Chapter 7: Reflection, transmission and standing waves

209

When considering incident sound from medium 1, the sound energy that is transmitted into medium 2 can be regarded as absorbed by medium 2. A very central concept in acoustics that describes the absorbing ability of a medium or a boundary is the absorption factor .. The absorption factor is defined as

i

r

i

ri

i

t

WW

WWW

WW === 1 , (7-27)

where Wi is the incident sound pressure, Wr is the reflected sound pressure and Wt is the transmitted sound power. Because the power can be expressed as W = Ix S, where S is the area and Ix is the intensity, which can be expressed as cpI x 0

2 /~ = according to (4-83), the absorption factor can be expressed as

2

2

2

,

, 1

11 R==+=i

r

ix

rx

pp

II . (7-28)

7.1.3 Propagation of plane waves in a three-dimensional space

Before analyzing the oblique incidence of a wave against a boundary, we consider how a wave can be described when its direction of propagation doesnt coincide with a coordinate axis. For sound propagation in the positive x-direction in a Cartesian coordinate system, (4-69) implies that

)(),( xktieptx = p , (7-29) where the (prime) symbol is used to distinguish that coordinate system from coming systems.

Figure 7-5 Plane wave propagation in the

positive x-direction. The wave fronts are surfaces joining points with identical phase.

y

x

W ave fronts

x'e G

y'e G

To describe multi-dimensi onal propagation, an unprimed coordinate system is introduced. In that system, for simplicity, we begin by studying the propagation in the xy-plane, in order to then generalize to three dimensions. The primed system has been rotated through an angle 1 about the z-axis relative to the unprimed, as shown in figure 7-6.

Chapter 7: Reflection, transmission and standing waves

210

Figure 7-6 Plane wave propagation described in two coordinate systems. One has been rotated through an angle 1 about the z-axis.

In a so-called orthogonal transformation, the description can be transformed from the primed to the unprimed system. The position vector

Gr to a point on the x-axis is indicated in the respective coordinate systems as yxx eyexexr

GGGG +== (7-30) i.e., yxxx eeyeexx

GGGG += , (7-31) where ),cos( jiji eeee

GGGG = in the transformation theory are usually called transformation coefficients, and are cosines of the angles between the base vectors ie

G and je

G. The

expression (7-31) can also be stated in the form

1111 sincos)90cos(cos yxyxx +=+= D , (7-32) and (7-29) transforms in the unprimed system to

)sincos( 11),,( kykxtieptyx =p . (7-33) To further generalize the discussion, a unit vector n

G is introduced to designate the

direction of propagation; it is expressed the respective coordinate systems as

yyxxx enenenGGGG +== . (7-34)

From (7-34), applying the orthogonality relations 1= xx ee GG and 0= yx ee GG , it follows that

1cos),cos( === xxxxx eeeen GGGG , (7-35)

1sin),cos( === yxyxy eeeen GGGG . (7-36) The wave number vector is defined as

nkkGG = , (7-37)

ey

y

x

Wave fronts

1

x

y

rxeGGyeG G

xeG

Chapter 7: Reflection, transmission and standing waves

211

with a magnitude k = /c, and a direction nG identical to the direction of propagation; it can be expressed as yxyyxx ekekenenkk

GGGGG11 sincos)( +=+= . (7-38)

Thus, the components of the wave number vector, i.e., its x and y-axis projections, are

1coskk x = , (7-39) 1sin kk y = , (7-40) respectively, and we conclude that the most general form of the solution becomes

)(),( rktieptrGGG = p , (7-41)

or in component form

)(),( ykxkti yxeptr = Gp . (7-42) In three dimensions, it follows by analogous logic that

)()( =),( zkykxktirkti zyxepeptr =

GGGp , (7-43) where = rk GG constant, (7-44) constitute surfaces of constant phase. Entering (7-43) into the wave equation (4-43)

2

2

22

2

2

2

2

2 1t

pcz

py

px

p=

++

(7-45) provides the condition

222 zyx kkkckk ++=== G . (7-46)

That condition is an important relation that will be utilized in the discussion that follows.

7.1.4 Oblique incidence on a boundary between two fluid media

In order to analyze what happens when a plane acoustic wave with a certain angle of incid-ence i reaches the bounding surface between two fluid media, it is necessary to supplement the types of boundary conditions used up to this point. These boundary conditions, which require continuity of pressure and particle velocity across the boundary surface, are supplemented with the condition that the incident, reflected, and transmitted waves have the same periodicity along the boundary surface, i.e., the plane x = 0 in figure 7-7.

Comment [UC1](4-43)