**Specular**** reflections of transient
pressures from finite width plane faces**

**C. S. Clay**

*Department of Geology and
Geophysics, Weeks Hall, **University** of **Wisconsin**, **Madison**, **Wisconsin** **53706*

**Dezhang**** Chu**

*Department of Applied Ocean
Physics and Engineering Woods Hole Oceanographic Institution, Woods **Hole**, **Massachusetts** **02543*

**Saimu**** Li**

*Ocean **University** of **Qing** Dao**, Qing
Dao, Shandon& People's Republic of **China*

( Received 9 June
1992; accepted for publication 21 June 1993 )

The
reflections of transient pressure signals from hard (perfectly reflecting)
finite width plane facets have been studied theoretically and experimentally.
The experiments and theory used point sources, and sound pressures were
observed in the specular direction. Theoretical
solutions used the Helmholtz-Kirchhoff integral and
incident spherical waves in the Fresnel
approximation. The results of these laboratory experiments are compared to
numerical evaluations of the integrals. Applications to acoustical oceanography
and to architectural acoustics are: the specular
reflection of waves from a reflecting facet on the seafloor and the specular reflection of sound waves from sound deflectors
above a performer on a stage.

PACS
numbers: 43.30.Gv, 43.20.Fn, 43.20.Bi, 43.SS.Fw

INTRODUCTION

Most computations of waves scattered and reflected at
rough interfaces use either the incident spherical waves (usually in the Fresnel approximation) or the incident plane-wave
approximation (Fraunhofer). The reflection and
scattering of waves by a finite-dimensioned plane facet gives the problem in
its elemental form. We will study the transition from the Fraunhofer
region, where the dimensions of a facet are less than the first Fresnel zone, to the Fresnel
region where the facets are larger than the first Fresnel
zone. Neubauer and Dragonette
made a set of laboratory measurements of the reflections of effectively continuous
wave pressures from blocks of various "hard" materials in water.l They got excellent
agreement between the measurements and evaluations of the HelmholtzKirchhoff
integral.

As an extension of the usual solutions for harmonic
waves, we calculate the amplitudes of transient pressures reflected from finite
width plane facets. Figure 1 shows various sizes of facets superimposed on the Fresnel zones. As a simple approximation, there are two
regions for facet reflection amplitudes. In Fig. 1(a) and (c), both dimensions
of the facets are larger than the first Fresnel zone
and we expect the wave to reflect with full amplitude. Then, the dimensions of
the facet are effectively infinite. In the second region, Fig. 1(b) and (d),
one facet dimension is less than the width of the first Fresnel
zone and reflection amplitudes are reduced.

Clearly the reflection and diffraction of waves at
finite width strips of facets are coupled, and reflected components from the
Helmholtz-Kirchhoff integral are only part of the
solution. Tolstoy published a pair of papers that give solutions for the
diffractions by a hard (perfectly reflecting) truncated wedge, strip, and
sound barriers[2,3]. However, he was primarily
interested in the multiple diffraction part of the solution. In the wedge
assembly or facet ensemble methods for calculating waves diffracted and
reflected at rough seafloors and sea surfaces, one has both diffracted and
reflected components. Diffracted components come from the intersections of
the plane facets, i.e., wedge crests and troughs. Specularly
reflected components come from the facets. We deal with the specularly
reflected components in this paler. Computations of the diffraction components
are in Refs. 8-14.

We mention applications of our results to
architectural acoustics, concert halls, and the design of "acoustical adjustments."
Many halls have acoustic devices over the stage that are
intended to reflect and scatter sound into the audience. These devices are
known as "reflectors," "clouds," and
"deflectors." To be effective, the reflecting devices must reflect
sound over a wide range of frequencies. The wavelengths of lowest frequencies,
the distance of the reflecting devices above the performers, and spherical wave
theory control the minimum dimensions of the reflectors.

I.
AMPLITUDES OF REFLECTIONS FROM A PLANE FACET

We
use the Helmholtz-Kirchhoff integral to make
quantitative computations of the effects of facet widths on reflection
amplitudes. Since the Helmholtz-Kirchhoff integral
is for continuous waves, we suppress the time dependence and calculate the
pressure *P(f) *as a function of
frequency f. Expressions for the Helmholtz-Kirchhoff
integral are in Neubauer and Dragonette,1 Clay and Medwin (1977),15 and Clay (1990).14 Expressions from Clay
and Medwin are designated as GM. After making the Kirchhoff approximation, the Helmholtz-Kirchhofl'
integral is (C-M A10.5.7)

Fig.4 Fig.5

FIG.4.[ on left] Relative
reflection amplitudes of transient signals ιg(u,∞)ι

FIG. 5 [on right].
Laboratory experimental geometry and a pressure signal. (a) The experiments
were made in air. The spark source and pressure receiver are nearly colocated. The facet is made of plaster wall board and the
Boor is cement. The facet length 120 cm is effectively infinite. (b) Typical
pressure signal. The direct arrival was suppressed. The reflection from the
facet is called "facet" and the reflection from the Boor is labeled "floor." Diffraction arrivals are
relatively small for this geometry.

III.
EXPERIMENTS AND THEORY

Figure 5 shows a laboratory experiment and a typical signal.lg Relative to air, plaster wall board is very hard
and dense and the reflection coefficient is nearly unity. These plaster
materials damp vibrations and are good for laboratory acoustic experiments.
The data were taken near vertical incidence and ψ=0. The part of the signal labeled
"Facet" was isolated from the rest of the signal to study the
reflection from the 4-cm-width facet. The reference pressure or *p*∞*( t) *was
achieved by replacing the small facet by a large sheet of plaster sheet board
at the same range. An example of an unfiltered *p*∞*(**t) *is shown
in Fig. 6(a). We delayed digitization and recording of the signal until just
before the signal arrived. The corresponding spectrum is labeled
"source-system response" in Fig. 6(c). We digitally recorded *p*∞*(**t)
*and then used FFT-filter-IFFT operations to calculate
pressures for a set of signals having peak frequencies that range from 3 to 20
kHz. An example of filtered *p*∞*(**t) *for a peak
frequency of 3 kHz is shown in Fig. 6(b) and its spectrum is shown in Fig.
6(c). The filtered domain signals were time shifted for display. The filtered *p*∞*(**t)
*resembles a damped sine wave. Reflected sound
pressures from the facets were processed the same way and filtered signals were
computed for the pair, *p(**t) *and *p*∞*(t). *The
wavelengths λ*p
*that correspond to spectral peaks were computed for
each pair of filtered signals. The λp range, and facet widths were used
to compute a and the experimental values of g exp( u,* *∞) shown in Fig. 7.

Theoretical calculations of g(u,
∞),
also shown in Fig. 7, used the damped sine wave signal ( 33 ) where the damping
constant was

*T=1/fP. *(38)

The
theoretical curve of g(u, oo
) was computed as a function of *u, *i.e.,
*fP** *using (32) and (36). The experimental
values of g exp( u, ∞) are a little larger than
the theoretical curve between u=1 and 1.5.

IV.
SOUND DEFLECTORS IN MUSIC HALLS

Sound deflectors are often placed over the stage in
concert halls. As sketched in Fig. 8, the deflector reflects some of the upward
traveling sound waves from a performer on the stage
into the seating area. The stage overhead and ceiling are not shown. Since
deflectors are part of the stage, the designs can be rather decorative.
Depending on the acoustical problems with the room, the sound deflectors may
also be intended to reinforce specific frequency ranges. Sometimes designers
use a "rule of thumb" that the deflectors scatter and reflect all
waves that have λ less than the dimensions of the
deflector. We believe that the finite facet theory gives an improved "rule
of thumb."

The pressure facet reflection factor ( 21) gives the pressure reflected by a facet relative to
the pressure reflected by

ACKNOWLEDGMENTS

This paper is taken from a tutorial that was written
after Clay's conversations with John Preston and others at the

1.
W. G. Neubauer and L. R. Dragonette,
"Monostatic reflection from a rigid
rectangular plate," J. Acoust. Soc. Am. 41,
656-661 (1967).

2.
I. Tolstoy, "Diffraction by a hard truncated wedge ans
a strip," IEEE J. Ocean Eng. 14, 4-16 (1989).

3.

4.
W. A. Kinney, C. S. Clay, and G. A. Sandness,
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theory and the facet ensemble method," J. Acoust.
Soc. Am. 73, 183-194 (1983).

5.
J. C. Novarini and H. Medwin,
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from a periodic assemblage of wedges: Comparisons with theories of
diffraction gratings;' J. Acoust. Soc. Am. 73,
183-194 (1983).

6.
H. Medwin, "Shadowing by finite noise
barriers," J. Acoust. Soc. Am. 72, 1005-1013
(1982).

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183-194 (1983).

8.
M. A. Biot and I. Tolstoy, "Formulation of
wave propagation in infinite media by normal coordinates with an application
to diffraction," J. Acoust. Soc. Am. 29,
381-391 (1957).

9.
Tolstoy and C. S. Clay, *Ocean
Acoustics.' *Theory *and Experiment
in Underwater Sound 2nd Edition *(American Institute of Physics, Now York,
1987), Appendix 5 is Ref. 8.

10.
S. Li and C. S. Clay, "Sound transmission experiments from an impulsive
source near rigid wedges," J. Acoust. Sac. Am.
84, 2135-2143 (1988).

11.
D.

12.
D.

13.
D.

14.
C. S. Clay, *Elementary Exploration
Seismology *(Prentice Hall, Eng1ewood Cliffs, NJ, 1990), Chap. 14 and
Appendix C1.2.

15. C.S. Clay and Herman Medwin,
*Acoustical Oceanography *(Wiley, New
York, 1977), Sees. 10.4 and 10.5 and Appendix AI0.5.

16.
M. Abramowitz and *Nand6ook of Mathematical Functions, *Appl.
Math. Ser. 55 (National Bureau of Standards, U.S. Government Printing Office,
Washington, DC, 1965).

17.
C. S. Clay and W. A. Kinney, "Numerical computations of time domain
diffractions from wedges and reflections from facets," J. Acoust. Soc. Am. 83, 2126-2133 (1988).

18.
S. Li and C. S. Clay, "Experimentation of time domain di9'raedons from
wedges and reflections from facets" (in Chinese), Acta
Acustica 17, 56-64 (1992 ).
Contains comparisons of the reflections of transient pressures from finite
dimensioned facets and the theory in Ref. I7. The agreement is good.

19.
A. W. Trorey, "A simple theory for seismic
diffractions," Geophysics 35 762-784 (1970). Gives an impulsive evalution of the Helmholtz - The Kirchhoff
integral, also known as the Rabinowitz solution [M.
Born agreement is good. and E. Wolf, *Principles
of Optics *(Pergamon, Oxford, 1965), Sec. 8.9].

20.
M. Jebsen and H. Medwin,
"On the failure of the Kirchhoff assumptions
in backscatter," J. Acoust. Soc. Am. 72,
1607-1611 (1982).

From : The Journal of the
Acoustical Society of America 94(1993)October, No.4, Woodbury,