Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles
In the context of wave propagation through gratings of periodically distributed obstacles, we use analytical approaches to derive explicit formulas for the scattering parameters in the low-frequency approximation. The geometric shape of the obstacles is arbitrary. Numerical solution of the main in...
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irk-123456789-1769842021-02-10T01:26:30Z Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles Scarpetta, E. Sumbatyan, M.A. In the context of wave propagation through gratings of periodically distributed obstacles, we use analytical approaches to derive explicit formulas for the scattering parameters in the low-frequency approximation. The geometric shape of the obstacles is arbitrary. Numerical solution of the main integral equations for assigned shapes will provide some graphs, and comparison with previous results will show that some of these are incorrect or hold under very restrictive conditions. При розглядi поширення хвиль через ґратку перiодично розповсюджених перешкод за допомогою аналiтичного пiдходу отримано явнi формули для параметрiв розсiяння при низькочастотнiй апроксимацiї. Геометрична форма перешкод є довiльною. На основi чисельного розв’язку основних iнтегральних рiвнянь для вибраних форм побудовано графiки, а порiвняння з попереднiми результатами показує, що деякi з них не пiдтверджуються, а iншi мають мiсце за досить суворих обмежень. 2003 Article Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles / E. Scarpetta, M.A. Sumbatyan // Нелінійні коливання. — 2003. — Т. 6, № 4. — С. 511-524. — Бібліогр.: 15 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176984 517.9 en Нелінійні коливання Інститут математики НАН України |
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In the context of wave propagation through gratings of periodically distributed obstacles, we use analytical
approaches to derive explicit formulas for the scattering parameters in the low-frequency approximation.
The geometric shape of the obstacles is arbitrary. Numerical solution of the main integral equations for
assigned shapes will provide some graphs, and comparison with previous results will show that some of
these are incorrect or hold under very restrictive conditions. |
| format |
Article |
| author |
Scarpetta, E. Sumbatyan, M.A. |
| spellingShingle |
Scarpetta, E. Sumbatyan, M.A. Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles Нелінійні коливання |
| author_facet |
Scarpetta, E. Sumbatyan, M.A. |
| author_sort |
Scarpetta, E. |
| title |
Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles |
| title_short |
Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles |
| title_full |
Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles |
| title_fullStr |
Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles |
| title_full_unstemmed |
Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles |
| title_sort |
low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles |
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Інститут математики НАН України |
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2003 |
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http://dspace.nbuv.gov.ua/handle/123456789/176984 |
| citation_txt |
Low-frequency propagation of plane scalar waves through a periodic array of arbitrary volumetric obstacles / E. Scarpetta, M.A. Sumbatyan // Нелінійні коливання. — 2003. — Т. 6, № 4. — С. 511-524. — Бібліогр.: 15 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT scarpettae lowfrequencypropagationofplanescalarwavesthroughaperiodicarrayofarbitraryvolumetricobstacles AT sumbatyanma lowfrequencypropagationofplanescalarwavesthroughaperiodicarrayofarbitraryvolumetricobstacles |
| first_indexed |
2025-07-15T14:56:27Z |
| last_indexed |
2025-07-15T14:56:27Z |
| _version_ |
1837725267232030720 |
| fulltext |
UDC 517.9
LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES
THROUGH A PERIODIC ARRAY OF ARBITRARY VOLUMETRIC
OBSTACLES
ПОШИРЕННЯ НИЗЬКОЧАСТОТНИХ СКАЛЯРНИХ ХВИЛЬ
НА ПЛОЩИНI ЧЕРЕЗ ПЕРIОДИЧНИЙ МАСИВ ДОВIЛЬНИХ
ПЕРЕШКОД
E. Scarpetta
Univ. Salerno
84084 Fisciano (SA), Italy
M. A. Sumbatyan
Research Inst. Mech. and Appl. Math.
Stachki Prospect 200/1, Rostov-on-Don 344090, Russia
In the context of wave propagation through gratings of periodically distributed obstacles, we use analytical
approaches to derive explicit formulas for the scattering parameters in the low-frequency approximation.
The geometric shape of the obstacles is arbitrary. Numerical solution of the main integral equations for
assigned shapes will provide some graphs, and comparison with previous results will show that some of
these are incorrect or hold under very restrictive conditions.
При розглядi поширення хвиль через ґратку перiодично розповсюджених перешкод за допомо-
гою аналiтичного пiдходу отримано явнi формули для параметрiв розсiяння при низькочас-
тотнiй апроксимацiї. Геометрична форма перешкод є довiльною. На основi чисельного розв’яз-
ку основних iнтегральних рiвнянь для вибраних форм побудовано графiки, а порiвняння з попе-
реднiми результатами показує, що деякi з них не пiдтверджуються, а iншi мають мiсце за до-
сить суворих обмежень.
1. Introduction. The importance of scattering problems through gratings of obstacles variously
distributed inside a medium is well known in many practical applications regarding mechani-
cal, acoustic or electromagnetic sciences. The books of Krautkramers [1] and Jones [2] can be
usefully referred to for a survey of the researches devoted to such topics. These investigati-
ons can be performed following either purely numerical or analytical points of view, the latter
ones being more involved when regular (periodic) distributions of obstacles are concerned;
see [3] where a number of references for both these approaches can be found. In every case,
it is assumed very often that the frequency of the incident wave (perhaps together with the
characteristic size of the obstacles) is small, giving typically rise to a weak interaction regime in
which approximate results can be established.
Of course, analytical methods are valuable since they provide explicit formulas for the
relevant scattering parameters, from which a correct limit for low frequencies can be easily
extracted. Actually, up to today, this has been made only for thin (slit-type) obstacles; see [2,
4 – 8] for an outline of such analytical procedures.
Some authors also claimed explicit results for low frequencies in the case of rectangular
obstacles periodically distributed [9, 10]. Of course, this case (geometrically equivalent to a
c© E. Scarpetta, M. A. Sumbatyan, 2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4 511
512 E. SCARPETTA, M. A. SUMBATYAN
screen of finite thickness with regular gaps) is more simple with respect to that of arbitrarily
shaped obstacles, because it allows an (analytical) approach in terms of Fourier series [3, 11].
Nevertheless, we will show below that such results (echoed in [11]) are incorrect.
For an arbitrary (volumetric) shape, some interesting analytical results were obtained by
Lamb [4], Twersky [12] and Miles [13]. These authors operate starting from an a priori known
solution of the problem regarding an ideal incompressible fluid flow around a single obstacle of
the given shape, solution which is clearly independent of frequency. Unfortunately, this method
works well under the assumption of weak reflection between the wave and the obstacle, thus
requiring not only a low frequency, but also the smallness of the obstacle (actually, we will show
below in concrete examples that, even in the low-frequency range, the Lamb – Twersky – Miles
approximation holds good enough for only very small obstacles).
Besides what already has been mentioned about previous results, the aim of the present
paper is to construct analytically a correct low-frequency limit from explicit formulas in scatte-
ring (scalar) problems involving periodic gratings of arbitrary volumetric obstacles. Particular
attention will be devoted to re-considering the case of rectangular scatterers, which admits an
alternative procedure and thus permits an interesting comparison between the two methods of
solution (along with reciprocal testing of efficiency).
Even if the framework can be directly applied to similar problems in acoustics or electro-
magnetism, we prefer to address the problem in elastic context, in which the scalar wave field
and obstacles concerned are interpreted as anti-plane displacement and special defects in the
(solid) structure, respectively.
2. Mathematical formulation. We consider the anti-plane normal penetration of a harmonic
plane wave of SH-type into an unbounded two-dimensional elastic medium in which an infinite
periodic array of equal defects with arbitrary shape is present (see Fig. l). Denoted by d the
period of this vertical grating, the natural symmetry of the problem entitles us to restrict the
attention to a single layer 0 < y2 < d containing an obstacle D; let l be the regular line
surrounding D (to which can be applied the plane Green theorem). In the assumed harmonic
regime, all field variables have the common factor e−iωt, that will be omitted in the sequel.
Thus, the incident (scalar) wave of unitary amplitude has the form eiky1 and gives rise to
a scattered (stationary) field ϕsc(y1, y2)satisfying the Helmholtz equation throughout the layer
except D:
∆yϕsc + k2ϕsc = 0, ∆y = ∂2/∂y2
1 + ∂2/ϕy2
2. (2.1)
Above, k is the wave number and ω the circular frequency; of course, ω/k gives the (transverse)
wave speed of the material in concern. Writing
ϕ(y1, y2) = ϕsc(y1, y2) + eiky1 (2.2)
for the total wave field, which also satisfies the Helmholtz equation, the periodicity requires
that
ϕ(y1, 0) = ϕ(y1, d),
∂ϕ
∂y2
(y1, 0) =
∂ϕ
∂y2
(y1, d). (2.3)
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LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES THROUGH A PERIODIC ARRAY . . . 513
Fig. 1. Normal penetration of a plane scalar wave into a periodic array of ar-
bitrary defects.
Moreover, in view of the low-frequency regime to be considered, we can assume that the total
field in the far left and right parts of the layer be expressed in the following (one-mode) form:
ϕ(y1, y2) ∼
{
eiky1 +Re−iky1 , as y1 → −∞;
Teiky1 , as y1 → +∞,
(2.4)
where R and T represent the reflection and transmission coefficients of the structure, respecti-
vely; cf. [6 – 8, 11, 13]. These are the scattering parameters for which (approximate) formulas
are searched showing explicitly the dependence on the frequency.
Since the contour of the defects cannot sustain (tangential) stress, we put, as basic boundary
condition of the problem, the vanishing of the normal derivative of the total field along l,
∂ϕ
∂n
∣∣∣∣∣
l
= 0. (2.5)
Now, we need to consider an appropriate Green’s function Φ = Φ(y;x) for the given layer
(henceforth, we will find convenient to put simply y or x for the pair (y1, y2) or (x1, x2)); it
should satisfy the nonhomogeneous Helmholtz equation
∆yΦ(y;x) + k2Φ(y;x) = −δ(y1 − x1)δ(y2 − x2), (2.6)
along with the periodicity conditions
Φ(y1, 0;x) = Φ(y1, d;x),
∂Φ
∂y2
(y1, 0;x) =
∂Φ
∂y2
(y1, d;x). (2.7)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
514 E. SCARPETTA, M. A. SUMBATYAN
To this aim, we resort to a work by Van den Berg and Voorman [14], where in a similar context
(but addressed to purely numerical results) such a function is constructed as
Φ(y;x) =
ieik|y1−x1|
2kd
+
+∞∑
n=1
e−βn|y1−x1| cos[(2πn/d)(y2 − x2)]
βnd
,
(2.8)
βn =
√
(2πn/d)2 − k2.
Of course, the aimed low-frequency approximation entitles us to consider real and positive
numbers βn , so that the following asymptotic expression of Φ holds at large distances from the
defects’ array:
Φ(y;x) ∼ ieik|y1−x1|
2kd
(as x1 or y1 → ±∞). (2.9)
Considering the boundary condition (2.5), classical results of potential theory [2, 15] can be
used to provide the scattered field in the layer (except D) with the following integral representati-
on over the contour l in terms of the normal derivative of the Green’s function:
ϕsc(x) = ϕ(x)− eikx1 =
∫
l
[
ϕ(y)
∂Φ
∂ny
(y;x)
]
dly, x is outside of D (2.10)
(ny is the external unit normal to D). Aiming to let x1 → ±∞ in Eq. (2.10), we can calculate
∂Φ
∂ ny
from Eq. (2.9) as follows:
∂Φ
∂ny
≈ ∂Φ
∂y1
n1(y) ≈ eik|y1−x1|
2d
n1(y) sign (x1 − y1) (as x1 → ±∞). (2.11)
Therefore, by Eq. (2.10) when x1 → −∞ or +∞ and Eq. (2.4)1 or (2.4)2, we get, respectively,
ϕsc(x) ≈ −
∫
l
ϕ(y)
eik(y1−x1)
2d
n1(y)dly ≈ Re−ikx1 , as x1 → −∞, (2.12a)
ϕsc(x) ≈
∫
l
ϕ(y)
eik(x1−y1)
2d
n1(y)dly ≈ (T − 1)eikx1 , as x1 → +∞, (2.12b)
from which the following expressions for the reflection and transmission coefficients are easily
derived:
R = − 1
2d
∫
l
ϕ(y)eiky1n1(y)dly, T = 1 +
1
2d
∫
l
ϕ(y)e−iky1n1(y)dly. (2.13)
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LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES THROUGH A PERIODIC ARRAY . . . 515
Note that the dependence on k is also involved (implicitly) in ϕ(y), whose restriction to l
actually remains the only unknown of the problem. Thus, in order to find a (boundary) integral
equation to be solved for ϕ, we let x → Y ∈ l in Eq. (2.10) and invoke well-known properties
of the ”double layer” integral in the potential theory, to obtain
1
2
ϕ(Y )−
∫
l
[
ϕ(y)
∂Φ
∂ny
(y;Y )
]
dly = eikY1 , Y = (Y1, Y2) ∈ l, (2.14)
which gives the searched equation (cf. [14]).
3. Explicit results for the scattering coefficients in the low-frequency limit. We now aim to
derive the (analytical) limit of Eqs. (2.13), (2.14) as ω ∼ k → 0. Our strategy is that to neglect
terms of order O(k2) wherever they appear. We firstly need to evaluate the limiting form of the
Green’s function from Eq. (2.8); putting there βn ≈ 2πn/d, we get
Φ(y;x) ' ieik|y1−x1|
2kd
+
1
2π
∞∑
n=1
e−(2πn/d)|y1−x1| cos[(2πn/d)(y2 − x2)]
n
=
=
ieik|y1−x1|
2kd
+
|y1 − x1|
2d
−
− 1
4π
ln
{
2
[
cosh
2π(y1 − x1)
d
− cos
2π(y2 − x2)
d
]}
, as k → 0, (3.1)
where the summation formula
∞∑
n=1
e−nξ cos(nζ)
n
=
ξ
2
− 1
2
ln[2(cosh ξ − cos ζ)] (3.2)
has been used. As a consequence, the limit of
∂Φ
∂ny
=
∂Φ
∂y1
n1(y) +
∂Φ
∂y2
n2(y) in Eq. (2.14) can
be obtained as follows:
∂Φ
∂ny
(y;Y ) ' 1
2d
(
1− eik|y1−Y1|
)
sign (y1 − Y1)n1(y)−
− 1
2d
sinh [(2π/d)(y1 − Y1)]n1(y) + sin[(2π/d)(y2 − Y2)]n2(y)
cosh[(2π/d)(y1 − Y1)]− cos[(2π/d)(y2 − Y2)]
, y, Y ∈ l, (3.3)
from which, with error O(k2), we finally get
∂Φ
∂ny
(y;Y ) = − ik
2d
(y1 − Y1)n1(y)−K0(y, Y ), as k → 0, (3.4a)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
516 E. SCARPETTA, M. A. SUMBATYAN
where the kernel
K0(y, Y ) =
1
2d
sinh [(2π/d)(y1 − Y1)]n1(y) + sin[(2π/d)(y2 − Y2)]n2(y)
cosh[(2π/d)(y1 − Y1)]− cos[(2π/d)(y2 − Y2)]
, y, Y ∈ l, (3.4b)
is real and does not contain k.
Thus, the main integral equation (2.14) at k → 0 can be written (with error O(k2)) as:
1
2
ϕ(Y ) +
∫
l
K0(y, Y )ϕ(y)dly = 1 + ikY1 −
ik
2d
(g1 − g0Y1), Y ∈ l, (3.5a)
where
g0 ≡
∫
l
ϕ(y)n1(y)dly, g1 ≡
∫
l
y1ϕ(y)n1(y)dly. (3.5b)
Now, writing formally the left-hand term above as
(
1
2
I +K0
)
ϕ(Y ), it is clear by linearity that
if one can solve the two integral equations(
1
2
I +K0
)
h0(Y ) = 1,
(
1
2
I +K0
)
h1(Y ) = Y1, Y ∈ l, (3.6)
both of them real and not containing k, then a solution of Eq. (3.5) can be constructed as
ϕ(y) =
(
1− ik
2d
g1
)
h0(y) +
ik
2d
(2d+ g0)h1(y), y ∈ l. (3.7)
To find constants g0 and g1, let us integrate twice Eq. (3.7) with respect to l after multiplying by
n1(y) and y1n1(y). We get, respectively,
g0 =
(
1− ik
2d
g1
)
H00 +
ik
2d
(g0 + 2d)H10, (3.8a)
g1 =
(
1− ik
2d
g1
)
H01 +
ik
2d
(g0 + 2d)H11, (3.8b)
where
H00 =
∫
l
h0(y)n1(y)dly, H10 =
∫
l
h1(y)n1(y)dly, (3.9a)
H01 =
∫
l
y1h0(y)n1(y)dly, H11 =
∫
l
y1h1(y)n1(y)dly (3.9b)
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LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES THROUGH A PERIODIC ARRAY . . . 517
are real constants not containing k, that can be calculated after solving the auxiliary integral
equations (3.6). Solution of the linear system (3.8) gives, letting k → 0 and neglecting terms
O(k2) in numerator and denominator,
g0 =
H00 + ikH10
1 + (ik/2d)(H01 −H10)
, (3.10a)
g1 =
H01 + ik[H11 + (1/2d)(H00H11 −H10H01)]
1 + (ik/2d)(H01 −H10)
. (3.10b)
On the other side, letting k → 0 and using Eqs. (3.5b), Eqs. (2.13) become (with error
O(k2))
R = − 1
2d
(g0 + ikg1), T = 1 +
1
2d
(g0 − ikg1). (3.11)
By substituting g0 and g1 from Eqs. (3.10), we easily get
R = −H00 + ik(H10 +H01)
2d+ ik(H01 −H10)
, T =
H00 + 2d
2d+ ik(H01 −H10)
(3.12)
(neglecting terms O(k2) in the numerator).
Of course, from a physical standpoint, the reflection coefficient should vanish when k does,
and this implies that the constantH00 must be null identically1. Thus, the above explicit formulas
finally become in the present low-frequency approximation :
R = − ik(H10 +H01)
2d+ ik(H01 −H10)
, (3.13a)
T =
2d
2d+ ik(H01 −H10)
. (3.13b)
We remark that Eqs. (3.13) hold without any assumption of smallness of the defects; cf. [13].
4. Case of rectangular obstacles. The above results hold for an arbitrary geometric shape
of the defects (this shape being involved only in Eqs. (3.6) and (3.9)), and of course can be as
well applied to rectangular defects. Such a particular case has a special importance in scattering
problems since it formally amounts to consider a screen of finite thickness inside the medium
having regular gaps periodically distributed. Apart from approaches of numerical or engine-
ering type in acoustic context [9, 10], this case has been treated analytically in [11] by means of
suitable Fourier expansions of the wave field in the three main regions of the layer (vertically
centered around the opening between adjacent obstacles, rather than around the obstacle itself;
see Fig. 2). By imposing the continuity of the wave field and of the stress’ zx-component through
neighbouring regions, we arrived at the following formulas for the reflection and transmission
1This will be confirmed by numerical calculations (Section 5).
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
518 E. SCARPETTA, M. A. SUMBATYAN
Fig. 2. Normal penetration of a plane scalar wave into a periodic array of rect-
angular defects a× (d− 2b).
coefficients (defined as in Eqs. (2.4) in the context of the same far-field propagation of one-
mode type):
R = 1− G+ +G−
2ikd
, t =
G+ −G−
2ikd
, (4.1a)
G± =
−2S±
1−
[
cos(ka)∓ 1
2bk sin(ka)
+
1
ikd
]
S±
, S± =
b∫
−b
s±(y2)dy2, (4.1b)
where (−b, b) is the opening between adjacent rectangles, of which a is the length, and s±(y2)
are two (real) functions solving the integral equations
b∫
−b
s±(t)
{ ∞∑
n=1
1√
(πn)2 − (kd/2)2
cos
[
2πn
d
(y2 − t)
]
+
+
ch
√
(πna/b)2 − (ka)2 ∓ 1√
(πn)2 − (kb)2 sh
√
(πna/b)2 − (ka)2
cos
[πn
b
(y2 − t)
]}
dt = 1, |y2| < b.
(4.2)
Now, we aim to derive the low-frequency limit for these formulas. Constants S± in Eqs. (4.1)
depend on k via the kernels of Eqs. (4.2), so that to extract correct behaviour ofG±, R, T , when
k → 0, we need first to study this limit for Eqs. (4.2). Neglecting there terms O(k2) under
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LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES THROUGH A PERIODIC ARRAY . . . 519
square roots, and using the summation formula
∞∑
n=1
cos
2πnξ/d
n
= − ln
∣∣∣∣2 sin
πξ
d
∣∣∣∣ , (4.3)
we easily get the integral equations, free of frequency parameter, to be satisfied by the functions
s± in the low-frequency approximation,
1
π
b∫
−b
s±(t)
{ ∞∑
n=1
ch(πna/b)∓ 1
n sh(πna/b)
cos
[πn
b
(y2 − t)
]
− ln
∣∣∣∣2 sin
π(y2 − t)
d
∣∣∣∣
}
dt = 1, |y2| < b.
(4.4)
Let us go back to Eqs. (4.1b) for constants G±, in which the (real) values for S± are now
taken from the solutions of Eqs. (4.4), so that the dependence on k actually appears only in the
square bracket. To get the limit as k → 0, we must consider G+ and G− separately; in G− we
have
1−
[
cos(ka) + 1
2bk sin(ka)
+
1
ikd
]
S− = 1− cot (ka/2)
2bk
S− − 1
ikd
S− ' −
(
cot (ka/2)
2bk
+
1
ikd
)
S−,
which gives
G− ' 2S−
i(d/2)cot (ka/2) + b
ibkd
S−
=
2bkd
(d/2)cot (ka/2)− ib
, as k → 0. (4.5a)
On the contrary, in G+ we have
1−
[
cos(ka)− 1
2bk sin(ka)
+
1
ikd
]
S+ = 1 +
tan (ka/2)
2bk
S+ − 1
ikd
S+,
and therefore such a constant remains unchanged as
G+ = − 2bkd
(d/2)tan (ka/2) + ib+ bkd/S+
(4.5b)
also in the limit k → 0.
Using these values for G+, G−, formulas (4.1a) have the following limits as k → 0 (with the
same approximation as in Sect. 3):
R = −
ik
[
(d2/4)− b2 + (bd2/aS+)
]
bd/a− ik
[
(d2/4) + b2 + (bd2/aS+)
] , (4.6a)
T =
bd/a
bd/a− ik
[
(d2/4) + b2 + (bd2/aS+)
] (4.6b)
(cf. [10, 11], Sect. 4).
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520 E. SCARPETTA, M. A. SUMBATYAN
Fig. 3. Reflection coefficient |R| vs. frequency parameter kd/2 for circular
defects of diameter a = 0, 5d in the case of small frequencies: 1 —
analytical solution (3.13a), 2 — exact solution, 3 — Miles’ solution.
5. Numerical investigation and comparison with previous results. First of all, we should
remark that both the pairs of formulas (3.13) and (4.6) identically satisfy the well-known balance
of energies [15]
|R|2 + |T |2 = 1
with error O(k2). So, for what follows, it is sufficient to consider one only of the scattering
coefficients, for example, the reflection’s one.
By applying a direct numerical method to the integral equations (3.6) for some particular
shapes and disposition of the defects, we obtained values for all constantsH defined in Eqs. (3.9).
As the physics requires, H00 turns out to be always vanishing. Behavior of the function |R|
versus (small values of) the frequency parameter, from Eq. (3.13a), is reflected as line 1 in all
Figures 3 – 6. Validity of Eqs. (2.4) and (2.9) is guaranteed until kd/2 < π; cf. [13]. We can
observe a good agreement (better for smaller obstacles) with line 2 reflecting the exact soluti-
on, as derived numerically from Eqs. (2.14), (2.13)1; in any case, the initial slope is the same.
Miles’ solution [13] gives a (perfectly) linear behavior, reflected as line 3 in Figures 3 – 6, which
appears far from lines 1, 2 (indeed, farther for larger obstacles) and with different initial slope.
We note that Miles in [13] also constructed more approximate (not linear) formulas, but he used
a rather artificial procedure from both the mathematical and physical standpoints (see Sect. 3
in that paper).
Figures 7, 8 are concerned with defects of rectangular shape. In them, we compare the
analytical approach developed in this paper with the alternative one based on Fourier expansi-
on and continuity conditions [11], and expressed by Eqs. (4.6). The range of frequency has been
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LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES THROUGH A PERIODIC ARRAY . . . 521
Fig. 4. Reflection coefficient |R| vs. frequency parameter kd/2 for circular
defects of diameter a = 0, 9d in the case of small frequencies: 1 —
analytical solution (3.13a), 2 — exact solution, 3— Miles’ solution.
Fig. 5. Reflection coefficient |R| vs. frequency parameter kd/2 for elliptical
defects of semi-axes α1/α2 = 2 (α1 = 0, 5d) rotated at 45o, in the ca-
se of small frequencies: 1 — analytical solution (3.13a), 2 — exact solu-
tion, 3 — Miles’ solution.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
522 E. SCARPETTA, M. A. SUMBATYAN
Fig. 6. Reflection coefficient |R| vs. frequency parameter kd/2 for ellip-
tical defects of semi-axes α1/α2 = 2 (α1 = 0, 9d) rotated at 45o,
in the case of small frequencies: 1 — analytical solution (3.13a),
2 — exact solution, 3 — Miles’ solution.
Fig. 7. Reflection coefficient |R| vs. frequency parameter kd/2 for quad-
ratic defects with a = 0, 5d in the case of small frequencies: 1 —
exact solution, 2 — analytical solution (3.13a), 3 — analytical solu-
tion (4.6a), 4 — engineering solution [10].
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LOW-FREQUENCY PROPAGATION OF PLANE SCALAR WAVES THROUGH A PERIODIC ARRAY . . . 523
Fig. 8. Reflection coefficient |R| vs. frequency parameter kd/2 for rectan-
gular defects with a = 0,5 d, b = 0,1 d in the case of small frequencies:
1 — exact solution, 2 — analytical solution (3.13a), 3 — analytical
solution (4.6a), 4 — engineering solution [10].
enlarged to improve comparison. Equations (4.2) hold until kd/2 < π. Line 1 reflects the exact
solution, numerically obtained from Eqs. (2.14), (2.13)1 and practically coincident, in the gi-
ven range, with the exact and one-mode solutions developed in [11] (see lines 1, 2 in Fig. 2 of
that paper). Lines 2 and 3 reflect the low-frequency solutions analytically given by Eq. (3.13a)
and (4.6a), respectively; the values for S+ are derived by numerical solution of Eq. (4.4). We
can observe that the approach of [11], built up for only rectangular defects, is more precise
throughout the considered range. Line 4, finally, reflects the explicit solution derived in [10] by
means of an engineering approach confined to low-frequency; we could claim as incorrect this
solution, since the initial slope of the line is quite different from that of other lines (cf. also line
3 in Fig. 2 of [11]).
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Received 15.01.2003
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