Parametrization of acoustic boundary absorption and
dispersion properties in time-domain sourceÕreceiver
reflection measurement
Adrianus T. de Hoop,a兲 Chee-Heun Lam,b兲 and Bert Jan Kooij
Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 4 Mekelweg, 2628 CD Delft, The Netherlands
共Received 30 September 2004; revised 8 March 2005; accepted 25 May 2005兲
Closed-form analytic time-domain expressions are obtained for the acoustic pressure associated with the reflection of a monopole point-source excited impulsive acoustic wave by a planar boundary with absorptive and dispersive properties. The acoustic properties of the boundary are modeled as a local admittance transfer function between the normal component of the particle velocity and the acoustic pressure. The transfer function is to meet the conditions for linear, time-invariant, causal, passive behavior. A parametrization of the admittance function is put forward that has the property of showing up explicitly, and in a relatively simple manner, in the expression for the reflected acoustic pressure. The partial fraction representation of the complex frequency domain admittance is shown to have such a property. The result opens the possibility of constructing inversion algorithms that enable the extraction of the relevant parameters from the measured time traces of the acoustic pressure at different offsets, parallel as well as normal to the boundary, between source and receiver. Illustrative theoretical numerical examples are presented. © 2005 Acoustical Society of America. 关DOI: 10.1121/1.1954567兴
PACS number共s兲: 43.20.Bi, 43.20.El, 43.20.Px, 43.55.Ev 关JJM兴 Pages: 654–660
I. INTRODUCTION
In a variety of applications in acoustics共for example, in outdoor sound propagation, traffic noise analysis, jet-engine sound absorption in aircraft engineering and architectural acoustics兲, the analysis of the point-source excited reflection of sound waves by a boundary surface with certain absorp-tive and dispersive properties is of interest. In all these cases, the absorptive and dispersive properties of the boundary need characterization by a judiciously chosen set of param-eters. Following the pioneering paper by Ingard共1951兲, such a characterization goes via a local acoustic admittance, i.e., via a linear, time-invariant, causal, passive transfer function that links the normal component of the particle velocity on the boundary to the local acoustic pressure. For the canonical configuration consisting of a planar boundary, a monopole acoustic共volume injection兲 source and a monopole acoustic 共pressure兲 point receiver, we derive closed-form time-domain expressions for the received signal. For the same configura-tion and along similar lines, a recent paper共Lam et al., 2004兲 discusses some ad-hoc cases, where the boundary’s proper-ties are expressed via a complex-frequency domain Padé rep-resentation, the coefficients in which are matched to experi-mental data available in the literature. The approach via the Padé representation appears, however, to be limited to at most the Padé 共2,2兲 one. In the present paper, a more sys-tematic approach is followed where the complex-frequency domain characterization of the boundary admittance goes via
a partial-fraction representation that allows the incorporation of an arbitrary number of terms, each of them with an inter-pretable influence on the received signal. Amongst others, it is shown that, when source and receiver are both close to the boundary and the terms in the partial-fraction admittance representation meet a certain condition, large-amplitude os-cillatory surface effects can occur. Their amplitudes can even exceed the acoustic pressure values associated with the re-flection against a perfectly rigid boundary, a phenomenon that has also been reported elsewhere in the literature 共Wen-zel, 1974; Thomasson, 1976; Donato, 1976a, 1976b兲 and is confirmed by pertaining experiments共Daigle et al., 1996兲 as well as by computational finite-difference time-domain and finite element method studies 共Ju and Fung; 2002; Van den Nieuwenhoff and Coyette, 2001兲.
The analysis is carried out with the aid of an extension 共De Hoop, 2002兲 of the senior 共first兲 author’s modification of the Cagniard method 共Cagniard, 1962; De Hoop, 1960; De Hoop and Van der Hijden, 1984兲. It yields closed-form ana-lytic expressions for the time-domain acoustic pressure in the model configuration under investigation. Not only do these expressions reveal how the parameters governing the absorp-tion and dispersion properties of the reflecting boundary show up in the measured acoustic pressure, but they can also serve as benchmarks in further computational studies based on the numerical discretization of the acoustic wave equa-tions.
The methodology leans heavily on the use of the Schouten–Van der Pol theorem of the unilateral Laplace transformation 共Schouten, 1934, 1961; Van der Pol, 1934; Van der Pol and Bremmer, 1950兲. This theorem interrelates two 共causal兲 functions of time whose 共unilateral兲 Laplace
a兲Electronic mail: a.t.dehoop@ewi.tudelft.nl
b兲Presently at Laboratory of Circuits and Systems, Faculty of Electrical En-gineering, Mathematics and Computer Science, Delft University of Tech-nology, 4 Mekelweg, 2628 CD Delft, The Netherlands.
transforms are related such that the Laplace transform of the latter arises out of the Laplace transform of the former upon replacing the transform parameter s with a certain function
共s兲, where共s兲 belongs to the class of functions for which a causal time function corresponding to exp关−共s兲兴, with
艌0, exists.
The analysis can be carried out for an arbitrary number of terms in the partial-fraction characterization of the bound-ary’s acoustic admittance, each of them provided with its associated two parameters. This implies that a rather accurate tuning of the parameters to match the measured values of the admittance 共a procedure that is usually carried out in the frequency domain兲 can be achieved by incorporating a suffi-cient number of terms.
Some theoretical numerical examples illustrate how some physical phenomena can be attributed to certain ranges of the values of the parameters involved.
II. FORMULATION OF THE PROBLEM
Position in the configuration is specified by the coordi-nates兵x,y,z其 with respect to an orthogonal, Cartesian refer-ence frame with the originO and the three mutually perpen-dicular base vectors兵ix, iy, iz其 of unit length each; they form, in the indicated order, a right-handed system. The position vector is r = xix+ yiy+ ziz. The vectorial spatial differentiation operator is =ixx+ iyy+ izz. The time coordinate is t; dif-ferentiation with respect to time is denoted by t.
The acoustic wave motion is studied in the half-space D=兵−⬁⬍x⬍⬁,−⬁⬍y⬍⬁,0⬍z⬍⬁其, which is filled with a fluid with volume density of mass 0 and compressibility
0. The speed of sound waves in it is c0=共00兲−1/2. The acoustic wave motion is excited by an acoustic monopole point source with volume injection rate Q0共t兲 and located at r0=兵0,0,h其, with h艌0. We assume that Q0共t兲=0 for t⬍0. The acoustic pressure p共r,t兲 and the particle velocity v共r,t兲 then satisfy the first-order acoustic wave equations 共De Hoop, 1995, p. 44兲
p +0tv = 0, 共1兲
· v +0tp = Q0共t兲␦共r − r0兲. 共2兲 Causality entails that p共r,t兲=0 and v共r,t兲=0 for t⬍0 and all
r苸D. The acoustic properties of the planar boundary are
modeled via the local, linear, time-invariant, causal, passive acoustic admittance relation
vz共x,y,0,t兲 = − 共0c0兲−1Y共t兲ⴱ 共t兲
p共x,y,0,t兲, 共3兲
whereⴱ 共t兲
denotes time convolution and Y共t兲 is the boundary’s acoustic time-domain admittance transfer function, normal-ized with respect to the acoustic plane-wave admittance 共0c0兲−1 of the fluid. Figure 1 shows the configuration.
The acoustic wave field in the fluid is written as the superposition of the incident wave field to be denoted by the superscript i and the reflected wave field to be denoted by the superscript r. The incident wave field is the wave field that is
generated by the source and would be the total wave field in the absence of the boundary. Its acoustic pressure satisfies the scalar wave equation
2pi − c0−2t
2
pi= −0tQ0共t兲␦共x,y,z − h兲. 共4兲 From this equation we obtain 共see, for example, De Hoop, 1995, pp. 93–97兲 pi共r,t兲 =0t 2 Q0共t兲ⴱ 共t兲 Gi共r,t兲, 共5兲
in which the incident-wave Green’s function is Gi共r,t兲 =H共t − T0兲
4D0
for D0⬎ 0, 共6兲
with
D0=关x2+ y2+共z − h兲2兴1/2艌 0 共7兲
as the distance from the source to the receiver,
T0= D0/c0 共8兲
as the travel time from source to receiver and H共t兲 as the Heaviside unit step function.
III. THE COMPLEX SLOWNESS REPRESENTATION FOR THE ACOUSTIC WAVE FIELDS
The time invariance of the configuration and the causal-ity of the sound waves are taken into account by the use of the unilateral Laplace transform:
兵pˆ,vˆ其共r,s兲 =
冕
t=0⬁
exp共− st兲兵p,v其共r,t兲dt. 共9兲
The Laplace transform parameter s is taken positive and real. Then, according to Lerch’s theorem 共Widder, 1946兲 a one-to-one mapping exists between 兵p,v其共r,t兲 and their time-Laplace transformed counterparts 兵pˆ,vˆ其共r,s兲. The fluid is initially at rest, which has the consequence that the transfor-mation property t→s holds. Next, the complex slowness representations for兵pˆ,vˆ其共r,s兲 are introduced as
FIG. 1. Fluid-filled half-space with volume injection point source, acoustic pressure point receiver, and reflecting absorptive and dispersive boundary.
兵pˆ,vˆ其共x,y,z,s兲 = s2 42
冕
␣=−⬁ ⬁ d␣冕
=−⬁ ⬁ 兵p˜,v˜其共␣,,z,s兲 ⫻exp关− is共␣x +y兲兴d, 共10兲 where␣andare the wave slownesses in the x and y direc-tions, respectively. This representation entails the propertiesx→−is␣,y→−is. Use of the transforms in Eqs.共1兲–共4兲 yields for the incident wave
p
˜i共␣,,z,s兲 =0Qˆ0共s兲 2␥0
exp共− s␥0兩z − h兩兲, 共11兲 while for the reflected wave we write
p ˜r共␣,,z,s兲 =0Q ˆ 0共s兲 2␥0 R ˜ exp关− s␥0共z + h兲兴, 共12兲 in which ␥0共␣,兲 = 共c0−2+␣2+2兲1/2 with Re共␥0兲 艌 0 共13兲 is the wave slowness normal to the boundary and R˜ denotes the slowness-domain reflection coefficient. Use of Eqs.共11兲 and共12兲 in the complex slowness domain counterpart of the admittance boundary condition 共3兲, together with the prop-erty关cf. Eq. 共1兲兴
v
˜z= −共s0兲−1z˜ ,p 共14兲
Eqs.共11兲, 共12兲, and 共14兲 lead to
共␥0/0兲共1 − R˜兲 = 共0c0兲−1Yˆ共s兲共1 + R˜兲, 共15兲 from which it follows that
R ˜ =c0␥0− Yˆ共s兲 c0␥0+ Yˆ共s兲 = 1 − 2Yˆ共s兲 c0␥0+ Yˆ共s兲 . 共16兲
IV. SPACE-TIME EXPRESSIONS FOR THE ACOUSTIC WAVE FIELD CONSTITUENTS
The expressions for the time Laplace transformed re-flected wave field quantities are written as
pˆr共r,s兲 =0s2Qˆ0共s兲Gˆr共r,s兲, 共17兲 vˆr共r,s兲 = − sQˆ0共s兲 Gˆr共r,s兲, 共18兲 in which Gˆr共r,s兲 = 1 42
冕
␣=−⬁ ⬁ d␣冕
=−⬁ ⬁ R ˜ 1 2␥0 exp兵− s关i共␣x +y兲 +␥0共z + h兲兴其d 共19兲is the time Laplace transformed reflected-wave Green’s function. The time-domain counterparts of Eqs.共17兲–共19兲 are determined with the aid of an extension共De Hoop, 2002兲 of the standard modified Cagniard method共De Hoop, 1960; De Hoop and Van der Hijden, 1984兲. First, upon writing x = r cos共兲, y=r sin共兲, the transformation
␣= ip cos共兲 − q sin共兲,
= ip sin共兲 + q cos共兲, 共20兲
is carried out, which for the slowness normal to the boundary leads to ¯␥0共q,p兲=关⍀共q兲2− p2兴1/2, with ⍀共q兲=共c
0
−2+ q2兲1/2. Next, the integrand in the integration with respect to p is continued analytically into the complex p plane, away from the imaginary axis and, under the application of Cauchy’s theorem and Jordan’s lemma, the integration along the imaginary p axis is replaced by one along the hyperbolic path 共modified Cagniard path兲 consisting of pr+␥¯0共q,p兲共z + h兲=, together with its complex conjugate, for T1共q兲⬍ ⬍⬁, where T1共q兲=⍀共q兲D1 and D1=关x2+ y2+共z+h兲2兴1/2⬎0 is the distance from the image of the source to the receiver, while is introduced as the variable of integration. In the relevant Jacobian, the relation p /= i␥¯0/关2− T
1
2共q兲兴1/2 is used. Next, Schwarz’s reflection principle of complex func-tion theory is used to combine the integrafunc-tions in the upper and lower halves of the complex p plane, the orders of inte-gration with respect to and q are interchanged, and in the resulting integration with repect to q, that extends over the interval 0⬍q⬍共2/ D
1 2− c
0
−2兲1/2, the variable of integration q is replaced withdefined through q =共2/ D
1 2− c
0
−2兲1/2sin共兲, with 0艋艋/ 2. This procedure leads to
Gˆr共r,s兲 = 1 4D1
冕
=T1 ⬁ exp共− s兲Kˆr共r,,s兲d, 共21兲 in which Kˆr共r,,s兲 = 2 冕
=0 /2 Re冋
1 − 2Yˆ共s兲 c0¯␥0+ Yˆ共s兲册
d, 共22兲 with c0¯␥0=⌫1共r,兲 − i⌫2共r,兲cos共兲, 共23兲 ⌫1共r,兲 = c0共z + h兲/D12, 共24兲 ⌫2共r,兲 = c0共2− T1 2兲1/2r/D 1 2 , 共25兲is the reflected-wave kernel function and
T1= T1共0兲 = D1/c0 共26兲
is the travel time from the image of the source to the re-ceiver. Evaluation of the integral in the right-hand side of Eq. 共22兲 yields 共see the Appendix兲
Kˆr共r,,s兲 = 1 − 2Yˆ共s兲 兵关⌫1共r,兲 + Yˆ共s兲兴2+⌫2
2共r,兲其1/2. 共27兲 Since the right-hand side of Eq. 共27兲 is an analytic function of s in the right half 兵Re共s兲⬎0其 of the complex s plane, it has a causal time-domain counterpart Kr共r,, t兲 that van-ishes for t⬍0. In terms of the latter, Eq. 共21兲 leads to the time-domain expression Gr共r,t兲 =
冋
1 4D1冕
=T1 t Kr共r,,t −兲d册
H共t − T 1兲. 共28兲 To further separate in the second term on the right-hand side of Eq.共27兲 the influence of the configurational parameters of the measurement setup from the influence of the parametersassociated with the boundary’s acoustic admittance on the reflected field acoustic pressure, we make use of the Schouten–Van der Pol theorem of the unilateral Laplace transformation 共Schouten, 1934, 1961; Van der Pol, 1934; Van der Pol and Bremmer, 1950兲 and employ the Laplace-transform integral Formula共29.3.55兲 from Abramowitz and Stegun共1968, p. 1024兲, together with some elementary rules of the Laplace transformation to obtain
Yˆ共s兲 兵关⌫1共r,兲 + Yˆ共s兲兴2+⌫22共r,兲其1/2 = 1 −
冕
w=0 ⬁ KF共r,,w兲KˆY共w,s兲dw, 共29兲 in which KˆY共w,s兲 = exp关− Yˆ共s兲w兴H共w兲 共30兲 and KF共r,,w兲 = exp关− ⌫1共r,兲w兴兵⌫1共r,兲J0关⌫2共r,兲w兴 +⌫2共r,兲J1关⌫2共r,兲w兴其H共w兲, 共31兲 where J0and J1are the Bessel functions of the first kind and orders zero and one, respectively. Use of this result in Eq. 共27兲 yieldsKˆr共r,,s兲 = − 1 + 2
冕
w=0⬁
KF共r,,w兲KˆY共w,s兲dw. 共32兲 In terms of the共causal兲 time-domain counterpart KY共w,t兲 of KˆY共w,s兲 we end up with Kr共r,,t兲 = −␦共t兲 + 2
冋
冕
w=0 ⬁ KF共r,,w兲KY共w,t兲dw册
H共t兲. 共33兲 Note that in this expression the space-time configurational parameters of the fluid only occur in the kernel function KF共r,, w兲, while the parameters of Y共t兲 only occur in the kernel function KY共w,t兲. The space-time expressions for the reflected acoustic wave field quantities are from Eqs. 共17兲 and共18兲 finally obtained aspr共r,t兲 =0t 2 Q0共t兲ⴱ 共t兲 Gr共r,t兲, 共34兲 vr共r,t兲 = −tQ0共t兲ⴱ 共t兲 Gr共r,t兲. 共35兲
In Sec. V, an expression for KY共w,t兲 is obtained for the case where a partial fraction parametrization of the complex fre-quency domain acoustic admittance Yˆ共s兲 is used to specify the boundary’s acoustic dispersion and absorption properties.
V. PARTIAL-FRACTION PARAMETRIZATION OF THE COMPLEX FREQUENCY DOMAIN ACOUSTIC ADMITTANCE AND ITS COROLLARIES
In this section an expression for the kernel function KY共w,t兲, introduced via Eq. 共30兲, is constructed for the case where Yˆ共s兲 is parametrized through a partial fraction repre-sentation. Let Yˆ共s兲 =
兺
n=0 N Yˆ共n兲共s兲, 共36兲 with Yˆ共0兲共s兲 = Y⬁, 共37兲 Yˆ共n兲共s兲 = An s +␣n for n = 1,…,N. 共38兲Since the underlying assumption of such a representation is that Yˆ共s兲 arises as the causal response from a rational time differentiation operator with real-valued coefficients and a finite number of degrees of freedom, a number of properties hold共Kwakernaak and Sivan, 1991兲. First, Yˆ共s兲 has to be real and positive for s real and positive, which entails that Y⬁is real and艌0. Furthermore, Yˆ共s兲 has, in general, simple poles at s = −␣n 共n=1,…,N兲 that should be located in the left half of the complex s-plane. As to the terms Yˆ共n兲共n=1,…,N兲 two possibilities arise: either␣n共n=1,…,N兲 is real and 艌0 and the residues An共n=1,…,N兲 at the poles s=−␣n共n = 1 ,…,N兲 are real, or pairs of␣n 共n=1,…,N兲 are complex conjugate with positive real parts and the residues An共n = 1 ,…,N兲 at such pair of poles s=−␣n共n=1,…,N兲 are each other’s complex conjugate.共The case of higher-order poles is most easily handled by a limiting confluence procedure.兲 Equation共36兲 entails a representation of KˆY共w,s兲 of the form
KˆY共w,s兲 =
兿
n=0 N KˆY共n兲共w,s兲, 共39兲 with KˆY共0兲共w,s兲 = exp共− Y⬁w兲H共w兲, 共40兲 KˆY共n兲共w,s兲 = exp关− Y共n兲共s兲w兴H共w兲 for n = 1,…,N. 共41兲 The time-domain counterpart of Eq.共40兲 isKY共0兲共w,t兲 = exp共− Y⬁w兲H共w兲␦共t兲. 共42兲 To construct the time-domain counterpart of Eq. 共41兲 we again use the Schouten–Van der Pol theorem and employ Formula 共29.3.75兲 of Abramowitz and Stegun 共1968, p. 1026兲, together with some elementary rules of the time Laplace transformation to obtain
KY共n兲共w,t兲 = H共w兲␦共t兲 − exp共−␣nt兲
⫻共Anw/t兲1/2J1关2共Anwt兲1/2兴H共w兲H共t兲
for n = 1,…,N. 共43兲
the parameters兲, the time-domain counterpart of Eq. 共39兲 fol-lows as KY共w,t兲 = KY共0兲共w,t兲ⴱ 共t兲 KY共1兲共w,t兲ⴱ 共t兲 ¯ ⴱ 共t兲 KY共N兲共w,t兲. 共44兲 In this expression each of the factors contains only two pa-rameters, a property that can facilitate the parameter sensi-tivity analysis of the reflection measurement setup.
VI. PLANE-WAVE ADMITTANCE PARAMETRIZATION OF THE COMPLEX FREQUENCY DOMAIN
ACOUSTIC ADMITTANCE AND ITS COROLLARIES
In this section an expression for the kernel function KY共w,t兲, introduced via Eq. 共30兲, is constructed for the case where Yˆ共s兲 is parametrized through a plane-wave admittance expression, applying to a fluid with volume density of mass
1, compressibility1, normalized inertia relaxation function
␣ˆ1共s兲, and normalized compressibility relaxation function
ˆ
1共s兲. Accordingly, we write 共De Hoop, 1995, p. 42兲
YˆW共s兲 = Y1⬁关Xˆ共s兲兴1/2, 共45兲 in which Y1⬁=0c0
冉
1 1冊
1/2 =0c0 1c1 , 共46兲with c1=共11兲−1/2as the corresponding wave speed, is rep-resentative for the instantaneous response and
Xˆ 共s兲 =s +␣ˆ1共s兲 s +ˆ1共s兲
共47兲 is representative for the absorptive and dispersive properties. To construct the time-domain counterpart KW共w,t兲 of the corresponding kernel function
KˆW共w,s兲 = exp关− YˆW共s兲w兴 共48兲
we again use the Schouten–Van der Pol theorem and employ Formula 共29.3.82兲 of Abramowitz and Stegun 共1968, p. 1026兲 to obtain: KˆW共w,s兲 =
冕
u=0 ⬁ exp关− Xˆ共s兲u兴⌼共w,u兲du, 共49兲 where ⌼共w,u兲 = Y1⬁w 共4u3兲1/2exp冋
− 共Y1⬁w兲2 4u册
H共w兲H共u兲. 共50兲Since␣ˆ1共s兲 andˆ1共s兲 are system’s response functions of the linear, time-invariant, causal, passive type, Xˆ 共s兲 admits a partial-fraction parametrization of the type共36兲–共38兲 and the time-domain counterpart of exp关−Xˆ共s兲u兴 follows from Eq. 共44兲.
VII. SOME ILLUSTRATIVE NUMERICAL EXAMPLES
In the following, some illustrative numerical examples are presented. The source is placed at the boundary 共h=0兲. Two receiver positions are considered, viz. one at the bound-ary共r⬎0, z=0兲, i.e., the propagation takes place parallel to
the boundary, and one at the normal to the boundary through the source共r=0, z⬎0兲, i.e., the propagation takes place nor-mal to the boundary. With regard to the boundary’s acoustic admittance, two examples are discussed: 共A兲 the zero-order 共single-term兲 admittance and 共B兲 the first-order 共two-terms兲 admittance. Figure 2 shows the normalized incident-wave Green’s function as a function of normalized time 关cf. Eq. 共6兲兴.
A. Zero-order boundary admittance
For the zero-order boundary admittance we have
Yˆ共s兲 = Y⬁, 共51兲
which corresponds to the time-domain acoustic admittance
Y共t兲 = Y⬁␦共t兲 共52兲
and the time-domain boundary condition 关cf. Eq.共3兲兴 vz共x,y,0,t兲 = − 共0c0兲−1Y⬁p共x,y,0,t兲. 共53兲 This section mainly serves to illustrate the influence of Y⬁on the reflection problem. Figure 3 shows the normalized reflected-wave Green’s function as a function of normalized time关cf. Eqs. 共27兲 and 共28兲兴 at 共a兲 r=10 m, z=0 共propaga-tion parallel to the boundary兲 and 共b兲 r=0, z=1 m 共propa-gation normal to the boundary兲, for three different values of Y⬁. Note that for propagation parallel to the boundary the normalized Green’s function always starts at the value −1, irrespective of the value of Y⬁, while for propagation normal to the boundary the starting value is positive for Y⬁⬎1, zero for Y⬁= 1共admittance matched to the plane-wave value at normal incidence兲, and negative for Y⬁⬍1.
B. First-order boundary admittance
For the first-order boundary admittance we have 关cf. Eqs.共36兲–共38兲兴
Yˆ共s兲 = Y⬁+ A1
s +␣1, 共54兲
which we rewrite as
FIG. 2. Normalized incident-wave Green’s function 4D0Gias a function of normalized time t / T0.
Yˆ共s兲 = Y⬁s + z1 s + p1
, 共55兲
where −p1= −␣1 is the pole of Yˆ共s兲 and −z1 is the zero of Yˆ共s兲, both located in the left half of the complex s plane, and
A1= Y⬁共z1− p1兲 共56兲
is the residue at the pole. Equations共54兲 and 共55兲 correspond to the time-domain acoustic admittance
Y共t兲 = Y⬁␦共t兲 + A1exp共−␣1t兲H共t兲 共57兲 and the time-domain boundary condition关cf. Eq.共3兲兴
共t+ 1/v兲vz共x,y,0,t兲 = − 共0c0兲−1Y⬁共t+ 1/p兲p共x,y,0,t兲, 共58兲 wherev= 1 / p1is the velocity relaxation time andp= 1 / z1is the pressure relaxation time 共Christensen, 2003, pp. 17–19; Meinardi, 2002, p. 105兲. This section mainly serves to illus-trate the influence of v and p on the reflection problem. Therefore, we take Y⬁= 1, which implies matching to the plane-wave admittance at normal incidence.
Figure 4 shows the normalized reflected-wave Green’s function as a function of normalized time关cf. Eqs. 共28兲 and 共33兲兴 at 共a兲 r=10 m, z=0 共propagation parallel to the
ary兲 and 共b兲 r=0, z=1 m 共propagation normal to the bound-ary兲, for four different values of v, with p fixed. As Fig. 4共a兲 shows, strong oscillations occur at propagation parallel to the boundary, which phenomenon has been referred to in Sec. I. No such oscillations show up in the propagation nor-mal to the boundary, as Fig. 4共b兲 shows. It can be argued that this behavior can be inferred from Eq. 共31兲, where ⌫1 is related to the offset normal to the boundary and occurs in the damping exponential function, while⌫2is related to the off-set parallel to the boundary and occurs in the oscillating Bessel functions. Apparently, such an easy interpretation does not apply to Eq.共43兲, where for An⬎0 the Bessel func-tions are oscillatory, while for An⬍0 they change into modi-fied Bessel functions of the first kind that show a monotonic behavior.
VIII. DISCUSSION OF THE RESULTS
Via the combined applications of the modified Cagniard method and the Schouten–Van der Pol theorem of the unilat-eral Laplace transformation the time-domain acoustic pres-sure of the monopole共volume injection兲 point-source excited wave reflected against a locally reacting, absorptive and dis-FIG. 3. Normalized reflected-wave Green’s function 4D1Gras a function
of normalized time t / T1. Zero-order acoustic boundary admittance Y = Y⬁. Source at boundary共h=0兲; c0= 330 m / s.共a兲 Propagation parallel to bound-ary 共r=10 m, z=0兲, 共b兲 propagation normal to boundary 共r=0, z=1 m兲. Curves: 共- . -兲 Y⬁= 2.0,共- .. -兲 Y⬁= 1.0共matched to plane-wave value at normal incidence兲, 共- ... -兲 Y⬁= 0.5.
FIG. 4. Normalized reflected-wave Green’s function 4D1Gras a function of normalized time t / T1. First-order acoustic boundary admittance: 共t
+ 1 /v兲v=−共0c0兲−1Y⬁共t+ 1 /p兲p at boundary. Source at boundary 共h=0兲;
Y⬁= 1.0共matched to plane-wave value at normal incidence兲, c0= 330 m / s. 共a兲 Propagation parallel to boundary 共r=10 m, z=0兲, 共b兲 propagation normal to boundary 共r=0, z=1 m兲. Curves: 共- . -兲 v= 1.0⫻10−3s, p= 5.0 ⫻10−2s, A 1= −9.8⫻102s−1; 共- .. -兲 v= 2.0⫻10−3s, p= 5.0⫻10−2s, A1 = −4.8⫻102s−1; 共- ... -兲 v= 5.0⫻10−3s, p= 5.0⫻10−2s, A1= −1.8 ⫻102s−1;共- - -兲 v= 1.0⫻10−1s,p= 5.0⫻10−2s, A1= 1.0⫻101s−1.
persive boundary has been expressed as a multiple sequence of operations acting on the source signature. Each of the kernel functions in the expression contains separately the configurational parameters of the measurement setup 共loca-tion of source, receiver and boundary, and propaga共loca-tion through the fluid兲 and the parameters by which the absorp-tive and dispersive properties of the boundary can be char-acterized. Two parametrizations of the boundary’s complex frequency domain acoustic admittance have been discussed in detail: the partial-fraction parametrization and the plane-wave admittance parametrization. The explicit attribution of a sequence of parameters to their corresponding kernel func-tions is conjectured to play an illuminating role in the use of the reflection measurement setup to characterize共via an ap-propriate inversion algorithm applied to the measured values of the acoustic pressure兲 the absorption and dispersion prop-erties of the boundary, while the obtained expression itself is directly amenable to carry out the relevant parameter sensi-tivity analysis. It is noted that the multiple time convolutions that occur in the final expression for the acoustic pressure can numerically most profitably be evaluated through the use of the FFT algorithm.
ACKNOWLEDGMENT
The authors want to thank the 共anonymous兲 reviewers for their constructive criticism and their suggestions for im-proving the paper.
APPENDIX: EVALUATION OF THE INTEGRAL IN EQ.„22…
In this Appendix the integral occurring in Eq.共22兲 I = 2 Re
冋
冕
=0 /2 1 c0␥¯0+ Yˆ共s兲 d册
= 2 Re冋
冕
=0 /2 1 ⌫1− i⌫2cos共兲 + Yˆ共s兲 d册
= 2 冕
=0 /2 ⌫ 1+ Yˆ共s兲 关⌫1+ Yˆ共s兲兴2+⌫22cos2共兲 d, 共A1兲with关cf. Eqs. 共24兲 and 共25兲兴
⌫1= c0共z + h兲/D12, 共A2兲 ⌫2= c0共2− T1 2兲1/2r/D 1 2 , 共A3兲
is evaluated. Using the standard integral 2
冕
=0 /2 A A2+ B2cos2共兲d= 1 共A2+ B2兲1/2, 共A4兲 we obtain I = 1 兵关⌫1+ Yˆ共s兲兴2+⌫22其1/2 . 共A5兲This result is used in the main text.
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