quantitative modeling of reflected ultrasonic bounded beams and a

0885–3010/$25.00 © 2008 IEEE

2661IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. 55, no. 12, dEcEmbEr 2008

Abstract—The wavefields of bounded acoustic beams and pulses reflected from water-loaded plates are fully modeled with the phase advance technique. The wavefield produced at the source is propagated at any incidence angle using phase-shift modeling that incorporates the full analytic solution for the acoustic reflectivity at the interface. This approach pro-vides for the ready visualization of both the stationary mono-frequency beam wavefield and animation of the temporally bounded pulse. The model images are reminiscent of the clas-sic Schlieren photographs that first illustrated the nonspecular behavior of the reflected beams incident near critical angles. Various phenomena such as the lateral displacement and the null zone at the Rayleigh critical angle are recreated. A new approximation for this shift agrees well with that of the peak energy of the reflected beam. Similar effects are observed dur-ing the reflection of a bounded pulse. Although more com-putationally costly than existing analytic approximations, the phase advance technique can facilitate the interpretation of reflectivity measurements obtained in laboratory experiments. In particular, the full visualization allows for a better under-standing of the behavior of reflected waves at any angle of incidence.

I. Introduction

The reflectivity on an acoustic wave from the inter-face between a liquid and an elastic solid half-space

depends intimately on contrasts in the density and the mechanical properties of the 2 media. observations of the variation of the magnitude and phase of this reflectiv-ity with angle of incidence can in principle be used to reveal the solid’s elastic properties. However, despite a long history to this problem beginning with schoch’s early qualitative schlieren photographs [1], [2], actual quantita-tive experimental measurements remain rare [3]–[6]. one reason for this is that, in practice, the analysis of acoustic reflectivity is complicated greatly by nonspecular effects, particularly near critical incidence angles. These nonspec-ular and experimental geometry dependent effects do not allow simpler long-standing, analytic plane wave expres-sions to be employed directly; analysis must be carried out on a case-by-case basis. In this contribution, the newly de-

veloped analytic approximation of the phase shift is tested against the full wavefield technique.

The theory describing the relationships between the reflectivity of a monofrequency plane-wave and the physi-cal properties of the 2 media has long been known [7], [8] but, as is well understood, it does not adequately describe the reflectivity for the more realistic case of an incident wave produced by finite spatial-dimension and frequency bandwidth transducers. nonspecular behavior is seen near critical angles of incidence [9], [10], but the effects cen-tered near the rayleigh critical angle θR

c = arcsin(Vf /VR), where Vf and VR are the longitudinal fluid and the solid rayleigh wave speeds, respectively, are particularly striking. Generally, near this angle, the beam is split into a specular and a nonspecular lobe. The nonspecular lobe is apparently shifted along the interface. schoch [1], [2] first qualitatively captured this unexpected and complex behavior in schlieren photographs, with this work further expanded by later workers [3]–[6], [11]–[24]. one example of such a schlieren photograph taken by neubauer and dragonette [6] and shown in Fig. 1(a), for purposes of illus-tration only, for a bounded beam incident at θR

c displays the specular (s) and nonspecular (ns) lobes separated by a dark null zone. The schoch shift is the distance that the peak amplitude of the reflected wave along the solid’s surface is displaced from the specular reflection path.

These nonintuitive effects complicate the analysis of re-flection strength versus angle of incidence observations. For example, it is difficult or impossible to measure the shear wave critical angle θs

c = arcsin(Vf /VS), where VS is the speed of the transverse wave in the solid because it falls near or within the null zone. Further, because the wavefield generated by a transducer is band-limited in the wavenumber domain, the reflectivity curves are compli-cated and the critical angles are difficult to measure. one successful strategy is to measure only VP and VR through the associated critical angles and use well-known relation-ships to then calculate VS [6] although there have recently been claims that VS can be measured directly from the observed reflectivity [15].

The historical progression of the theoretical develop-ments for this problem has recently been reviewed [25] and only a brief overview is necessary for those readers who will not be familiar with the long background to this problem. schoch [1] first observed and, using schlieren photography, photographed a reflected ultrasonic beam near θR

c that appeared to be laterally shifted on reflection. He measured this shift, here referred to as Δ2, and de-

Quantitative Modeling of Reflected Ultrasonic Bounded Beams and a New Estimate of the

Schoch Shiftyoucef bouzidi and douglas r. schmitt

manuscript received February 26, 2008; accepted June 23, 2008. This work was funded by the nsErc discovery Grant Program and the can-ada research chairs Program to drs.

y. bouzidi is with divestco Processing, a division of divestco Inc., calgary, alberta, canada (e-mail: [email protected]).

d. r. schmitt is with the Institute for Geophysical research, de-partment of Physics, University of alberta, Edmonton, alberta, canada (e-mail: [email protected]).

digital object Identifier 10.1109/TUFFc.2008.981

veloped a first-order theory that predicted its magnitude [1], [2]. as noted by breazeale et al. [13], brekhovskikh [14] included the second derivative of the phase shift in an attempt to provide the distribution of the reflected amplitudes at the rayleigh angle. becker and richards [3] carried out a theoretical analysis that showed the in-fluence of the attenuation of the solid on the intensity of the reflection null, particularly that of the shear mode. bertoni and Tamir [12] developed an approximate method that appears to capture much of the character of the re-flectivity of a bounded Gaussian-profile beam near θR

c. Heuristically, their model suggests that the incident wave energy is partitioned into one portion that is specularly re-flected and another that is converted into a leaky rayleigh wave that then propagates along the fluid-solid interface. at θR

c, the phase of these 2 components of the energy are 180° out of phase and hence their destructive interference produces the null zone. breazeale et al. [11], [13] measured the beam from schlieren photographs displacement and found good agreement with this theory. claeys and le-

roy [16] further extended the ideas to develop an exact method to calculate the amplitude profile of a reflected beam given the incident profile. They placed no restric-tions on the shape of the incident profile. lately there has been renewed interest in this topic with studies of some specialized cases of corrugated surfaces [26], [27], with re-development of qualitative schlieren photography but en-hanced with digital imaging techniques [24], and with new developments in alternative methods of modeling [28].

This earlier work was limited to the study of the null and the nonspecular displaced maximum energy zones in part likely due to the limitations of the schlieren photog-raphy employed in early studies. as noted above, several authors worked on various approximations to predict the magnitude of the nonspecular lateral beam displacement, which is also called the schoch shift Δ2. These approxima-tions have difficulty in predicting the complex behavior of a narrow beam that, on reflection, exhibits complex patterns of alternating zones of low and high intensity. Here, an updated derivation for Δ2 that is applicable to a broader range of problems is derived and tested against full wave-field numerical calculations.

In this contribution, the propagation and reflection of such monofrequency bounded beams is first modeled nu-merically. This procedure has recently been successfully employed in modeling the transmission of an acoustic bounded pulse through a water-loaded plate [29] and in the study of a reflection of a long pulse emitted from a small transducer [24]. The results will also be necessary to interpret properly ultrasonic pulse reflectivity experiments produced under a real laboratory situation with finite di-mension ultrasonic transducers. a new expression that re-lies on a Taylor expansion for the lateral schoch shift of such beams is also developed. Images from the calculations for a variety of transducer dimensions and frequencies are given and compared with this approximated schoch shift. This is then further developed to show the behavior of a bounded pulse incident at the rayleigh angle.

II. monofrequency Plane Wave behavior

A. Geometry

The reference frame that will be referred to at all times is the ( , )x z cartesian coordinate system where x and z are unit vectors that denote the horizontal and vertical directions respectively; see Fig. 1(b). The reflection and transmission coefficients are all written in terms of vertical and horizontal wave numbers kz and kx, respectively. The components are related to each other through the wave number for the wave mode and in the fluid the longitudi-nal wave number is kf = (k x

2 + k z2 1 2) / = ω/Vf where ω

is the angular frequency and Vf is the speed of sound in the fluid. The projection of k onto the interface is kx = ksin(θP). The longitudinal and transverse wavenumbers in the solid are denoted kP and kS, respectively.

2662 IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. 55, no. 12, dEcEmbEr 2008

Fig. 1. (a) Example of a schlieren photograph showing the nonspecu-lar (ns) and specular (s) reflected lobes of a bounded beam from the interface between water and stainless steel. a beam is emitted from a 1.9 cm diameter piezoelectric ceramic transducer T and incident at the rayleigh critical angle θ. Frequency is not reported. Image adapted, with permission, from Fig. 2(b) in neubauer and dragonette [6]. copyright 1974, american Institute of Physics. (b) Problem geometry typical of a laboratory situation. T is the transmitter and r is the receiver.

B. Amplitude at a Liquid Solid Interface

The 2-d boundary value problem has been solved for a monofrequency plane wave with the reflection coefficient R given by (1) (see above) [13]. see the appendix, where ρ = ρf /ρs is the ratio of the fluid density over the solid den-sity. The normalized displacement of the reflected wave is described by:

u iR k k x k z erx rx rz

i k z k x trz rx= +( ) + -( ) ˆ ˆ ( )w (2)

The physical particle displacement is given by the real part of (2). The reflected energy fluxes and amplitudes for a case of water over glass are shown for purposes of illus-tration in Fig. 2; more details are given in the appendix.

C. Phase of a Reflected Plane Wave

Phase rotations on reflection affect the experimental measurements and will reappear as we continue through to successive levels of acoustic beam complexity. The phase angle of the reflection coefficient given in (1) may be subdivided onto 3 regions (I, II, and III) delimited by θP

c and θS

c as shown in Fig. 2(b) or, equivalently, by kP and kS. The phase rotation ξ = tan−1[Im(R)/re(R)] is non-trivial past the first critical angle θP

c once R(kx) becomes imaginary. Explicitly, this phase rotation for all incidence angles (see the appendix for details) in region I, II, and III is given in (3) (see next page).

The phase shift defined by (3) is continuous and exists for all angles of incidence. It is plotted for the illustrative water-glass case in Fig. 2(c). This shows that, although ξ(kx) is continuous, its first derivative does not exist at the critical angles. one additional point of interest is the angle θR

c where the phase rotation is 180°.

D. Nonspecular Displacement on Reflection

on reflection, the plane wave undergoes a phase ro-tation as part of the reflection coefficient. as shown in

2663boUzIdI and scHmITT: quantitative modeling of reflected ultrasonic bounded beams

R kk k k k k k k k k k

xx s p x s x x s p x( )

( ) ( ) ( ) ( )=

- + - - - -2 42 2 2 2 212 2 2

12 2 4 2 2r

112 2 2

12

2 2 2 2 212 2 2

12 2 42 4

( )

( ) ( ) ( ) (

k k

k k k k k k k k

x

x s p x s x x s

-

- + - - +

-

r kk k k kp x x2 2

12 2 2

12- -

-) ( )

(1)

Fig. 2. Plane wave modes generated at water-glass boundary. The prop-erties of the materials used are listed in Table I: (a) The normalized en-ergy distribution normal to the interface; (b) the normalized amplitudes; and (c) the phase rotation angles. The critical angles for this case are θP

c = 15.01°, θS

c = 25.69°, and θRc = 28.28°.

TablE I. Physical and mechanical Properties of Glass and Water.

Parameter Glass Water

density (g/cm3) 2490 1000VP (m/s) 5790 1500VS (m/s) 3460 n/aVR (m/s) 3165 n/a

Fig. 1(a), this phase shift induces an apparent lateral dis-placement of the main energy from its specular path. The geometric elements of the problem (Fig. 3) include the nonspecular lateral shift Δ2.

To first order, the displacement Δ2 along the interface of the reflected wavefield was given by schoch [1], [2]:

D 2 = -¶¶ =

x

q

( ).

kk

x

x k kx x

(4)

Where kθx is the projection of the specular incidence wave-number the x direction. In the case of a plane wave, this phase rotation is exact. The derivative of the phase angle exists at all angles with the exception of θP

c and θSc. at

these angles, the phase exhibits a sharp edge where the left and right derivatives are not equal. away from these angles the shift can be approximated (for a large bounded acoustic beam) by the first derivative of the phase angle of the reflection coefficient as given in (4).

It is useful to further examine this approximation. To do this, the geometry defined in Fig. 3 is employed, and this places the fluid-solid interface at z = h beneath the arbitrary reference z = 0 plane that lies in the fluid. let a plane wave be incident onto a fluid-solid boundary at an angle θP such that the steady-state wavefield is φ1(x,0) = e ik xx at t = 0 on the plane z = 0. Following brekhovskikh [14], at the fluid-solid interface, the wavefield is φ2(x,h) = e i k x k hx z( )+ at z = h and φ3(x,0) = R k ex

i k x k hx z( ) ( )+2 as the wave is reflected back to z = 0. Expressing R(kx) as

R k R k ex xi k x( ) ( ) ( )= x (5)

we have

f x3

20( , ) ( ) ( ( ))x R k exi k x k h kx z x= + + (6)

To find the point x + Δx at z = 0 where φ3(x,0) has the same phase as φ`(x,0) we must have

k x k h kx z xD = - -2 x( ) (7)

Taking the derivative on both sides of (10) with respect to kx we get

Dxkk

hkk

z

x

x

x= -

¶¶

-¶¶

2x( )

(8)

or

Dxkk

hkk

x

z

x

x= -

¶¶

2x( )

(9)

assume, temporarily, that the reflection coefficient is real, then the wave can be regarded as being virtually re-flected from the plane z = h without the influence of the reflection coefficient. In this case the displacement is sim-ply Δ1 = 2 (kx/kz)h, which is at the specular reflection path for a plane wave as given in brekhovskikh [14]. How-ever, when the reflection coefficient is complex, the re-flected wave amplitude is weighted by |r(kx)| and further shifted by the amount Δ2 = [∂ξ(kx)]/∂kx along the x-axis. This quantity is wavenumber, and hence frequency, depen-dent. because ξ(kx) is analytic in vicinity of qR

c that does not include the shear-critical angle, it can be expanded via a Taylor series and keeping only the first term we have [1], [2]

D 2 = -¶¶

» -x

x q( )

'( )kk

kx

xx (10)

where kθx is the wavenumber at the rayleigh critical angle qR

c . Eq. (9) then becomes


x

r

=

£

- - - -

--

0

2 2

21

4 2 2 2 2 212 2 2

12

2

:

tan( ) ( ) ( )

(

k k

k k k k k k k

k

x p

s x s x p x

x kk k k k k k k k k k k ks f x x p s x f x x s x2 4 2 2 2 2 2 2 2 2 4 2 8 216) ( ) ( )( )( ) (- + - - - - r --

é

ë

êêêê

ù

û

úúúú

³ ³

- - --

kk k k

k k k k k

ps x p

s x p f x

2

14 2 2

12 2 22

):

tan( ) ( )r

112 2 2 2 2 2

12 2 2

12 2

2 2 2

2 4

2

( ) ( ) ( )

( )

k k k k k k k

k k

x s x p x s x

x s

- - - -éëê

ùûú

- -- - -éëê

ùûú - - -4 2 2

12 2 2

12 2

2

2 2 2 8 2 2( ) ( ) ( ) (k k k k k k k k k kx p x s x f x s x pr ))

:

é

ë

êêêêê

ù

û

úúúúú

³

ì

í

ïïïïïïïïïïï

î

ïïïïïïïïïïï

k kx s

(3)

Fig. 3. Plane wave reflected at an interface between media fluid and a solid. on reflection, the wavefield is displaced by an additional amount of the schoch shift Δ2 along the interface due to the phase rotation angle of the reflection coefficient.

Dxkk

h kx

zx= -2 x q'( ). (11)

In the case of a monofrequency plane wave, only one wavenumber is present in the incident wavefield that leads to an exact displacement of the incident wavefield Δ = Δ1 + Δ2 (Fig. 3). However, for a beam that is bounded in space (as for a source transducer) and therefore band-limited in the wavenumber domain, each wavenumber will be shifted with a different amount on reflection from the interface. This spatial shift as a function of the wavenum-ber as shown in Fig. 4 exhibits large values in the vi-cinity of the rayleigh angle. consequently, the reflected wavefield exhibits various complex patterns of low- and high-intensity zones. These patterns will depend on the wavenumber content of the incident wavefield and will be illustrated via numerical modeling below.

In particular, the displacement of the maximum ampli-tude stripe will not be properly approximated by (4), es-pecially when the wavenumber bandwidth is large. Here a simple alternative approach is given to better estimate this displacement. First, (4) is used to compute the shift for each of the wavenumbers that most contribute to the maximum energy of the incident wavefield. Then, by tak-ing the weighted average of these shifts, we obtain a better approximation to the displacement of the maximum non-specular reflected amplitude stripe. let the bandwidth be delimited by [kx1kx2] and a k xq the amplitude at kθx. The weighted averaged displacement can then be written as

Dx

a k

a

k xk k

k

kk k

k

xx x

x

x

x x

x= - =

=

å

å

x '( )

.1

2

1

2 (12)

note that (12) is given at the interface and the displace-ment is taken relative to the specular point of reflection.

a numerical approach can be used to estimate the wavenumber bandwidth that contributes significantly to the energy of the bounded beam. First, the spatial ampli-tude spectrum A(kx) of the incident wavefield is computed and the wavenumber kθx at the maximum amplitude found (Fig. 5). The range of wavenumbers over which the aver-age is calculated is then taken here to be delimited by the wavenumbers kx1 and kx2 that lie on either side of kθx of the spatial amplitude spectrum, at which point the d2A(kx)/dkx

2 is a minimum, i.e., the point of minimum change in the tangent to A(kx). numerical experiments demonstrat-ed that the value of the average shift Δx changed little when a larger range was used; this criterion works well at least for the Gaussian beams employed in this study. an example of this bandwidth is delineated in Fig. 5.


Fig. 4. First-order schoch shift displacement Δ2 along a water-glass in-terface as a function of the angle of incidence at a frequency of 760 kHz.

Fig. 5. (a) Illustrative amplitude spectrum A(kx) for a Gaussian beam transducer with wavenumber kθx at the peak amplitude. The spatial wavenumber bandwidth over which the weighted average schoch shift is calculated is highlighted by the dark line and underlying gray area within the range [kx1, kx2]. (b) cumulative amplitude sum B(|kx2 – kx1|) showing inflection point that for the case of the Gaussian beam provides the point that delimits the bandwidth range [kx1, kx2] over which the averaging of the lateral displacement is calculated.

It is not clear how the bandwidth [kx1, kx2] should be determined, and here a numerical approach is taken. First, the spatial amplitude spectrum A(kx) of the inci-dent wavefield is computed and the wavenumber kθx at the maximum amplitude found, as shown in Fig. 5(a). define B(|kx2 – kx1|) to be the simple sum of all the amplitudes of A(kx) within the bandwidth [kx1, kx2] centered on kθx; then b may be plotted as a function of |kx2 – kx1| as the band-width [kx1, kx2] increasingly widens. as shown in Fig. 5(b) this function rapidly increases but then nearly flattens at a sufficiently large value of |kx2 – kx1|. numerical tests show that the value of B increases insignificantly past the inflection point found where d2 B(|kx2 – kx1|)/d(|kx2 – kx1|)2 is a minimum; and hence this criterion is used to select the integration bandwidth range [kx1, kx2] over which (12) is implemented. This range is highlighted in Fig. 5(a).

In the case of a bounded pulse, the weighted average would be taken over the area of the dominant amplitudes in the (ω,kx) domain.

III. monofrequency bounded acoustic beams

now that the basic theory behind plane wave propaga-tion and reflectivity has been discussed, it is important to next describe the reflectivity of monochromatic, but spa-tially bounded, acoustic beams. a monochromatic acous-tic beam, as is shown in Fig. 1(a), is a steady-state plane wave bounded in space. a spatially bounded monochro-matic propagating plane wave field φ(x,z) of width 2a cen-tered at the origin in the x direction is described by the integral [14]

fp

( , ) ( , )x k e dkxik x

xx0

12

0= -

-¥

+¥

ò F (13)

where Φ(kx,0) is the wavefield known (recorded in an ex-periment) on the plane z = 0 (Fig. 1) once transformed into the Fourier domain (kx,0) and is given by

F( , ) ( , ) .k x e dxxik x

a

ax0 0=

-

+

ò f (14)

Here the time dependence is left out for clarity. The wavefield can then be extended to the whole space ( , )x z using a phase-shifting method [30]

fp

( , ) ( , )x z k e e dkxik x ik z

xx z= - -

-¥

+¥

ò12

0F (15)

φ(x,z) is the wavefield everywhere in the space ( , ).x z as pointed out above we note that φ(x,z) can be any compo-nent (e.g., pressure or particle displacement) describing the wave as it propagates. Eqs. (13)–(15) describe a plane wave exactly when a → +∞. To find the wavefield inci-dent at the fluid-solid interface z = h we write

fp

( , ) ( , ) .( )x h k e dkxi k x k h

xx z=

-¥

+¥- +ò

12

0F (16)

The reflected wavefield can be expressed with a modu-lation by the reflection coefficient R(kx) at the reflecting interface before the inverse Fourier transform (16):

fp

( , ) ( ) ( , ) .( )x z R k k e e dkx zi k x k h ik z

xx z z= - + -

-¥

+¥

ò12

0F (17)

Eq. (17) describes the reflected wavefield everywhere above the reflecting interface. Eqs. (13)–(17) will be used to model a monofrequency bounded beam below.

a reflected bounded acoustic beam exhibits various patterns of intensity distribution near the rayleigh criti-cal angle θR

c. at this angle, the phase of the reflection coefficient is exactly 180°, and the reflected beam is appar-ently shifted a significant distance, relative to the wave-length, along the boundary. It is important to note that it is not the entire wavefield that is apparently shifted along the interface but rather the energy in the vicinity of the rayleigh angle that is greatly affected by the reflection coefficient’s phase angle. depending on the wavenumber content and bandwidth of the incident wavefield, various patterns of low- and high-intensity strips will occur in the reflected wavefield. Indeed, to some degree, the problem is more complicated than might be suggested by focus on the schoch shift alone.

The modeling procedure used to produce the results shown below of a bounded beam incident to a fluid-solid interface is as follows

1. assume a plane wave e ik xx incident to the interface at a given angle θ.

2. record the wavefield at the reference plane z = 0. 3. bound the wavefield within the projection of the

width 2a on the x-axis, the wavefield outside the bounds is set to zero.

4. Use (14) to transform the wavefield to the Fourier domain to obtain Φ(kx,0).

5. Use the phase shift method via (16) to phase ad-vance the wavefield in the fluid to the interface to obtain F( , ) ( )k ex

i k x k hx z0 + at z = h. 6. multiply by the reflection coefficient to obtain

R k k ex xi k x k hx z( ) ( , ) ( )F 0 +

7. Use the phase shift method via (16) to phase ad-vance the wavefield in the fluid to obtain F( , ) ( )k ex

i k x k hx z0 2+ at z = 0.

note: The extrapolated wavefield can be calculated by small z steps and each time the inverse Fourier trans-formed to obtain the wavefield at depth z.

The numerical modeling of a variety of reflected Gauss-ian bounded acoustic beams incident on a water-glass in-terface is shown in Figs. 6 to 8 to illustrate a variety of effects. The bounded beam widths in all cases given here are taken at the base of each beam. These scenarios include reflection away from critical angles for a single transducer (Fig. 6), reflection of different frequency beams


at the rayleigh angle for the same transducer aperture (Fig. 7), and reflection of different aperture beams with the same frequency at the rayleigh angle (Fig. 8). In these visualizations, the values of the amplitudes given are those of the signal’s time series s(t) amplitude envelope, or the modulus of the analytic signal f(t) = s(t) + iH[s(t)] where H[s(t)] is the Hilbert transform of s(t).

Fig. 6 displays the results of the above-described proce-dure of a bounded beam 5 cm wide and a frequency of 1 mHz reflected at 3 angles of incidence that lie within each

of the incidence ranges I, II, and III but also sufficiently far away from any of the 3 critical angles. as expected, the wavefield is reflected with no apparent shift along the in-terface for all these cases. It is to be noted that past P-critical angle q p

c, the phase of the reflected wavefield is affected by the reflection coefficients. but away from the critical angles, the phase rotation is small and does not appear readily on the reflected wavefield images.

as expected, the reflectivity is substantially more com-plicated near the rayleigh critical angle. The reflected wavefield produced by an 8 cm wide transmitter and in-cident at rayleigh angle of incidence of 28.28° is modeled for a variety of frequencies from 2 mHz to 0.15 mHz in Fig. 7. all of these beams experience the schoch shift that moves the beam to the right of the expected specular path. The first observation is that the higher frequency beams are less distorted on reflection. Further, at the higher frequencies, the schoch shift Δ2 is well explained by the traditional (4) and cannot be distinguished from the averaged shift (highlighted by the white dashed line) that employs (12). The two do not become noticeably dif-ferent until Fig. 7(c) at a frequency of 0.6 mHz, and they continue to diverge with decreasing beam frequency as the reflected beams becoming increasingly complex. The first hints of the zone of diminished amplitude, or trough, ap-pears in Fig. 7(d) close to the specular path. It should be noted that the amplitude does not disappear completely in this “null” zone as might be expected from the example schlieren photograph of Fig. 1. In many cases, the null zones shown in the schlieren photography likely result from insufficient sensitivity of the film [24]. The trough becomes more prominent in Fig. 7(e), and for the low-est frequency in Fig. 7(f) has evolved to a true null zone. These amplitude troughs and nulls follow close to but not exactly on the specular path.

although the high-frequency beams appear relatively well behaved, they still all are shifted laterally with re-spect to the specular path. The traditional schoch shift of (4) no longer agrees with the peak of the reflected beam, while the averaged schoch shift from (12) still gives a good estimation of where this maximum is through Fig. 7(e). at the lowest frequency in Fig. 7(f), the reflected beam is highly complex with several weak and strong zones. The averaged schoch shift from (12) appears to give a good es-timate of the shift but only in the vicinity of the interface. Interestingly, the reflected portion of the beam with maxi-mum amplitude diverges from the expected upgoing path shown by the white line, that is, the apparent angle of reflection no longer is the same as the angle of incidence.

In Fig. 8, three, 800-kHz (wavelength: 1.9 mm) Gauss-ian beams with transducer widths of 4 cm, 6 cm, and 8 cm are also reflected at the rayleigh angle. These images illustrate that narrower beams spread more than broad beams. In these cases the more traditional (4) displace-ment estimation is not far from the maximum amplitude but the new estimations via (12) give a better estimate, particularly in the last 2 examples.


Fig. 6. reflection of a large Gaussian acoustic beam of frequency 1 mHz and width of 5 cm at 3 angles of incidence away from critical angles: (a) at 10° within region I; (b) at 20 o within region II; and (c) at 35 o within region III. The reflection surface is a water-glass interface (see Properties in Table I). The thin arrow line shows the specular reflection path.

http://dx.doi.org/10.1109/TUFFC.2008.981/mm6

It is important to note that the schoch shift is only an apparent shift explained as a dispersive effect in the wave-number domain produced by the rapid change of the phase angle in the vicinity of the raleigh angle. Eq. (12) is based on this dispersive effect. The more that the wave energy is concentrated around the rayleigh angle, the greater will be the apparent shift along the interface. This concentration of energy around the rayleigh angle depends on the width of the beam that determines the spatial bandwidth. despite the fact that the reflection coefficient is independent of fre-quency, the derivative of its phase is frequency dependent.

consequently, the temporal frequency also affects these wavenumbers. When the width of the bounded acoustic beam is large compared with the wavelength, then most of its energy is concentrated around the rayleigh angle and the displacement can be predicted with good accuracy by the first derivative of the phase angle with respect to the wavenumber in the x direction. as indicated earlier by brekhovskikh [14], the width of the beam should be greater than the wavelength. a measure of the spreading expected for a given monofrequency beam producing a wavelength λ from a transducer of aperture a is:


Fig. 7. rayleigh incidence angle reflection behavior of a large Gaussian acoustic beam width of 8 cm but with varied frequencies of: (a) 0.15 mHz,; (b) 0.25 mHz; (c) 0.35 mHz; (d) 0.4 mHz; (e) 0.8 mHz; and (f) 1.2 mHz from a water-glass interface (see Properties in Table I). The incidence angle of the axis of the beam corresponds to the rayleigh angle of 28.28°. White arrows represent the average schoch shift as estimated using the formulation developed in (12). The black dashed arrow line represents schoch shift calculated using (4) but with only the single value for the peak wavenumber. note that in (e) and (f) the difference between these is small.


hl

=a

. (18)

consequently, the approximations employed here are valid as long as η ≪ 1. The examples shown here all sat-isfy this requirement. once η approaches unity, then the beam spreading is no longer so well behaved. The ratio η where the spreading is well behaved for the examples shown below is found to be at η = 0.075, as shown in Fig. 7(b). This behavior is clearly exhibited by Figs. 7 and 8, where more spreading occurs, particularly at lower temporal frequencies—Fig. 7(a) and (b)—and smaller ap-ertures—Fig. 8(a).

IV. bounded acoustic Pulses

The meaning given here to bounded acoustic pulses is that they are wavefields generated by exciting a transduc-er of a given shape and size by a time-limited pulse. This generates a pulse that is bounded in both time (and hence containing a range of temporal frequencies) and space (and hence containing a range of wavenumbers). The use of bounded acoustic pulses in laboratory experiments is convenient and hence a good understanding of their be-havior is necessary. let φ(x,0,t) be the wavefield generated by a line segment source located at z = 0. by analogy to the monochromatic bounded acoustic beam discussed above, the wave potential field of a bounded acoustic pulse can be written in the Fourier domain as

jp

w ww( , , ) ( , , )x t k e e dk dxik x i t

xx0

1

40

2= -

-¥

+¥

-¥

+¥

òò F

(19)

where Φ(kx,0,ω) is the wavefield known (recorded in an experiment) on the plane z = 0 once transformed into the 2-d (kx,ω) Fourier domain and is given by

F( , , ) ( , , ) .k x t e e dxdtxik x i tx0 0w j w= -

-¥

+¥

-¥

+¥

òò (20)

In the case of a pulse, the integration limits are re-duced to the spatial interval [−a,+a], which bounds the wavefield in space and to the time interval [−t,+t], which bounds the wavefield in time.

The wavefield described by (20) can now be extended to the whole space by forward continuation (or to z < 0 by backward continuation) similarly as for the monochromat-ic bounded beams. This continuation is achieved by the multiplication of each frequency component of φ(x,0,t) by the phase shift operator e ik zz- followed by the inverse Fou-rier transform in space and time. Then kz is simply the wavenumber in the z direction defined as kz = (k2 − kx

2)1/2 where k is the wavenumber in the medium for each frequency component present in the wavefield φ(x,0,t) and kx is the wavenumber in the x direction. The wavefield everywhere in the space ( , , )x z t is then given by

jp

w ww( , , ) ( , , ) .x z t k e e e dk dxik x i t ik z

xx z= - -

-¥

+¥

-¥

+¥

òò1

40

2F

(21)


Fig. 8. reflection of a Gaussian acoustic beam of frequency 800 kHz: (a) width 4 cm; (b) width 6 cm; and (c) 8 cm. The reflection surface is a water-glass interface (see Properties in Table I). The incidence angle of the axis of the beam corresponds to the rayleigh angle of 28.28 o in region III near the s critical angle of incidence. The beam appears to be displaced to the right along the interface. The dark dashed arrow shows the approximated displacement calculated via (4) and the light dashed arrow via (12).


The reflected wavefield at an interface located at a dis-tance h below the reference plane z = 0 is given by

f

pw ww

( , , )

( ) ( , , ) ( )

x h t

R k k e e dk dx xi k x k h i t

xx z

=

- +

-¥

+¥

-¥

+¥

òò1

40

2F ..

(22)

similar to the bounded acoustic beam, a bounded acoustic pulse will suffer the same phase rotation on re-flection when incident at the rayleigh angle. However, more complexity occurs as a result of the temporal fre-quency content. Each frequency is displaced a different amount according to (4); the broader the bandwidth, the greater the dispersion.

The composite displays of Fig. 9 attempt to describe this phenomenon. The color image within the diagram is the amplitude spectrum in the 2-d frequency-wavenum-ber (ω,kx) domain of the incident Gaussian bounded pulse wavefield. It is first useful to compare these spectra be-tween Fig. 9(a) and (b). The top and bottom spectra are

for the spatially wide (15 cm) and the spatially narrow (3 cm) Gaussian bounded pulses, respectively. The maxi-mum amplitudes of both beams are centered at θc

R. The wide and narrow pulses in real space are, conversely, nar-row and wide in the (ω,kx) domain as expected for Fourier theory. The angle of incidence (or equivalently horizontal wavenumber) dependent schoch displacement is the same for both cases and is given by (4); the composite of the dis-placement line graphs and the spectra illustrate first that the maximum displacement of the pulse does occur at the point corresponding to the peak amplitude of the signal (0.76 mHz) and at the wavenumber 1500 m−1. The goal of plotting this information together is to highlight that range of wavenumbers that contributes the most to the incident wavefield. These plots allow selection of the range of wavenumbers employed the averaging of the schoch dis-placement of (12). The range of wavenumbers centered at the predominant frequency of 0.76 mHz is found using the procedure described in section II-d above. (This is shown by the light gray area in the composite plots.) The 3 cm wide bounded acoustic pulse shown in Fig. 9(b) exhibits more spreading than the 16 cm wide bounded acoustic pulse. For the broader (large aperture) bounded acoustic pulse, the main energy is more localized in the vicinity of the rayleigh angle than that of the narrower (small aper-ture) bounded acoustic pulse.

Eqs. (19)–(22) are now used to model a bounded acous-tic pulse. The modeling procedure of a bounded pulse in-cident to a fluid-solid interface is as follows

1) assume a bounded acoustic pulse incident to the interface at a given angle θ.

2) record the wavefield at the reference plane z = 0, and make the record length long enough to make room for the reflected wavefield by zero padding to avoid wrap around in the Fourier domain. The re-cord length must exceed the travel time from z = 0 to z = h and back to z = 0 following the specular reflection path.

3) Use (20) to double transform the wavefield to the Fourier domain to obtain Φ(kx,0,ω)

4) Use the phase shift method via (21) to phase ad-vance the wavefield in the fluid to the interface to obtain F( , , ) ( )k ex

i k x k hx z0 w - + at z = h. 5) multiply by the reflection coefficient to obtain

R k k ex xi k x k hx z( ) ( , , ) ( )F 0 w - + .

6) Use the phase shift method via (21) to phase ad-vance the wavefield in the fluid back to z = 0 and obtainF( , , ) ( )k ex

i k x k hx z0 2w - + .

note: The extrapolated wavefield can be calculated by small z steps and each time inverse Fourier transform to obtain the wavefield at depth z.

The numerical modeling of a reflected Gaussian bound-ed acoustic pulse incident at θR

c on an interface of the hypothetical water-glass interface is illustrated in Fig. 10(a)–(d) in which the distribution of the amplitudes are


Fig. 9. (a) a composite display of (color image) a wide Gaussian acoustic pulse (width 15 cm) in (ω,kx) domain and (line graph) the correspond-ing shift along a water-glass interface at the peak frequency of 760 kHz. (b) a composite display of (color image) a narrow bounded acoustic pulse (width 3 cm) in the (ω,kx) (ω,kx) domain and (line graph) the corresponding shift along a water-glass interface at the peak frequency of 760 kHz.


shown for 3 different times. The reflected wavefield dis-plays its maximum energy away from the specular path as shown in Fig. 10(d). as for most of the monofrequency beam examples given above, the displacement estimated by (12) (color dashed line) correlates better than that cal-culated via (4) (dark dashed line) in this example. The estimated averaged displacement is calculated using the wavenumbers and temporal frequencies delimited by the white dashed line shown in Fig. 10(a). In an experiment, a receiver positioned on the specular path will record a drastic drop of amplitude. For bounded acoustic pulses, the temporal frequency content adds a complication to the spreading of the reflected wavefield because the displace-ment of each wavenumber present in the incident wave-field also depends on the temporal frequency. a wider fre-quency bandwidth will produce more spreading.

V. conclusions

a numerical modeling based on the phase advance tech-nique was employed to model spatially bounded acous-tic beams on reflection from an interface between a fluid and a solid. It is expanded to model acoustic bounded pulses (bounded both spatially and temporally) as well. The latter are important in laboratory experiments. on reflection, an incident-bounded beam or a bounded pulse exhibits a variety of patterns of high- and low-intensity zones that depend not only on the width of the beam but on the frequency content, as has long been known. The highest amplitude zone is displaced a distance of several wavelengths along the interface away from the specular reflection path. The traditional approximation of the dis-placement by the first derivative of the phase shift does not predict accurately the position of the center of the maximum amplitude zone for all types of Gaussian acous-tic beams and bounded acoustic pulses. In contrast, the numerically calculated displacement through an average of the displacements of the components that contribute the most to the maximum energy of the incident wavefield is in good agreement with a wide variety of Gaussian-bounded beams and Gaussian-bounded pulses of various width and frequency, although it becomes less accurate as the ratio between the wavelength of the acoustic wavefield and the transducer aperture increases. The numerical result of the phase advance technique reproduces even subtle features of the reflected wavefield with high fidelity.

appendix

The 2-d boundary value problem in the space ( , )x z of the reflectivity and transmissivity of acoustic waves at a fluid-solid interface located at the plane z = 0 for a P-wave impinging from the fluid (z < 0) has been solved (1) and can be found in many texts. However, explicit solu-tion of the phase shift is not reported in the literature and hence it is worth providing this here. It is also worth not-


Fig. 10. reflection of a Gaussian acoustic pulse of frequency 760 kHz and width of 8 cm. The reflection surface is a water-glass interface (see Properties in Table I). The incidence angle of the axis corresponds to the rayleigh angle of 28.28°. (a) 2-d Fourier Transform to (ω,kx) domain of the pulse with the area in (ω,kx) space employed in the calculation of the averaged schoch shift. snapshots of the pulse (b) immediately after being launched from the transducer and before reaching the water-glass interface; (c) during reflection from the interface; and (d) well after re-flection. The bounded acoustic pulse appears to be displaced along the interface and the pulse front broadened substantially. The black line with the arrow point up is the path of the specular reflection, whereas the black and the light dashed lines show the path of the reflected peak pulse amplitude estimated using schoch’s original approximation (4) and the averaged displacement (12), respectively. The averaged displacement agrees well with the peak amplitude of the reflected pulse. These 3 up-ward-going paths are the same in Fig. 10 (b)–(d) but are denoted only

in Fig 10(b).



ing that the normalized amplitude for the reflected wave remains the same for the potential, the displacement, par-ticle velocity, pressure, and particle acceleration. no mat-ter what is measured (e.g., particle displacement, velocity, or acceleration) in a reflection experiment we obtain the same value as long as we normalize the reflected wave field to the measured direct arrival.

as above, θP, q pc, and q s

c are the incident, the P-critical, and s-critical angles, respectively. For the reflec-tion coefficient, we distinguish 3 regions. region I is where the angle of incidence is below the P-wave critical angle (q < q p

c); region II is where the angle of incidence is

between the P-wave and s-wave critical angles (q sc > θ

> q pc); and region III is where the angle of incidence is

beyond the s-wave critical angle ( ).q q> sc In region I,

the reflection coefficient is always real, and thus the phase angle is zero. In region II, all terms of the reflection coef-ficient in (a1) are real except for those that contain a purely imaginary wave number in the z direction of the P-wave in the solid. In this region we can write for pur-poses of brevity

Ra i b ca i b c

=+ -+ +

( )( )

, (a1)

where a, b, and c are always real and are given by

a k k

b k k k k k

c k k k

x s

x p s x x

s x p

= -

= - -

= -

( )

( ) ( )

( )

2

4

2 2 2

2 212 2 2

12 2

4 2 212r (( ) .k k x

2 212-

-

(a2)

Therefore, the phase angle in region II is given by (a3) (see above).

In region III, all terms of (a1) are real except for those that contain a wave number component in z direction of the P and s waves. These wave numbers are both pure imaginary numbers. consequently, we have

Ra b ica b ic

=+ -+ +

. (a4)

Therefore, the phase angle in region III is given by (a5) (see above).

acknowledgments

The authors thank the associate editor and two anony-mous reviewers for their constructive advice on ways to improve the manuscript and figures.

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xr

=- - - -

--tan

( ) ( ) ( )

( ) (1

4 2 2 2 2 212 2 2

12

2 2 4

2 2

2

k k k k k k k

k k ks x s x p x

x s22 2 2 2 2 2 2 2 4 2 8 2 216- + - - - - -

é

ë

êê

k k k k k k k k k k kx x p s x x x s x p) ( )( )( ) ( )rêêê

ù

û

úúúú (a3)

xr

=- - - - - --tan

( ) ( ) ( ) ( )14 2 2

12 2 2

12 2 2 2 2 2

122 2 4k k k k k k k k ks x p x x s x p (( )

( ) ( ) ( )

k k k

k k k k k k k

x s x

x s x p x s x

2 212 2

2 2 2 2 212 2 2

122 4

-éëê

ùûú

- - - - 22

2

2 2 2 8 2 2éëê

ùûú - - -

é

ë

êêêêê

ù

û

úúúúú( ) ( )k k k k kx s x pr

(a5)

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Youcef Bouzidi is a research geophysicist at di-vestco Processing, a division of divestco Inc. (for-merly Geo-X systems ltd.) based in calgary. He joined the company in 2002. He received an engi-neering degree in geophysics from the algerian Petroleum Institute in 1980, an ms degree in geo-physics from columbia University, new york, in 1983, and a Ph.d. degree in geophysics from the University of alberta, canada, in 2003.

He is a registered professional geophysicist with the association of Professional Engineers,

Geologists, and Geophysicists of alberta (aPEGGa) and is an active member of the society of Exploration Geophysicists (sEG) and the ca-nadian society of Exploration Geophysicists (csEG).

His areas of research are signal processing with a focus on seismic depth imaging, rock physics, and wave propagation in porous media.

Douglas Schmitt obtained his Ph.d. degree in geophysics from the california Institute of Tech-nology and his b.sc. degree in physics from the University of lethbridge, canada. He worked for Texaco canada resources as a geophysicist and carried out postdoctoral research at stanford Uni-versity. He is currently a professor in the depart-ment of Physics at the University of alberta, holds the canada research chair in rock Physics, and is director of the Institute for Geophysical re-search. His research interests focus primarily on

field and laboratory experiments related to the determination of earth material properties.


quantitative modeling of reflected ultrasonic bounded beams and a

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