Reflexion/ Transmission of a plane wave on a plane interface

(1)

Reflexion/ Transmission of a plane wave on a plane interface

F. Rahmi

^a

, H. Aklouche

^a

, N. Bedrici-Fraï

^b,c

, Ph. Gatignol

^b

, C. Potel

^d

a Laboratoire Dynamique Des Moteurs et Vibroacoustique, Université M. B. de Boumerdès 35 000, Algérie.

b Laboratoire Roberval, UMR CNRS 6253, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne Cedex, France.

c ESTACA, BP 76121, 53061 LAVAL Cedex 9, France.

d LAUM, UMR CNRS 6613, Université du Maine, 72 085 LE MANS Cedex 9, France.

* Corresponding author : Tel : +213 774 65 75 44; Fax : + 213 24 91 30 99; Email : rahmifahim@gmail.com Received: 23 May 2011, accepted: 30 September 2011

Abstract

On a plane interface between two elastic half-space, P and SV waves propagating in the

( ) ^{x z} ^,

pare related by Snell's law and the law of continuity of displacement components

u

^x_and

u

_z and constraints

σ

^zz_and

σ

_zx on both sides of the interface. An incident wave P or wave SV generates two P or SV reflected waves and two transmitted waves P or SV. The four continuity equations are written in the form of a matrix multiplied by a vector transmission-reflection coefficient, defined for potential movement of the particles. For an planar boundary between fluids with different characteristic impedances, there is continuity of

u

z_and

σ

_zz on both sides of the interface and the shear

σ

^zx in the medium must vanish at the interface (fluid media involving only perfect no viscosity, so that was normal stresses, not shear stress

σ

_xz

= 0

). As soon as the angle of incidence exceeds a critical value of incidence, the wave for which the value of incidence is greater than

30

^° becomes evanescent. The reflection- transmission coefficients become complex.

Keywords : P waves, incidence, reflection, transmission.

1. Introduction

The non destructive characterization of structures has grown considerably in recent years. The ultrasonic methods have become the preferred tool for non destructive evaluation of mechanical properties of materials [1]. They also have the advantage of being applicable to a wide range of materials. Surface waves were a long time the subject of extensive studies which had applications in both non- destructive tests in signal processing [1, 2]. Much research has been conducted on the interaction of such waves with surface discontinuities but most on the reflection and transmission. The elastic waves that result from moving particles propagate only in material media, so that electromagnetic waves propagate in a vacuum also. It was possible to address immediately the propagation of elastic waves in a fluid because this medium is a set of free particles; their properties are expressed using scalar parameters: density

ρ ,

coefficient of compressibility

χ ,

mean free path (average distance traveled by a particle between two collisions). The propagating waves are fully described by a scalar, pressure, or potential expansion of the displacement or velocity [1, 2, 3].

In summary, in a perfect fluid :

- The polarization of the wave, that is to say, the particle motion is necessarily longitudinal, parallel to its wave vector, the absence of viscosity preventing any shearing motion;

- The speed of propagation is expressed by

1 ;

c = ρχ

- The Poynting vector indicating the direction of energy propagation is parallel to the wave vector;

- The polarization of reflected and transmitted waves, on both sides of a surface separate two media of different impedances, and that of the incident wave. Their amplitudes and propagation directions are given by the Snell-Descartes in which only are involved the impedances of the media and the angle of the incident wave.

- The wave continues to propagate when the distance between a maximum and minimum pressure becomes the order of magnitude of the mean free path of particles.

2. Reflection / transmission at a plane interface

Consider the interface between two homogeneous fluids of different velocities

( c

1_and

c

2

) ;

when changing propagation environment, changing the characteristics of a plane wave is particularly interesting. The change in speed causes a specular reflection of the wave in the first medium (in a direction symmetrical to the normal at the point of incidence) and a refraction of the wave in the second medium at an angle given by famous law of Snell.

A progressive plane wave acoustic pressure

p )

a

which is of the form

f t )

1

( − n r c r r

_a

.

1

) ,

maintained by a source located at infinity (for z tends to −∞

),

propagates in a half-space fluid (density

ρ

1 and speed

c

1

),

bounded by a planar interface located at

z

=

0,

separating it from another half- space

ρ

² velocity of

c

²

)

(Figure 1).

Journal of New Technology and Materials JNTM

Vol. 01, N°00 (2011)33-38 OEB Univ. Publish. Co.

(2)

F T th m T in a a m m 2 w

⎛ ⎜

⎝

C (e e

p ) v

^)r

W T o

ρ

B c

Figure 1. Refle fluids w The axes are c he incident w makes an angle The interaction nterface gene amplitude

B

)1

angle

θ

¹

′

_with

middle F2 am making an angl 2. 1. Writing th The prop written F1

2 2

x z c

⎛ ∂ ∂

+ −

∂ ∂

⎝

Conditions at equal sound each medium)

( )

1

, ;

p x z t ) =

( )

1

, ; . v x z t n

^)r ^r

Where

n r

is th The projectio outside sources

1

v . p

t n ρ ^∂ + ^∂

∂ ∂

)r r Because the p

ommute,

ection / transm with different ch chosen so that wave is located

e

θ

¹_{with the}

n of the obliqu erates a refle

(and directi the axis

( ) ^Oz

mplitude

A )

2

le

θ

²

′

_{with the}

he problem in pagation equa

2

(

2 2 1 1

1 P x ,

c t

∂ ⎞

∂ ⎠ ⎟ )

the boundar pressures nor are written as

( )

2

, ; , p x z t )

( )

2

, ;

v x z t

=

^)r

he normal to th on on the no s in the mediu

1

0 p n =

∂

)

projection ope

mission at the i haracteristic im t the direction d in the plane

axis

( ) ^Oz

_(o

ue incident pl cted wave in on of propag ), and a wave (and directio e axis

( ) ^Oz ^).

n the middle F ation in the

)

; 0, , z t = ∀ ∀ x

ry (interface rmal and pec

follows:

, 0,

∀ x z = ∀

) ^{. , ,} ⁿ

^r

^∀ ^{x z} ⁼

he interface

(

ormal

n r

of um F1 is writte erators on sca

interface betw mpedances.

of propagatio e

( ^Oxz ) ^,

_an

obliqualy incid lane wave with n the middle

gation making transmitted in on of propaga

F1

fluid medium

0,

z t

∀ ≤ ∀

F1 / F2)

z

culiar velocitie

∀ t

(2-a)

0, ∀ t

_(2-b)

)

here n r r = e

z

Euler's equa en:

(3) alar

n r

and

∂

34

ween on of

nd it dent).

h the e F1

g an n the ation

m is

(1)

= 0

es in

) ^.

ation

∂ ∂ t

w

A

W

(

¹

1

. v ρ ^∂ t

∂ )r

Similarly, in th

(

²

2

. v n ρ ^∂ t

∂ )r

By using equa written, equiva

(

1 1

1 p , ;

n x z ρ

∂

∂ ) Or

(

1 1

1 p , ;

z x z ρ

∂

∂ )

The reflected not a conditi infinity).

2. 2. Writing t The pro written as F2

2 2

x z

⎛ ∂ ∂

+ −

⎜∂ ∂ ⎝

The bounda identical to The medium than at

z

=

0

3. Solutions o The pressu pressure field can be written

( ) (

1

, ;

_a

p x z t ) = p )

(7)

And the field

( )

2

, ;

p ) x z t =

(8)

Where

n n

^{r r}_a

,

propagation i denotes the po The conditio

)

₁

n p

n + ∂

∂

r )

he middle F2,

)

₂

n p

n + ∂

∂

r )

ations (4), the alently

)

²

2

; 1 p

t ρ n

= ∂

∂ )

)

²

2

; 1 p

t ρ z

= ∂

∂ )

d wave propag ion of Somm the problem in opagation equ

2

(

2 2 2 2

1 P x

c t

∂ ⎞

∂ ⎠ ⎟ )

ary conditions o those F2 containing

0,

write a cond of the problem ure field

p

⁾¹_in

incident

p

⁾^a

n as follows

( ) ( x z t ^{, ;} + p x z )

b

^,

F2 is written i

2 2

2

. f t n

c

= ⎛ ⎜ −

⎝ ) r

2

n and n

^rb ^r _d

incident, refle osition vector n (2-a) equal

1

= 0

boundary con

( ^{x z t} ^{, ; ,} )

^∀

^x

( ^{x z t} ^{, ; ,} )

^∀

gates to infinit merfeld for th in the middle F uation in the

)

, ; 0, , x z t = ∀ x

s at the inte written in g neither sourc dition of no re m

n the middle F and

p

⁾^b_refle

)

1 1

; n

^a

. z t f t

c

= ⎛ ⎜ −

⎝ ) r

in the middle

2

r ⎞

⎟ ⎠ r

denote the c ected and tr

OM uuuur = xe r

x

+

l pressures

p

⁾

(4- ndition (2-b) c

, 0, x z

= ∀

t

₍

, 0,

x z t

∀ = ∀ ty (note that t he existing fie

F2

e fluid mediu

0,

z t

∀ ≤ ∀

₍₆

erface

z = 0

n §. 2.

ce nor border turn.

F1 is the sum o ected pressure

1

1 1

. r n r

_b

. c g t c

⎞ ⎛

+ −

⎟ ⎜

⎠ ⎝

r r r

)

onditions of ransmitted an

z

. + ze r

p

⁾1_and

p

⁾₂_a

(4-a)

-b) can be

(5-a)

(5-b) this is eld at

um is

6)

0

_are

1.

other

of the e field

r ⎞

⎟ ⎠ r

wave nd

r

at the

(3)

R in F a

t

O o

n

T

n

B a w

s

A

θ

T It la w v o 3 v a m a im T d si th w p a p In p w

Reflexion/ Trans nterface

z

= Following the f and their argum

1 a

. t n r t

− r r

c

= − Or, noting

n

^x

on the axis

( ^O

1 1

xa xb

n x n x c

=

c

This implies

1 1

xa xb

n n

c

=

c

= By expressin according to a written

1

1 1

sin sin

c c

θ

₌

θ

As a result, sinc

1

=

1

θ θ ′

They are reflec t should be n aws do not r wave, and seco vector propaga of Snell, and no 3. 1. Slowness

Slowness vector

m

^r

= n

^r

are collinear an match in rever analogous to mportant role The slow sur direction, the ince the vecto he Snell's law without calcu propagate in acoustic rays propagation).

n fluid mediu propagation, th with the plane

smission of a pl

0,

_whatever functions

f )

1

,

ments :

1 b

.

n r n

c t

− r r= −

xa

, n

xb

and n

)

Ox

_vectors

n

2 2

, , n x

x

c x

= ∀

2 2

n

x

c

ng the com

angles

θ θ

¹

,

¹

′

1 2

2

sin c

θ

′

θ

=

ce

θ

¹

and θ

cting equal ang noted firstly th require the a ondly it is the ation direction ot vice versa.

surface surface (L) is

C

led to a fix nd

mC

=

1,

_, rse pole O a the surface in the proble rface of a m solutions of th or of the incid

to the surface lation, the v

the one an (that is to um, the speed he slow surface

of incidence

(

Case

lane wave on a p r are the val

1 2

g and f ) )

w

2 2

. , , n r x z

c

∀ =

r r

2

n

x _{the respe}

,

2

a b

n n and n

0, z

= ∀

t

mponents

n

2

, and θ

_eq

θ

1

′

are betwee gles of inciden hat the establi assumption o e conservation ns on interface

the location o xed point O.

surface area a and power 1.

of optical i ems of reflecti material own

he wave equa dent wave is k s of delays for vectors of t nother, their

say, the dir is the same fo

e is a sphere a

( ^Oxz )

_{here i}

1 2

:

c < c

plane interface lues of varia which are iden

0, t

= ∀ ₍ ective compon

2

,

(9-b)

(9-c)

xa

,

xb

n n and

quations (9-c)

(10)

en

0

_and

π 2

(11) nce and reflect

ishment of Sn of monochrom n of the projec e that involves

of the ends of Since

m and r

and slow speed Slowness sur indices, plays ion and refrac supplies for ation. Accordi known, simply r the two mate the waves c polarization ection of en or any directio and its intersec is a circle.

JNTM (201

35

ables.

tical,

(9-a) nents

2

n

x

are

2 ,

tion.

nell's matic ction laws

f the

d C r

ds to rface, s an ction.

any ngly, y add erials could and nergy on of ction

w

A 1)

Figu Noting that ea projection on reflected and the plane of infinite F1 and waves propaga parallel to th incidence for (Figure 3. Wh amount to the of slow circle angles of refle

Figure 3. Usin of the slown 3. 2. Harmon The sour a vector wav pressure

p

⁾_a

,

Are

1

a a

k

r =

k n

r =

1

b b

k

r =

k n

r =

2 2 2

k

r =

k n

r =

ure 2. Slownes ach term of e n the x-axis v transmitted w incidence, su d F2, can view ating in each m he axis

Oz

θ

1_{on the a}

here

c

¹

< c

²

e right provide es F1 and F ection and tran

ng the intersec ness surface of nic solution

rce to infinity ve number as

, p and p

⁾_b ⁾2

1

n

a

c ω

_r

1

n

b

c ω

_r

2 2

c n ω

_r

s surface case equation (10) vector slowne waves, using th urface delays w, graphically medium. It suf correspondin axis

Ox ,

_the so

1

₁

1 c >

es an intersecti 2 and conseq nsmission.

ction with the f each semi-inf

is monochrom ssociated with

can be define

F. Rahm

1 2

.

c < c

corresponds t ess of the inc he intersection of each of some propert ffices to draw ng to the ang amount

sin θ

2

1 ). c

^Delayin

ion with each quently, yield

plane of incid finite medium

matic pulsation h each plane ed,

(12-a) (12-b) (12-c)

mi et al

to the cident n with

semi- ties of a line gle of

1

c

1

θ

ng this curve ds the

dence [3].

n

ω ,

wave

)

(4)

36

And the use of a solution can be written

( ) { ( ) }

( ^{p x z t}

⁾ⁱ

^{, ;}

⁼

^A

⁾ⁱ

^exp

⁻

^{i k OM}

^rⁱ

^.

^uuuur⁻

^ω ^t )

( ^{, ;} )

¹

^exp { ( ^. ) }

a a

p ) x z t = A ) − i k OM r uuuur − ω t

(13-a)

( ^{, ;} )

¹

^exp { ( ^. ) }

b b

p ) x z t = B ) − i k OM r uuuur − ω t

(13-b) And

( ) { ( ) }

2

, ;

2

exp

2

.

p ) x z t = A ) − i k OM r uuuur − ω t

(13-c) According to equations (9-c) written in paragraph (§ 3) that come from writing the boundary conditions at the interface

0,

z

= written for all values of x and t, the projections on the interface

( ) ^Ox

vector wave number

k k and k r r

_a

,

_b

r

2

are equal :

=

2

=

xa xb x x

k = k k k

(14) The laws of Snell (10) and equal angles of incidence and reflection (11) who sell require, in the F1 environment, equality (at sign) projections on the axis

( ) ^Oz

vector wave number

k and k r

_a

, r

_b

:

=

1

za zb z

k − k = k

(15) In summary we write :

1

a x x z z

k r = k e r + k e r

(16-a)

1

b x x z z

k r = k e r − k e r

(16-b)

2 x x z2 z

k r = k e r + k e r

(16-c) And

x z

OM = xe + ze

uuuur r r

(16-d) The formula for the pressure field in the middle F1

( ) ( ) ( )

⁽ ¹ ⁾ ⁽ ¹ ⁾

1

, ; , ; , ;

1 i k x k z t^x ^z 1 i k x k z t^x ^z

a b

p x z t = p x z t + p x z t = Ae )

⁻ ^{+ −}^ω

+ Be )

⁻ ^{− −}^ω

) ) )

(17)

And that F2 is in the middle

( )

⁽ ² ⁾

2

, ;

2 ^{i k x k}^x ^z ^z ^t

p ) x z t = A e )

⁻ ⁺ ⁻^ω

(18) Equality of pressures (2-a)

z = 0

leads to

( ) ( ) ( )

1 1 2

x x x

i k x t i k x t i k x t

A e )

⁻ ⁻^ω

+ B e )

⁻ ⁻^ω

= A e )

⁻ ⁻^ω

Hence

1 1 2

A ) + B ) = A )

(19-a) Or

p p

1 R T

− ) + ) =

(19-b)

1 1

R )

p

= B A ) )

(20-a)

2 1

T )

p

= A ) ) A

(20-b) Are the coefficients of reflection and transmission of

pressure amplitude.

The partial derivatives with respect to

z

pressures can be written

( ) {

¹ ¹

}

⁽ ⁾

1

1 1 1

, ;

_z ^{ik z}^z ^{ik z}^z ^{i k x}^x ^t

p x z t ik A e B e e

z

ω

− −

∂ −

= − +

∂

) ) )

(21-a)

And

( )

² ⁽ ⁾

2

2 1

, ;

_z ^ik^z ^z ^{i k x}^x ^t

p x z t ik A e e z

ω

− −

∂ = − −

∂

) )

(21-b) Their report in the condition (4-b) equality normal speeds (4-b)

z = 0

leads to

( )

⁽ ⁾ ⁽ ⁾

1 2

1 1 2

1 2

= , , 0,

x x

i k x t i k x t

z z

ik ik

A B e

^ω

A e

^ω

x z t

ρ ρ

− − − −

− +) ) − ) ∀ = ∀

Or

( )

1 2

1 1 2

1 2

z z

k k

A B A

ρ

^{− +} ^{= −}

ρ

) ) )

(22-a)

Hence

1 2 1

z z z

p p

k k k

R T

ρ

⁺

ρ

⁼

ρ

) )

(22-b) Solving the system of two equations (19) and (22) with two unknowns

R and T )

^p

)

^p

leads to

2 2 1 1

z z

p

z z

k k

R k k

ρ ρ

− +

= +

)

(23-a) And

1 1

2 2 1 1

2

_z

p

z z

T k

k k

ρ

ρ ρ

= +

)

(23-b)

By replacing

k

z1

and k

z2 by their respective expressions in terms of

k

1

, , θ

1

k and

2

θ

2

1

= cos

1 1

k

z

k θ

(24-a)

And

2

=

2

cos

2

k

z

k θ

(24-b) And introducing the characteristic impedances

Z and Z

¹ ²

of the two media F1 and F2

1

=

1 1

Z ρ c

(25-a) And

2

=

2 2

Z ρ c

(25-b)

(5)

R T w

R

A

T

)

O

R

A

T

)

E tr o F

‘’

a tr 3 tr d [1 a c w h 4

ρ

E

ρ

Reflexion/ Trans The coefficien written

2

cos

p

cos

R θ

θ

=− )

And

2

p

cos

T

⁼

θ

)

Or

2 2

cos

p

cos R Z

=

Z

)

And

2

p

cos T Z

=

Z

)

Examples of ransmission of of incidence For an angle

’rupture’’ appe angle for the ransmitted wav 3. 3. Evanescen

In the ransmitted w decreases expo

1, 2, 3]. The to a reflection giv

oefficient. Th wave produce horizontally pr 4. Numerical r Example 1 :

1

2000 Kg ρ

=

Figure 4. M tran Example 2 :

1

3000 Kg m ρ

=

smission of a pl nts of reflectio

2 2

2 2 1

cos cos Z Z

θ θ

θ

+ +

1 1

2 1

cos cos

Z Z

θ θ

+

2 1

s co

Z Z θ θ

− +

2 2

2 1

cos

s cos

Z Z

θ θ

+ changes in f pressure amp

are presen of incidence ears in these c e interface c ves are evanes nt waves

case where wave becomes onentially with otal reflection ven by the argu he sum of the

es a standing ogressive.

results :

3

,

1

750 m c

=

m

Magnitude and nsmission coeff

3

,

1

750 m c

=

m

lane wave on a p n and transm

1 1

Z Z θ

θ

Z

1

1 1

s s θ

θ

s θ

1 coefficients o plitude, depen nted in the

e equal to

3

curves. This a considered a scent.

1 2

c > c

_an

s evanescent h a distance f phase shift ac ument of the c incident wave g wave total

,

2

2500 m s ρ

=

K

d phase of the fficient for Exa

,

2

1000 m s ρ

=

K

plane interface mission (23) ca

(2

(26-

(27-

(27-b of reflection nding on the a e figures be

30

^° [1, 2, 3 angle is the cr above which

d

θ θ

¹

>

_c

,

: its ampli from the inter ccompanied

χ

complex reflec e and the refle lly vertically

3

,

2

15 Kg m c

=

reflection and ample 1

3

,

2

150 Kg m c

=

JNTM (201

37

an be

26-a)

-b)

-a)

b) and angle elow.

3], a ritical

the

the itude rface

χ

_by

ction ected and

500 m s

d

00 m s

W

w

W 1)

Figure 5.

tran Example 3 :

1

1000 Kg m ρ

=

Figure 6.

tran With : R: Re T: T Using the con certain angle o for the interfa 3], there is no the slow curve waves in the become evan parallel to th decreases exp The value

arcsin θ

c

=

This correspo When

c

¹

> c

is no critical a

Magnitude an nsmission coe

3

,

1

750 m c

=

m s

Magnitude an nsmission coef eflection coeff ransmission co nstruction of of incidence n ace F1/F2 con o longer inters

e of the medi middle F2 nescent: they he interface w ponentially in t

of the critic

1 2

c . c

onds to

θ

^c²

=

2

,

c

constructio angle on the in

nd phase of the efficient for Ex

.

,

2

1000 s ρ

=

Kg

nd phase of the fficient for Ex ficient;

oefficient;

Figure 3, it ap noted

θ

^c_and

nsidered (whe sects the line p

ium (Figure 7 are no longe see their ene whereas the p

the direction o cal angle

θ

^c

2 . π

on of Figure 7 nterface consid

F. Rahm

e reflection an xample 2

3

,

2

1500 g m c

=

e reflection an ample 3.

ppears that fr called critical ere

c

¹

< c

²

)

parallel to

Oz

7-a). Correspo er propagating ergy propagati

pressure amp of

z

_increasin

θ

c_{is given}

7-c. shows that dered.

mi et al

nd

m s

nd

rom a angle [1, 2,

z

_with

nding g and ing in litude ng.

by :

there

(6)

4 p d p tr th m re c

θ

re p re w im th m b im th e a

Figure 7. Int slowness surf

4. 1. Discussion In the id perfectly flat, w different from portion of th

ransmitted in t he reflected w modulus less t

eflecting medi ommunity, we

θ

c at which tr eflection coeff phenomenon

eflection coef when close to

mpedances of he second m modified; in pa be slightly less

mpedance is v he F1 commu energy loss, t amplitudes of

tersection with face for an inte

1 2

,

c < c

_–

on of results deal case whe

while the seco those acoust he energy of the second me wave will be ass than or equal ium has a velo e have seen t ransmission is ficient is then of total reflec fficient decrea

vertical it dep f two media.

medium the articular the c s than 1. Fina very small or unity, the incid the reflection waves reflect

h the plane of i erface F1/F2 – c)

c

1

> c

2_[3

ere the interfa ond medium

tic

( ρ

1

,c

1

)

f the inciden edium. The co signed a reflec l to 1 [1, 2, ocity greater th that there exis impossible, th equal to 1 [1, ction). At the ases sharply w pends only on In the presen phenomena coefficient of t ally, if the refl very large co dent wave will coefficient ted and incid

incidence of th –a)

c

1

< c

2

,

_–

3].

face is consid has characteri and

( ρ

2

,c

2

nt wave will onsequence is ction coefficien

3, 4, 5, 6]. If han that of th sts a critical a he modulus o

2, 3 , 4, 5, 6]

critical angle with the angle, n the character nce of dampin will be slig total reflection flecting mediu

mpared to th l reflect almos (the ratio of dent) will equ

38

he –b)

dered istics

) ^,

_a

l be s that

nt of f the he F1 angle f the (the e the

and ristic ng in ghtly n will um is at of st no f the ual 1

w

A J

regardless of t The reflection the second me wave is given transmitted wa result simply sides of the conservation o is equal to the 5. Conclusion When the circles F1 and incidence

θ

¹

from which th evanescent. T equal to 1 mo presence of evanescent tr interface, whi depth (when z References [1] L. E. Kins Fourth editio pages, ISBN 0 [2] E. S. Kreb 216, 2007.

[3] C. Potel, M 352 pages, 20 [4] C. Potel, J 6152-6161, 19 [5] C. Potel, Acustica-Acta [6] C. Potel, J.M. Gherbez 2, 1128-1135,

the angle [1, 2 n coefficient o edium, affectin n by

T = + 1

ave can excee reflects greate interface, and of energy and e sum reflected n

speeds of w d F2 are such t rated

θ

^c_call

he transmitted The reflectanc

odule (total re acoustic ene ransmitted wa ile its amplitu z increases), pe

sler, A. R. Fre n, John Wile 0-471-84789-5 bes and P. F. D M. Bruneau, E

06.

J.F. De Bellev 995.

J.F. De Bel Acustica, 82, S. Devolder, zza, O. Leroy,

1999.

, 3].

of transmission ng the amplitu

.

+ R

_{The pre}

ed that of the i er continuity o d in no way v

can easily sho d and transmit

wave propagati that

c

¹

< c

²

,

ed critical ang d wave in the m

ce in the mi eflection), but ergy in the ave propagate ude decreases

erpendicular t

ey, A. B. Copp ey & Sons, In 5.

Daley, Geophy Ed. Ellipse coll val, J. of Appli lleval, E. Gen 5, 738-748, 19 A.U. Rehman , M. Wevers,

n of the pressu ude of the refr essure level o incident wave.

of pressure on violates the la ow that the int

tted intensities

ion in

c

¹_an

there is an an gle for this inte middle F2 bec

iddle F1 bec there is alway medium F2.

es parallel to exponentially to the interface

pens, J. V. San nc, New york

ys. J. Int, 170, lection Techn ied Physics, 7 nay, Ph. Gat 996.

n, J.F. De Be J. Appl. Phys

ure in racted of the

. This n both aw of tensity s.

nd

c

²

ngle of erface comes comes ys the The o the y with

e.

nders, k. 548

205–

nosup, 7, 12, tignol, lleval, s., 86,

Reflexion/ Transmission of a plane wave on a plane interface