φ

⊥,⊢

_n

(r) =

Z

+∞

0

Φ

⊥,⊢

n

(k)

J

n

(k r) k

d

k .

(6)

Notethe dependen e on the parity of

n

in(5) be ause the azimuth of the wave-ve tor

−k

is

α+π

.

2.2 THT of a rst-order tensor eld

A ve tor eld

u

and its 2D Fouriertransform

u

ˆ

are written in ylindri al oordinates:

u(x) =





u

r

(x)

u

θ

(x)

u

z

(x)





and ˆ

u(k) =





i

U

k

(k)

i

U

α

(k)

U

z

(k)





=





i i

1



⋆U(k) ,

(7)

the hange-of-basis matri esbeing (see Fig.1):

R =

cos θ −sin θ 0

sin θ cos θ 0

0

0

1

!

and

S =

sin α cos α 0

−cos α sin α 0

0

0

1

!

,

(8)

and

⋆

denoting the elementwise produ t.

Coe ients i in (7) ensure that ea h omponent

U

ξ

of

U

satises

U

ξ

(−k) = U

∗

ξ

(k)

( omplex onjugate) if the ve tor eld

u

is real-valued.

The 2D transformof the ve tor eld

u

is:

ˆ

u(k) =

Z

+∞

0

 1

2π

Z

π

−π

e i

k r

sin(α−θ)

S

T

_{R u(x)}

d

θ



r

d

r ,

(9)

Its azimuthal Fourier series expansiongives:

u(x) = u

0

(r) +

+∞

X

n=1

Cos(nθ) ⋆ u

n

⊢

(r) + Sin(nθ) ⋆ u

⊥

n

(r) ,

(10)

where

Cos(a)=

cos a

sin a

cos a

,

Sin(a)=

sin a

−cos a

sin a

.

Aftersome algebra, (4), (9) and (10)yield the azimuthal Fourierseries expansion of

U(k)

:

U(k) = U

0

(k) +

+∞

X

n=2

even n

Cos(nα)⋆U

n

⊢

(k) + Sin(nα)⋆U

⊥

n

(k)

+

i

+∞

X

n=1

odd n

−Cos(nα)⋆U

⊥

n

(k) + Sin(nα)⋆U

n

⊢

(k) ,

(11) Ea h ve tor

U

⊥,⊢

n

is the

n

th

-order ve tor Hankeltransform [3℄ [4℄[5℄of the ve tor

u

⊥,⊢

n

:

U

⊥,⊢

n

(k) =

Z

+∞

0

J

_n

_{(k r) u}

⊥,⊢

_n

_{(r) r}

d

r

|

{z

}

m

z

}|

{

u

⊥,⊢

n

(r) =

Z

+∞

0

J

_n

_{(k r) U}

⊥,⊢

_n

_{(k) k}

d

k ,

(12)

where the matrix

J

n

is dened by:

J

_n

(a) =









J

n+1

(a)−J

n−1

(a)

2

J

n+1

(a)+J

n−1

(a)

2

0

J

n+1

(a)+J

n−1

(a)

2

J

n+1

(a)−J

n−1

(a)

2

0

0

0

J

n

(a)









.

(13)

NotethattheTHTofareal-valuedeldisreal-valuedandthatthePlan herel-Parsevalidentity [3℄be omes:

Z

+∞

0

u

n

(r)

· v

n

(r) r

d

r

=

Z

+∞

0

U

n

(k)

· V

n

(k) k

d

k .

(14)

2.3 THT of a se ond-order tensor eld

A matrix eld

m

and its2D Fouriertransform

m

ˆ

are writtenin ylindri al oordinates:

m

_{(x) =}

" m

rr

(x) m

rθ

(x) m

rz

(x)

m

θr

(x) m

θθ

(x) m

θz

(x)

m

zr

(x) m

zθ

(x) m

zz

(x)

#

(15) and

ˆ

m

_{(k) =}

" M

kk

(k) M

kα

(k)

i

M

kz

(k)

M

αk

(k) M

αα

(k)

i

M

αz

(k)

i

M

zk

(k)

i

M

zα

(k) M

zz

(k)

#

=

1 1

i

1 1

i i i

1

!

⋆M(k) ,

(16) su h that

M

ξζ

(−k) = M

∗

ξζ

(k)

if

m

is real-valued. The 2D transformof the matrix eld

m

is:

ˆ

m

_{(k) =}

Z

+∞

0

 1

2π

Z

π

−π

e i

k r

sin(α−θ)

S

T

_{R m(x) R}

T

_S

d

θ



r

d

r .

(17)

Its azimuthal Fourier series expansionyields:

m

_{(x) = m}

₀

_{(r) +}

+∞

X

n=1

Cos(nθ) ⋆ m

⊢

n

(r) +

Sin(nθ) ⋆ m

n

⊥

(r) ,

(18.a)

where

Cos(a)=

cos a sin a cos a

sin a cos a sin a

cos a sin a cos a

!

(18.b)

and

Sin(a)=

sin a −cos a sin a

−cos a sin a −cos a

sin a

−cos a sin a

!

.

(18. )

Withthe same approa h asabove, the matrix Hankel transform an be dened as follows:

M

⊥,⊢

n

(k) =

Z

+∞

0

J

_n

(k r) : m

⊥,⊢

n

(r) r

d

r

|

{z

}

m

z

}|

{

m

⊥,⊢

n

(r) =

Z

+∞

0

J

_n

_{(k r) :M}

⊥,⊢

_n

_{(k) k}

d

k ,

(19)

where the olon denotes the tensor produ t su h that

( T :

P )

ij

= Σ

_k,l

T

ijkl

P

kl

,

T

and

P

being fourth-order and se ond-order tensors, respe tively. The fourth-order symmetri tensor

J

n

is dened by (

(a)

 is omitted in

J

ζ

(a)

):

J

_n,··11

(a) =









−J

n−2

+2J

n

−J

n+2

4

J

n−2

−J

n+2

4

0

J

n−2

−J

n+2

4

J

n−2

+2J

n

+J

n+2

4

0

0

0

0









,

(20.a)

J

_n,··22

(a) =









J

n−2

+2J

n

+J

n+2

4

−J

n−2

+J

n+2

4

0

−J

n−2

+J

n+2

4

−J

n−2

+2J

n

−J

n+2

4

0

0

0

0









,

(20.b)

J

_n,··33

(a) =









0 0 0

0 0 0

0 0

J

n









,

(20. )

J

_n,··23

_{(a) = J}

_n,··32

T

_{(a) =}









0 0

J

n−1

+J

n+1

2

0 0

−J

n−1

+J

n+1

2

0 0

0









,

(20.d)

J

_n,··13

(a) = J

_n,··31

T

(a) =









0 0

−J

n−1

+J

n+1

2

0 0

J

n−1

+J

n+1

2

0 0

0









,

(20.e)

J

_n,··12

(a) = J

T

n,··21

(a) =









J

n−2

−J

n+2

4

−J

n−2

+2J

n

−J

n+2

4

0

−J

n−2

−2J

n

−J

n+2

4

−J

n−2

+J

n+2

4

0

0

0

0









.

(20.f)

3 Appli ation to elastodynami s

Letus applynow theTHT to the displa ementeld

u

and the stress tensor

σ

inanisotropi elasti mediumof mass density

ρ

, longitudinaland transverse velo ities

c

L

,

c

T

, submitted toan external for edensity

f

.

3.1 Formulation in the

( k , z , ω )

domain

Be ause the stress tensor is symmetri ,the matrix Hankel transform an berewritten by using Voigtnotation:

σ

_n

⊥,⊢

(r, z, ω) =

























σ

⊥,⊢

n,rr

σ

_n,θθ

⊥,⊢

σ

⊥,⊢

n,zz

σ

_n,θz

⊥,⊢

σ

⊥,⊢

n,rz

σ

_n,rθ

⊥,⊢

























=

Z

+∞

0

J

n

(k r)

























Σ

⊥,⊢

_n,kk

(k, z, ω)

Σ

⊥,⊢

n,αα

(k, z, ω)

Σ

⊥,⊢

n,zz

(k, z, ω)

Σ

⊥,⊢

n,αz

(k, z, ω)

Σ

⊥,⊢

_n,kz

(k, z, ω)

Σ

⊥,⊢

_n,kα

(k, z, ω)

























k

d

k ,

(21)

the (non-symmetri ) matrix

J

n

(a)

being:



























−J

n−2

+2J

n

−J

n+2

4

J

n−2

+2J

n

+J

n+2

4

0

0

0

J

n−2

−J

n+2

2

J

n−2

+2J

n

+J

n+2

4

−J

n−2

+2J

n

−J

n+2

4

0

0

0

−J

n−2

+J

n+2

2

0

0

J

n

0

0

0

0

0

0

−J

n−1

+J

n+1

2

J

n−1

+J

n+1

2

0

0

0

0

J

n−1

+J

n+1

2

−J

n−1

+J

n+1

2

0

J

n−2

−J

n+2

4

−J

n−2

+J

n+2

4

0

0

0

−J

n−2

−J

n+2

2



























.

(22)

One an demonstrate that, by using the THT, the Hooke's law (23) and the Newton's se ond law (24)

(e.g., [9℄[10℄) an be simplyexpressed inthe

( k , z , ω )

-domain:

1

ρ

Σ

⊥,⊢

n

=

















0 0 c

2

L

−2c

2

T

0 0 c

2

L

−2c

2

T

0 0

c

2

L

0 c

2

T

0

c

2

_T

0

0

0 0

0

















∂

z

U

⊥,⊢

n

+ k

















c

2

_L

0 0

c

2

L

−2c

2

T

0 0

c

2

L

−2c

2

T

0 0

0

0 0

0

0

−c

2

T

0

c

2

T

0

















U

⊥,⊢

n

,

(23)

− ρω

2

U

⊥,⊢

n

=





0 0001 0

0 0010 0

0 0100 0





∂

z

Σ

⊥,⊢

n

+ k





−10000 0

0 0000

−1

0 0001 0





Σ

⊥,⊢

_n

+ F

⊥,⊢

_n

.

(24)

3.2 The standard 2D problem

Notethat(23)and(24)areindependentfrom

n

and,remarkably,exa tlythesameasforthestandard2Dproblem (

y

-invariant)inCartesian oordinates, 

(k, z, ω)

 being omittedin

U

⊢,⊥

2D

(k, z, ω)

and

Σ

⊢,⊥

2D

(k, z, ω)

:

u(x, z, ω) =

Z

+∞

0

sin kx

cos kx

cos kx

!

⋆

U

⊢

_2D

+

− cos kx

sin kx

sin kx

!

⋆

U

⊥

_2D

d

k ,

(25) and

σ(x, z, ω) =

Z

+∞

0













cos kx

cos kx

cos kx

cos kx

sin kx

−sin kx













⋆

Σ

⊢

2D

+













sin kx

sin kx

sin kx

sin kx

−cos kx

cos kx













⋆

Σ

⊥

2D

d

k .

(26)

3.3 Solutions

In any ase, (23) and (24) leads tothe ordinary dierential system (

2D

,

n

,

⊢

and

⊥

omitted):



















c

_T

2

∂

z

2

U

_k

− k (c

2

L

−c

T

2

) ∂

z

U

z

+ (ω

2

−k

2

c

L

2

) U

k

=

−

_ρ

1

F

k

c

2

T

∂

z

2

U

_α,y

+ (ω

2

−k

2

c

_T

2

) U

_α,y

=

−

1

ρ

F

α,y

c

2

L

∂

z

2

U

_z

+ k (c

_L

2

−c

_T

2

) ∂

_z

U

_k

+ (ω

2

−k

2

c

_T

2

) U

_z

=

−

1

ρ

F

z

.

(27)

The solutionsof (27) withoutsour e-term are:

U

±

P

(k, z, ω) = a

±

P

(k, ω)

e

∓

i

k

z,L

(k,ω) z

k

0

±

i

k

z,L

(k, ω)

!

,

(28)

U

±

SV

(k, z, ω) = a

±

SV

(k, ω)

e

∓

i

k

z,T

(k,ω) z

±

i

k

z,T

(k, ω)

0

k

!

,

(29)

U

±

SH

(k, z, ω) = a

±

SH

(k, ω)

e

∓

i

k

z,T

(k,ω) z

0

k

0

!

,

(30)

where the axial wavenumber is:

k

z,X

(k, ω) =



























sign(ω)

s

ω

2

c

2

X

− k

2

_{, k < c}

X

|ω|

−

i

s

k

2

₋

ω

2

c

2

_X

, k > c

X

|ω|

.

(31)

The latter solutions orrespond in the 2D ase to the standard P, SV and SH waves, respe tively. In the

initial problem, for ea h azimuthal wavenumber

n

and for ea h azimuthal dire tion

⊢

or

⊥

, equivalent waves are dened and shouldbeworthy ofthorough study. Thus, they are used belowto al ulatetheresponse ofthe

half-spa e

(z > 0)

toa surfa e for e-sour e

f

0

(x, ω)

lo atedonthe boundary

(z = 0)

.

3.4 Response of a half-spa e to a surfa e for e-sour e

After an azimuthal Fourier series expansion and a THT, the problem to solve in the

( k , z , ω )

-domain is to nd the oe ients

a

⊢,⊥

n,P

,

a

⊢,⊥

n,SV

,

a

⊢,⊥

n,SH

su h that the THT of the displa ement

u

⊢,⊥

n

(r, z, ω)

(dened in (10)) is rewritten as follows:

U

⊢,⊥

n

(k, z, ω) =





k

e

−

i

k

z,L

z

i

k

z,T

e

−

i

k

z,T

z

0

0

0

k

e

−

i

k

z,T

z

i

k

z,L

e

−

i

k

z,L

z

k

e

−

i

k

z,T

z

0











a

⊢,⊥

_n,P

a

⊢,⊥

_n,SV

a

⊢,⊥

_n,SH





 ,

(32)

and that the boundary ondition:

F

⊢,⊥

_0,n

(k, ω) = ρ c

2

T





2

i

k k

z,L

k

2

_−k

2

z,T

0

0

0

i

k k

z,T

k

2

−k

2

z,T

2

i

k k

z,T

0











a

⊢,⊥

_n,P

a

⊢,⊥

_n,SV

a

⊢,⊥

_n,SH





 .

(33)

is satised. This problem iseasy tosolveif the determinantof the square matrix in(33)is dierent from zero,

i.e. the Rayleighwaves are not ex ited:

F

⊢,⊥

0,n

(k,

±c

R

k) =

→

PSfragrepla ements

x

z

surroundingmedium

ir ular half- ylinder

transdu er

Figure2: A half- ylinder surrounded bya half-inniteelasti mediumand ex ited atits end.

3.5 Case of a ylinder embedded in an elasti media

The previous se tion summarizes the rst step to al ulate, by using a standard s attering approa h, the eld

generated into a ir ular half- ylinder surrounded by a half-innite elasti medium by a surfa e for e-sour e

lo atedat itsend (see. Fig.2).

Indeed,thein identeldinthe ylindri alwaveguideisgivenby(32)and(33). Inase ondstep,thisin ident

eld isexpressed inthe

(r, k

z

, ω)

-domainto al ulatethe ree ted eld(guidedwave)and thetransmitted eld (radiated intothe surrounded medium). More details are given ina related paper[7℄.

Consequently,the THT is relevant formodeling ir ularproblems asthe latter be auseea h ase

hara ter-ized by anazimuthalwavenumber

n

and anazimuthaldire tion

⊢

or

⊥

anbetreatedseparatelyand issimilar to the standard 2D problem.

4 Referen es

[1℄ R. Bra ewell, The Fourier transform anditsappli ations, 3 rd

edition, M Graw-HillS ien e Engineering,1 st

editionin 1965, NewYork,1986,ISBN 978-0-07-007015-8.

[2℄ I.N.Sneddon,Z. Olesiak,W.Nowa kiandG.Eason,Appli ation of integral transformsin the theory of elasti ity, Springer-Verlag,Wien,NewYork,1975.

[3℄ W. C.ChewandJ.A. Kong,Resonan eofnonaxialsymmetri modesin ir ular mi rostripdiskantenna, J.Math. Phys., vol.21,pp.25902598,1980.

[4℄ S. M.Ali,W.C.ChewandJ.A.Kong,Ve torHankeltransformanalysisofannular-ringmi rostripantenna, IEEET rans-a tionsonAntennasandPropagation,vol.30, pp.637644,1982.

[5℄ W. C.Chewand T.M.Habashy, Theuseofve tortransformsin solvingsomeele tromagneti s attering problems, IEEE Transa tionsonAntennasandPropagation,vol.34,pp.871879,1986.

[6℄ E.Kausel,FundamentalSolutionsinElastodynami s: ACompendium,CambridgeUniversityPress,2006,ISBN0-521-85570-5.

[7℄ S. Yaa oubi, E. Du asse, M. Des hamps and L. Laguerre, Cal ulation of the ultrasoni eld generated by a transdu er non-axiallylo atedat the endof a ylindri alwaveguidesurrounded byanelasti medium, in ICU'09,SantiagoofChile, 2009.

[8℄ E.T.WhittakerandG.N.Watson,A Courseof ModernAnalysis,4 th

edition,CambridgeUniversityPress,NewYork,1927, ISBN 978-0-521-09189-3.

[9℄ F. J. Fedorov, Theory of elasti waves in rystals, Plenum Press / Kluwer A ademi Publishers, New York, 1968, ISBN 0-30630-309-4.

[10℄ B.A.Auld,A ousti FieldsandWaves inSolids,2 nd