Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://num-meth.srcc.msu.ru/english/zhurnal/tom_2006/ps/v7r103.ps Äàòà èçìåíåíèÿ: Fri Jan 13 13:24:54 2006 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:59:54 2012 Êîäèðîâêà:

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 23
UDC 538.975;621.382.13:535
LOW­FREQUENCY APPROXIMATION AND THE CHOICE OF OPTIMAL ELASTIC
PARAMETERS FOR TWO­LAYER BLAST PROTECTION JACKETS
A. V. Tikhonravov 1 , M. K. Trubetskov 1 , N. A. Winfree 2 , and J. H. Kang 2
Low­frequency approximation of elastic wave transmittance for two­layer structures is proposed.
An accuracy of the approximation in a range up to several kilohertz is studied. Based on this
approximation, the problem of the optimal elastic parameter choice for two­layer blast protection
jackets is considered.
1. Introduction. A blast wave produced by an explosion can result in lung injury characterized by diffuse
contusions and haemorrhage of the alveoli [8]. The injury is caused by the wave's impingement upon the chest
wall, rather than its propagation through the upper respiratory system [2]. The severity of the injury typically
increases with the pressure amplitude of the blast wave and with the loading rate [4, 10]. Cumulative injury
can be caused by repeated exposure to weaker blast events [7, 9].
Two of the authors of this paper (Winfree and Kang) have been involved in developing a jacket to protect
against lung injury caused by blast waves impinging upon the chest. Our approach is that a jacket of one or more
layers acts as a frequency filter, altering the wave transmitted into the chest by attenuating some frequencies of
the incident wave and amplifying others [11]. This approach was also used by Cooper and colleagues [5]. Modeling
all materials as linearly elastic, the transmission behavior through the jacket can be predicted analytically.
Using this method, we consistently predicted that any two­layer design we considered would amplify across
some range of low frequencies. This amplification is also apparent in Cooper's results. We began to suspect that
it was impossible to find any two­layer combination that would not have this presumably undesirable response.
Other work we were involved in also required designing multi­layered structures as frequency filters for
elastic or acoustic waves. Optimizing a layered media to achieve a particular transmission behavior is a difficult
problem, but it has been much studied for thin­film coatings in optics. In the simplest cases, the propagation
of elastic waves through layered media and the propagation of light through thin films are both described
by the one­dimensional wave equation. Thus, the work of Winfree and Kang in elastic layered media led to
a collaboration with the other authors of this paper (Tikhonravov and Trubetskov), experts in optimization
methods for thin­film coatings. The goal of the collaboration was to develop software for the optimization of
elastic layered media to achieve specified transmission or reflection behaviors in acoustics and elastic wave
propagation.
The analysis presented here was conducted in the course of our collaboration. It supports previous findings
that practical two­layer jackets (where practical means that the jacket must utilize reasonable materials and
cannot be too heavy or thick) do not attenuate all low frequencies and indeed amplify some frequencies. This
result holds if the following assumptions are met:
--- all materials are linearly elastic and isotropic;
--- both the input media and structure to be protected (in our case, respectively the air and the chest) can
be modeled as half­spaces;
--- all layers remain in contact, with no gaps opening between them;
--- each layer of the protective structure is thin compared to the wavelength of a low­frequency harmonic
wave in the material (we use f = 2000 Hz as an estimate for this frequency); this is typically the case because
of bulk and weight constraints that limit the thickness of the layers;
--- the characteristic material impedance of the structure to be protected (in our case, the chest) is greater
than the impedance of the media from which the blast wave is incident; thus, the analysis can be applied to
situations such as a blast wave impinging on a building or to a ship's hull subject to an underwater explosion;
on the other hand, the analysis does not hold for divers subject to underwater explosions or noises, because the
impedance of the person is similar to that of the water;
1 Research Computing Center of Moscow State University, 119992, Moscow, Russian Federation; e­mail:
tikh@srcc.msu.ru, trub@srcc.msu.ru
2 Dominca, Inc., 12111 Ranchitos Rd, NE, Albuquerque, NM, USA; e­mail: nancy@dominca.com,
joe@dominca.com
c
fl Research Computing Center of Moscow State University

24 Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su)
--- the characteristic material impedance of the layer closest to the structure is less than that of the structure,
and the impedance of the outer layer is much greater than that of the incident media;
--- the impedance of the layer farthest from the structure is less than the impedance of the structure (this
would be a ``soft'' jacket).
While the last assumption seems very restrictive, in truth the analysis also holds for many cases when the
impedance of the outermost layer is greater than that of the structure (for example, if the outer layer is metal
and the structure is the chest); we just have not categorized the combinations of soft inner with stiff outer layers
for which the analysis holds versus when it does not. In practice, we have never been able to devise any two­layer
jacket subject to our weight and bulk constraints for which amplification is not predicted at low frequencies.
The analysis also demonstrates how a two­layer jacket can be optimized to achieve the greatest attenuation
over a range of higher frequencies. As a particular example, we optimize a design --- subject to constraints of a
fixed total thickness and areal density --- to achieve the maximum attenuation over 1 to 3 kHz. This frequency
interval was chosen because Cooper and colleagues suggested that these are the components of an incident blast
wave that are most responsible for lung injury [4, 5]. In brief, optimization requires that the outer layer be as
heavy as allowed, while the inner layer should be a material with a low modulus and should be as thick as can
be tolerated.
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
xxxxxxxxxxxxx
...
......
xxxxxxxxxx xxxxxxx xxxxxxxxxx
PSfrag replacements
Incident wave, u i
Transmitted wave, u t
Reflected wave, u r
Input media
Output media
j = N + 1
j = N
j = 2
j = 1
j = 0
hN
h 2
h 1
xN
xN-1
x 2
x 1
0 x
x x x x x x x x x x x x x x x x x x
Layers
Layer indices
1
Fig. 1. Schematic of layered structure consisting of N layers. Properties of the input layer will be denoted by
the index i (for ``input''), but properties of other layers will be denoted by their index j. In this paper, we
consider only a two­layer structure, so that N = 2
2. Basic equations for elastic waves in layered structures. Both the input and output media will
be modeled as semi­infinite isotropic media (see Fig. 1). Waves propagate from the input to the output media
across one or more layers considered to be isotropic and infinite in two dimensions. The waves are assumed to
be longitudinal waves propagating normal to the interfaces.
The only component of displacement is in the x­direction; displacement is a function of the x­coordinate
but not of the y­ or z­coordinates. Therefore, denote displacement by u = b u(x; t). The deformation gradient fl
and the velocity V are
fl = @u=@x ; (1)
V = @u=@t : (2)
Assuming that body forces can be ignored, conservation of linear momentum reduces to
@oe
@x = ae @V
@t ; (3)
where ae is density and oe is the normal stress in the x­direction.
For linearly elastic isotropic solids, oe and fl are related by
oe = F fl ; (4)
where F is the longitudinal modulus of the material related to the more familiar moduli of isotropic materials
by F = –+2¯ = E (1 \Gamma š)
(1 + š)(1 \Gamma 2š) : Here – and ¯ are the Lam'e moduli, E is Young's modulus, and š is Poisson's
ratio. The Lam'e modulus ¯ is also known as the shear modulus.
Equations (1), (2), and (4) can be used to express the conservation of linear momentum in terms of u,
resulting in the one­dimensional wave equation for displacement: @ 2 u
@t 2 = c 2 @ 2 u
@x 2 , where
c =
p
F=ae (5)

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 25
is the longitudinal wave velocity in the material.
Impedance z is defined by z = oe=V . In the case of a plane wave, the impedance can be shown to be
z = \Sigmaae c, where the choice of sign depends upon the direction of propagation of the wave. The quantity ae c is a
material property called the characteristic impedance and will be denoted by an upper case Z:
Z = ae c : (6)
A common approach in studying the reflective and transmissive properties of layered structures is to
perform analysis in the frequency domain [3]. This is done by considering the incident wave as a superposition
of harmonic plane waves. Because all of the basic equations are linear, all waves in a layered structure can be
considered as harmonic waves with the same frequency as that of the incident wave. This approach is equivalent
to the application of the Fourier transform analysis to all time­dependent quantities of interest.
With the frequency­domain approach, it is important to note that computational formulas depend on the
choice of sign in a time factor of the Fourier transform and on the choice of the direction of the normal to a
layered structure (the choice of the direction of the x­axis in Figure 1). A detailed discussion of this subject can
be found on the first pages of the book [6].
Using this approach, the incident particle velocity and stress are superpositions of harmonic waves V i and oe i ,
each of angular frequency ! and with the form
V i (!; x; t) = V I exp
\Gamma
ik(x \Gamma xN ) + i!t
\Delta
; oe i (!; x; t) = oe I exp
\Gamma
ik(x \Gamma xN ) + i!t
\Delta
;
where V I and oe I are amplitudes of the waves, k is a wavenumber (its relation to other parameters is discussed
later), and i = p
\Gamma1 when it appears by itself (as opposed to its use as a subscript to indicate a property of the
incident media). The dependence on x in the form ik(x \Gamma xN ) is taken here for convenience, so that V I and oe I
are amplitudes at the boundary between the input media and the layered structure (see Figure 1). A similar
expression can be written for the incident displacement u i , but we will not need it.
Similarly, the transmitted waves are superpositions of harmonic waves of the forms
V t (!; x; t) = VT exp (ikx + i!t) ; (7)
oe t (!; x; t) = oe T exp (ikx + i!t) : (8)
Here VT and oe T are amplitudes of the transmitted waves at the boundary of the output medium and the layered
structure.
Finally, the reflected waves are superimposed of harmonic waves that propagate in the opposite direction
to the incident and transmitted waves:
V r (!; x; t) = VR exp
\Gamma
ik(x \Gamma xN ) \Gamma i!t
\Delta
; oe r (!; x; t) = oe R exp
\Gamma
ik(x \Gamma xN ) \Gamma i!t
\Delta
:
Again, the dependence on x is specified in a form that corresponds to the amplitudes of reflected waves taken
at the boundary between the input media and the layered structure.
Our goal is to find the stress transmission coefficient T and stress reflection coefficient R of the layered
structure, where we use the definitions
T = oe T =oe I ; R = oe R =oe I : (9)
In the following, we use two different recurrent techniques for calculating the transmission and reflection
coefficients T and R of a layered structure, the so­called matrix technique and the admittance technique. To
show consistency between these techniques, the following subsections provide a concise description of major
steps of the derivations for the reflection and transmission coefficients.
2.1. Solution by the matrix technique. We restate Eq. (3) for convenience:
@oe
@x
= ae
@V
@t
; (10)
This is a first order partial differential equation for stress oe and velocity V . Another first­order equation is
obtained by differentiating Eq. (4) with respect to time and making use of Eqs. (1) and (2) to eliminate fl,
resulting in
@oe
@t
= F @V
@x
: (11)

26 Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su)
In Eqs. (10) and (11), V and oe are functions of two variables, namely x and t. We transit to the frequency
domain by representing them as a superposition of the terms V (x) exp(i!t) and oe(x) exp(i!t) with various
angular frequencies !. In fact, each pair of functions V (x) and oe(x) depends on the parameter !, but, as it is
usually done, we omit the direct indication of the dependence of V and oe on ! for conciseness.
Equations (10) and (11) give the system of ordinary differential equations for all pairs of these functions:
doe
dx
= i!aeV ; dV
dx
= i!
aec 2 oe : (12)
Here the wave speed c is given by Eq. (5).
Solutions of Eqs. (12) inside the jth layer can be written in the form
oe (x) = oe (x j \Gamma1 ) cos !
c j
(x \Gamma x j \Gamma1 ) + i ae j c j V (x j \Gamma1 ) sin !
c j
(x \Gamma x j \Gamma1 ) ;
V (x) = i
ae j c j
oe(x j \Gamma1 ) sin !
c j
(x \Gamma x j \Gamma1 ) + V (x j \Gamma1 ) cos !
c j
(x \Gamma x j \Gamma1 ) ;
(13)
where ae j and c j are the density and wave velocity of the material of layer j and x j \Gamma1 is the coordinate of the
leftmost boundary of this layer in Figure 1.
Now put x = x j in Eqs. (13), so that we have x j \Gamma x j \Gamma1 = h j on the right­hand side of these equations (h j
is the thickness of the jth layer). In this case, Eqs. (13) describe how the pair (oe, V ) is transformed from the
left boundary to the right boundary of the jth layer. We introduce the value ' j = !
c j
h j , which will be called
the phase thickness of the jth layer. We can write the above­mentioned transformation in the matrix form
`
oe
V
'
x=x j
=
0
@
cos ' j i Z j sin ' j
i
Z j
sin ' j cos ' j
1
A
`
oe
V
'
x=x j\Gamma1
; (14)
where Z j is the characteristic impedance of the jth layer as defined by Eq. (6).
By analogy with optics, the matrix on the right­hand side of Eq. (14) can be called the characteristic matrix
of the jth layer [6].
Provided the layers remain in contact, the functions oe(x) and V (x) are continuous across the layer bound­
aries: oe(x +
j ) = oe(x \Gamma
j ) and V (x +
j ) = V (x \Gamma
j ). Thus, the characteristic matrices enable the recursive calculation
of these functions, working from the leftmost boundary between the output medium and the first layer to the
rightmost boundary between the last layer and the input medium in Figure 1. As a result, we have
`
oe
V
'
x=xN
= M
`
oe
V
'
x=0
; (15)
where M is the product of characteristic matrices of all layers:
M =
1
Y
j=N
0
@
cos ' j i Z j sin ' j
i
Z j
sin ' j cos ' j
1
A : (16)
At the leftmost and rightmost boundaries of the layered structure we have
oe
fi fi
x=0 = oe T ; V
fi fi
x=0 = VT ; oe
fi fi
x=xN = oe I + oe R ; V
fi fi
x=xN = V I + VR ; (17)
where the indices I, T , and R are used to denote quantities related to the incident, transmitted, and reflected
waves and taken at the respective boundaries of the layered structure.
Now we need a relation between stress and velocity in the output media. Substituting Eqs. (7) and (8) into
Eqs. (12) gives koe T = !ae 0 VT ; kVT = !
ae 0 c 2
0
oe T , where ae 0 and c 0 are the density and longitudinal wave velocity
of the output medium. From these we have that k = !=c 0 in the output medium and
oe T = Z 0 VT ; (18)
where Z 0 = ae 0 c 0 is the characteristic impedance of the output medium.

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 27
Using the subscript ``i'' to denote elastic parameters of the input medium, we similarly obtain that k = !=c i
in the input medium and
oe I = Z i V I ; oe R = \GammaZ i VR ; (19)
where Z i = ae i c i is the characteristic impedance of the input medium.
Equations (17), (18), and (19) together with the matrix equation (15) allow us to express the transmission
and reflection coefficients (9) through the elements of the matrix M . We will denote these elements as m 11 ,
m 12 , m 21 , and m 22 . Expressions for T and R are as follows:
T = 2
m 11 + m 12 =Z 0 + m 21 Z i +m 22 Z i =Z 0
; (20)
R = m 11 + m 12 =Z 0 \Gamma m 21 Z i \Gamma m 22 Z i =Z 0
m 11 + m 12 =Z 0 + m 21 Z i +m 22 Z i =Z 0
: (21)
Expressions (20) and (21) together with Eq. (16) are basic formulas of the matrix technique. In the case
when there are no layers between the input and output media, we have m 11 = m 22 = 1, m 12 = m 21 = 0, and
Eqs. (20) and (21) are reduced to the expressions presenting the transmission and reflection coefficients of the
boundary between input and output media:
T = 2Z 0
Z 0 + Z i
; R = Z 0 \Gamma Z i
Z 0 + Z i
: (22)
2.2. Solution by the admittance technique. Now consider the admittance technique. It will be seen
that it is similar to the impedance recurrent technique presented in Brekhovskikh's book [3]. We introduce
admittance A as the ratio of particle velocity to stress: A = 1=z = V =oe.
The admittance depends on the x­coordinate and it is a continuous function of x inside the layered structure,
including the boundaries between the layers. This follows from the continuity of V and oe. Taking Eqs. (13) at
x = x j and dividing the second equation by the first one, we obtain
A(x j ) = A(x j \Gamma1 ) + i=Z j tan ' j
1 + iZ j A(x j \Gamma1 ) tan ' j
: (23)
This formula gives a recurrent algorithm for calculating the admittance from the leftmost boundary of the
layered structure to its rightmost boundary. According to Eqs. (17) and (18), at the left boundary
A(0) = 1=Z 0 : (24)
This equation gives the initial condition for the recurrent formula (23).
Using Eqs. (9), (17), and (19), it is easy to obtain that at the right boundary of the layered structure
A(xN ) = 1
Z i
1 \Gamma R
1 + R
: (25)
Solving Eq. (25) with respect to R, we come to the equation expressing the amplitude reflection coefficient R
through the admittance A(xN ), which is often called the input admittance:
R = 1 \Gamma Z i A(xN )
1 + Z i A(xN ) : (26)
Equations (23), (24), and (26) are basic formulas of the admittance technique providing a recurrent algo­
rithm for calculating R. The amplitude transmission coefficient T can also be calculated using the admittance
technique. Here we provide only the final formula for T without details of its derivation:
T = 2=Z i
1=Z 0 + 1=Z 1
N
Y
j=1
A(x j ) + 1=Z j
A(x j ) + 1=Z j+1
exp(\Gammai' j ) :
3. Low­frequency approximation for the intensity transmission coefficient of a two­layer struc­
ture. In this section, instead of the angular frequency ! we will use the frequency f = !
2ú
, which is a more
common parameter in many practical applications.

28 Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su)
The transmission coefficient T (f) of a two­layer structure is calculated by the formula (20) through the
elements of the matrix product (16), where N = 2 and
' 1;2 = 2úfh 1;2
c 1;2
= 2úh 1;2
– 1;2
: (27)
Recall that h 1;2 , c 1;2 , and ae 1;2 are thicknesses, longitudinal velocities, and material densities of the first and
second layers of the structure. We have introduced – = c=f , which is the wavelength of a wave of frequency f
in a material with wave speed c.
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
c
y
,
k
H z
N
o
r
m
a
l
i
z
e
d
t
r
a
n
s
m
i
s
s
i
o
n
l
o
g
,
B
T ransmittanceacrossasoft2­layerjacket,Design
A
Layer1=2.2mmFirmFoam( f
1
=0.059)
Layer2=5.8mmFabric( f
2
=0.353)
0.03dB
0.32dB
Attenuation
Amplification
f
f
n
max
Fig. 2. Exact normalized transmittance predicted for the initial design of a soft two­layer jacket (this is
called ``Design A'' in the figure title). The layer materials and their thicknesses are provided in the figure.
The input media is always air and the output media is always water (meant to represent the chest). The
vertical scale is log 10 Ü (f). The labels at f = 1 and 3 kHz indicate Ü (f) at these frequencies: they are a
convenience in evaluating the performance of the design over the 1 to 3 kHz target range. Note that this
design amplified the transmission of the incident wave over this band and in fact provides no attenuation
at any frequency covered in this plot. More of the energy in the incident blast wave over these frequencies
is passed into the chest than if no jacket were worn!
Let us introduce the normalized intensity transmission coefficient by the equation Ü (f) =
fi fi fi fi
T (f)
T (0)
fi fi fi fi
2
. For
conciseness, this may simply be called the normalized transmittance.
At f = 0, the matrix elements m 11 and m 22 equal 1, while the matrix elements m 12 and m 21 are zeroes.
Thus, it follows from Eq. (20) that T (0) equals the transmission coefficient across the uncovered interface of
the input and output media (see Eqs. (22)). Thus, the normalized transmittance is the ratio of the intensity
transmission coefficients when the layers are used versus when they are not : this is a measure of how much the
layers increase the transmission (Ü ? 1) or decrease the transmission (Ü ! 1) compared to having no layers
between the incident and output media.
Figure 2 presents a plot of normalized transmittance calculated from Eq. (20) for a soft two­layer blast
protection covering at the boundary between air and the chest wall (modeled as a half­space with the wave
speed and density of water). The layer closest to the chest is a ``firm'' foam, and the second layer is fabric. The
material properties used are presented in the table. The thicknesses of the layers were chosen to give a total
thickness of 8 mm and a total areal density of 0.49 g/cm 2 , values similar to those for a soft ballistic jacket. The
materials and layer thicknesses for this and all examples to follow are provided in each figure. Also provided
in each figure are the phase thicknesses ' 1 and ' 2 for a harmonic wave of frequency f = 2000 Hz. Because
the jacket layers are thin compared to a wavelength – = c=f at this frequency, ' 1 and ' 2 are small values for
all of our examples and we can replace trigonometric functions of them with approximations valid for small
quantities. These approximations will be crucial in the analysis.

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 29
Material parameters used for calculations, arranged in order of decreasing
modulus F . Also shown are wavelengths – = c=f at f = 2 kHz. Values for
copper, water, and air are from Appendix B of [1]. The fabric is a ballistic
fabric: its density and wave speed across several fabric layers were measured
at Dominca. Wave speed was measured by finding the transit time across
32 layers of the fabric, total thickness approximately 6 mm, using ultrasonic
transducers with center frequencies of 500 kHz. The foams are two different
crosslinked, closed­cell polyolefine foams: their wave speeds and densities
were measured at Dominca
ae c Z = aec F = aec 2 –, mm
Material g/cm 3 km/s MPa/(m/s) GPa at f = 2 kHz
Copper 8.93 4.99 44.6 222 2495
Water 1.10 1.4 1.54 2.156 700
Fabric 0.80 0.208 0.167 0.0346 104
Firm Foam 0.0961 0.4590 0.0441 0.0202 230
Soft Foam 0.0320 0.2910 0.0093 0.0027 146
Air 0.0013 0.329 0.000428 0.00014 165
The frequency dependence presented in Figure 2 has some special features. At low frequencies, the normal­
ized transmittance Ü increases with frequency. It reaches its first maximum Ü max at f = fn and decreases at
f ? fn , the subscript n being chosen to indicate that this is a natural or resonance frequency. There are other
maxima at frequencies higher than those plotted in Figure 2.
Assume now that we wish to maximize the attenuation over 1 to 3 kHz. The design of Figure 2 amplifies
these components of the transmitted signal. If we are to block rather than enhance transmission over these
frequencies, we need to move f n to a frequency below 1 kHz. To accomplish this, we derive an approximate
expression for Ü (f) that will be easier to understand.
As indicated in Figure 2, the phase thicknesses ' 1;2 are small at f = 2 kHz (the center of our frequency
band of interest). Thus, we shall consider an approximate expression for the normalized transmittance obtained
by using the first terms of trigonometric function expansions in the layer characteristic matrices M j . Let us
write the characteristic matrices of the first and second layer in the forms
M 1 ú
/
1 \Gamma ' 2
1 =2 iZ 1 (' 1 \Gamma ' 3
1 =6)
(i=Z 1 ) (' 1 \Gamma ' 3
1 =6) 1 \Gamma ' 2
1 =2
!
; M 2 ú
/
1 \Gamma ' 2
2 =2 iZ 2 (' 2 \Gamma ' 3
2 =6)
(i=Z 2 ) (' 2 \Gamma ' 3
2 =6) 1 \Gamma ' 2
2 =2
!
:
The product of these matrices gives the matrix M whose elements are used in the expression (20) for T .
Recalling that the phases ' 1 and ' 2 are proportional to the frequency f (see Eq. (27)), it can be seen that the
matrix elements m 11 and m 22 are real and are expressed as series of terms with even powers of frequency f ,
while the matrix elements m 12 and m 21 are imaginary and are represented as series with odd powers of f . To
calculate Ü (f), one should take a squared modulus of T from which we conclude that the denominator of Ü (f) is
a series of terms with even powers of f . From the definition of the normalized transmittance, we have Ü(0) = 1.
On the whole, this means that at low frequencies the normalized transmittance Ü (f) can be represented in the
form
Ü (f) ú 1
1 + fff 2 + fif 4 : (28)
We shall call this form a low­frequency approximation of the normalized transmittance.
The coefficients ff and fi in Eq. (28) have been expressed as functions of the properties of the layers in the
appendix and some constraints that ensure that ff ! 0 are also discussed there. In the following, we will only
need to work with Eq. (28) in its current form in order to make some practically useful conclusions.
If the coefficient ff in Eq. (28) is negative, then Ü (f) is a monotonically increasing function of f at low
frequencies: the jacket amplifies the transmission of low frequencies. This is just as observed in Figure 2.

30 Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su)
According to Eq. (28), the maximum value of Ü (f) is attained at
fn ú
r
\Gamma ff
2fi (29)
and this value is
Ü max ú 1
1 \Gamma ff 2 =(4fi)
: (30)
Equations (29) and (30) allow the parameters ff and fi to be expressed by the values f n and Ü max . We have
that ff = \Gamma 2
f 2
n
Ü max \Gamma 1
Ü max
, fi = 1
f 4
n
Ü max \Gamma 1
Ü max
.
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
n
c
y
,
k
H
z
Layer1=2.2mmFirmFoam( f
1
=0.059)
Layer2=5.8mmFabric( f
2
=0.353)
f
E
x
a
c
t
.
=
5
6
6
0
H
z
,
=
2
.
0
4
d
B
1
+
(
-
1
)
(
1
-
(
)
)
2
T ransmittanceacrossa2­layerjacket,Design
A
2
0.03dB
0.32dB
6
maxmax
max
max
f
f
n
/
f
n
l o
g
,
d
B
Normalizedtransmittance
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
l
o
g
,
d
B
Layer1=2.2mmFirmFoam( f
1
=0.059)
Layer2=5.8mmFabric( f
2
=0.353)
E
x
a
c
t
.
=
5
6
6
0
H
z
,
=
2
.
0
4
d
B
T
ransmittanc eacros
s a2­layer
jack et,Design
A
2
6
max
isexact.
F
r
e
q
u
e
n
c
y
f
-
f
f
n
f
n
(
)
4
f
n
,
Normalizedtransmittance
,kHz
Fig 3. The approximation of Eq. (31) compared with
the exact plot of Ü (f). In the approximation, we take
Ü max and fn from the exact solution
Fig 4. The approximation Ü (f) ú
i fn
f
j 4
from
Eq. (32) compared with the exact plot of Ü (f). In
the approximation, we take fn from the exact solu­
tion
Using these equations, we can rewrite Eq. (28) in the form
Ü (f) ú Ü max
1 + (Ü max \Gamma 1)
\Theta
1 \Gamma (f=fn ) 2
\Lambda 2 : (31)
Figure 3 compares the exact normalized transmittance from Figure 2 with its low­frequency approximation
calculated in accordance with Eq. (31). The agreement is good for f ! f n but less appropriate for higher
frequencies. If we cared to pursue a better approximation, we would include at least the f 6 terms in the
denominator of Eq. (28).
In this and all of our examples, Ü max AE 1. Thus, we will further approximate that ff ú \Gamma2=f 2
n and fi ú 1=f 4
n .
For f ? fn , the last term in the denominator of Eq. (28) plays a dominant role, and we can further approximate
Ü (f) ú
` f n
f
' 4
for f ? fn : (32)
This approximation is presented by the dashed curve in Figure 4. Though this approximation is truly a poor
description of Ü for this jacket concept, it will be useful nonetheless in modifying the design and it will become
an increasingly better match to Ü for f ? f n as we do so.
It is worthwhile to note that the low­frequency approximation considered in this section is valid not only
for two­layer structures but also for arbitrary multilayer structures, providing that the sum of phase thicknesses

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 31
of all layers is small. This follows from the fact that considerations of this section are independent of the number
of structure layers.
4. Considerations on the optimal choice of layer parameters. In this section we will optimize a
two­layer jacket design subject to the constraints that the total thickness and areal density must remain fixed
at 8 mm and 0.49 g/cm 2 , respectively. The materials we will consider for the jacket are described in the table.
It will be useful to define the following ratios between the impedance of the output media and the
impedances of each of the other materials:
n 1 = Z 0
Z 1
; n 2 = Z 0
Z 2
; n i = Z 0
Z i
: (33)
The methods in this section are based on the admittance recurrent technique described in Section 2.2. We
will suppose that the layer phase thicknesses ' 1 and ' 2 are small enough, so that we can use the first order
approximations for tan ' 1;2 , i.e., we can put tan ' 1 = ' 1 and tan ' 2 = ' 2 in Eq. (23). For the first layer,
Eq. (23) together with the definition of n 1 from Eq. (33) gives
A(x 1 ) = (1=Z 0 ) + i' 1 =Z 1
1 + i' 1 =n 1
: (34)
Minimizing transmittance requires that the output medium and the first layer have a high contrast in
impedances. According to this supposition, n 1 AE 1 and we will ignore the term ' 1 =n 1 in the denominator of
Eq. (34), leaving
Z 0 A(x 1 ) = 1 + in 1 ' 1 : (35)
For the second layer, Eq. (23) gives A(x 2 ) = A(x 1 ) + i' 2 =Z 2
1 + iA(x 1 ) Z 2 ' 2
: Using Eq. (35) along with the definition
of n 2 from Eq. (33), we can rewrite this as follows:
Z 0 A(x 2 ) = 1 + in 1 ' 1 + in 2 ' 2
1 + i (1 + in 1 ' 1 ) ' 2 =n 2
: (36)
Equation (26) allows the amplitude reflection coefficient R to be expressed through the admittance A(x 2 ).
Using the definition of n i from Eq. (33), we can write the expression for R in the following form:
R = n i \Gamma Z 0 A(x 2 )
n i + Z 0 A(x 2 ) : (37)
If we assume that there are no energy losses in the layered structure, then (Z i =Z 0 )jT j 2 = 1 \Gamma jRj 2 and we
have from Eq. (37) that
jT j 2 = 4n 2
i Re
\Phi
Z 0 A(x 2 )
\Psi
h
n i + Re
\Phi
Z 0 A(x 2 )
\Psi i 2
+
h
Im
\Phi
Z 0 A(x 2 )
\Psi i 2 : (38)
It is seen now that to find the normalized transmittance Ü (f), one should separate the real and imaginary
parts of the admittance A(x 2 ). It follows from Eq. (36) that
Re
\Phi
Z 0 A(x 2 )
\Psi
= 1 + ' 2
2
\Gamma
1 \Gamma (n 1 =n 2 )' 1 ' 2
\Delta 2
+ (' 2 =n 2 ) 2 ; (39)
Im
\Phi
Z 0 A(x 2 )
\Psi
= (n 1 ' 1 + n 2 ' 2 )
\Gamma
1 \Gamma (n 1 =n 2 )' 1 ' 2
\Delta
\Gamma ' 2 =n 2
\Gamma
1 \Gamma (n 1 =n 2 )' 1 ' 2
\Delta 2 + (' 2 =n 2 ) 2
: (40)
We will assume that the impedance of the output medium is significantly greater than the impedance of the
input medium, i.e., that n i AE 1 in Eq. (38). For f = 0 we have from Eq. (36) that Z 0 A(x 2 ) = 1 and Eq. (38)
gives
fi fi T (0)
fi fi 2
= 4n 2
i =(n i + 1) 2 ú 4.
Obviously, jT j 2 will be much greater than this value when Re
\Phi Z 0 A(x 2 )
\Psi is much greater than 1 and
Im
\Phi Z 0 A(x 2 )
\Psi is still a relatively small value. At low frequencies, Re
\Phi Z 0 A(x 2 )
\Psi will be great when the first

32 Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su)
term in the denominator of Eq. (39) equals to zero. Thus, it is natural to suppose that Ü (f) =
fi fi T (f)
fi fi 2
=
fi fi T (0)
fi fi 2
is close to its maximum value when
(n 1 =n 2 ) ' 1 ' 2 = 1 : (41)
Substituting here ' 1 , ' 2 from Eq. (27) and n 1 , n 2 from Eq. (33), we find that Eq. (41) is satisfied when
f = f n = c 1
2ú
r ae 1
ae 2 h 1 h 2
= 1
2ú
p
F 1 =h 1
p
ae 2 h 2
= 1
2ú
r
k 1
m 2
; (42)
where m 2 = ae 2 h 2 is the mass per unit area of the outermost layer and k 1 =
p
F 1 =h 1 is the ``spring constant''
per unit area of the innermost layer. Thus, our assumptions have led us to a lumped­parameter model: Eq. (42)
is the equation for the natural frequency f n of a point mass m 2 connected to a fixed wall by a massless spring
with spring constant k 1 . Though this approximation will not be particularly good for our first jacket design,
we can still use it to understand how to move f n to lower frequencies and decrease the transmission over some
range of frequencies greater than f n .
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
n
c
y
,
k
H
z
Layer 1=2.2mmFirmFoam( f
1
=0.059)
Layer 2=5.8mmFabric
( f
2
=0.353)
f
E
x
a
c
t
.
=
5
6
6
0
H
z
,
=
2
.
0
4
d
B
-
T ransmittanceacrossa2­layerjacket,Design
A
2
0.03dB
0.32dB
6
maxmax
f
f
n
f
n
(
)
4
f
n
,
-
1
2
m
-
2
k
1
-
=
H
z
7
1
0 7
l
o
g
,
d
B
Normalizedtransmittance
(
)
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
q
u
e
n
c
y
,
k
H
z
Layer 1=2mmSoftFoam( f
1
=0.086)
Layer 2=6mmFabric
( f
2
=0.363)
f
E
x
a
c
t
.
=
H
z
,
2
5
6
8
=
2
.
d
B
5
2
-
T ransmittanceacrossa2­layerjacket,DesignB
2
0.15dB
6
maxmax
f
f
n
f
n
(
)
4
f
n
,
-
1
2
m
-
2
k
1
-
=
H
z
2
6
7 6
l
o
g
,
d
B
Normalizedtransmittance
(
)
f
f
Fig 5. The approximation Ü (f) ú
i f n
f
j 4
from
Eq. (32) compared with the exact plot of Ü (f). In the
approximation, we take f n = 1
2ú
r
k 1
m 2
from Eq. (42)
Fig 6. As in Figure 5, but the design has been revised
so that a softer foam is used behind the fabric
In Figure 5, we approximate fn by this equation and plot the approximation Ü (f) = (f n =f) 4 for the design
considered in Figures 2 to 4. We find that Eq. (42) predicts that f n = 7107 Hz, while the maximum value
of Ü (f) is really attained at 5650 Hz (see Figure 2). The discrepancy indicates that describing the outer shell
as a point mass and the inner shell as a massless spring is not adequate for this design. Of course, the curve
Ü (f) = (f n =f) 4 remains a poor description of Ü .
Though not accurate for this design, Eq. (42) still provides useful guidance in how to modify the design to
achieve lower transmittance across our 1 to 3 kHz target. The first step is to force the resonance frequency f n
below our target range. By inspection of Eq. (42), we see that this can be done by increasing the total mass in
the outer shell, by selecting a lower­modulus material for the inner shell, or by increasing the thickness of the
inner layer.
In Figure 6, a softer, less dense foam has been used for the inner layer. To maintain the same areal density
and thickness as in the first design, the outer layer has become slightly thicker and the inner layer slightly
thinner. The result is that f n has indeed decreased to 2568 Hz. If predicted by Eq. (42), f n = 2676 Hz, a
significantly improved match compared to that for the prediction for the previous design. Overall, however, the
approximation Ü (f) = (f n =f) 4 for f n is still not very good. Worse, the peak transmittance now falls within the
range over which we aim to decrease the transmittance!

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 33
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
n
c
y
,
k
H
z
Layer1=1.9mmair( f
1
=0.073)
Layer2=6.1mmFabric( f
2
=0.368)
f
E
x
a
c
t
.
=
H
z
,
6
1
5
=
2
.
d
B
9
2
-
T ransmittanceacrossa2­layerjacket,DesignC
2
6
maxmax
f
f
n
f
n
(
)
4
f
n
,
-
1
2
m
-
2
k
1
-
=
H
z
6
1 6
l
o
g
,
d
B
Normalizedtransmittance
(
)
f
f
-
-
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
n
c
y
,
k
H
z
Layer1=7.5mmFirmFoam( f
1
=0.206)
Layer2=0.5mmcopper( f
2
=0.001)
f
E
x
a
c
t
.
=
H
z
,
3
9
2
0
=
2
.
d
B
2
3
-
T ransmittanceacrossa2­layerjacket,DesignD
2
6
maxmax
f
f
n
f
n
(
)
4
f
n
,
-
1
2
m
-
2
k
1
-
=
H
z
4
0
3 8
l
o
g
,
d
B
Normalizedtransmittance
(
)
f
f
0.06dB
0.77dB
Fig 7. As in Figure 5, but air is used instead of foam
behind the fabric
Fig 8. In this and the remaining figures, the outer
layer of the jacket is copper instead of fabric. Here it
is backed by the firm foam of Figure 5. Compared to
that figure, we have been able to move f n to a lower
frequency, but there is still no attenuation over our 1
to 3 kHz target band
Continuing, we replace the inner layer by air. Again, the outer layer becomes a little thicker, and the inner
layer a little thinner if we hold the total mass and thickness constant. The predicted transmissivity is plotted
in Figure 7. Note that the approximation Ü (f) = (f n =f) 4 has improved for f ? fn and is a reasonable match
over much of our 1 to 3 kHz range of interest. Our lumped­mass approximation for the natural frequency has
also improved: Eq. (42) predicts that f n = 613 Hz, a very good match to the exact value of f n = 612 Hz. By
moving the natural frequency to a lower value, we have finally achieved attenuation over our 1 to 3 kHz target
range.
A problem with all of these designs is that the inner layer is so thin that, in reality, we might expect it
to ``bottom out,'' violating our assumption of linear elasticity. With the mass and thickness constraints we are
enforcing, nearly the entire jacket must be fabric, leaving little thickness available for the inner layer. In addition,
we may be able to better block the transmission across our target range if we can increase the thickness of the
inner layer, effectively reducing the spring constant k 1 .
To increase the thickness of the inner layer, let us consider using a more dense material for the outer layer.
We will choose copper and pair it in turn with the firm foam, the soft foam, and the air that we have just
considered, still fixing the areal density and total thickness. This lets us replace the six or so millimeters of
fabric by only 0.5 mm of copper. The predicted exact transmittance is shown for each combination in Figures 8
to 10 along with the approximation of Eq. (32) for Ü (f), using f n computed from Eq. 42. The approximation
of Ü (f) for frequencies just above the first resonance remains poor when the firm foam is used, but is better
when the soft foam or air is used, situations for which the lumped­parameter model becomes more appropriate.
Again, the usefulness of the approximation is in systematically guiding the design modifications in order to
achieve additional attenuation over the target range of frequencies.
Of the three designs with copper, only the copper backed by air provides significant attenuation over
the entire target range. Indeed, copper paired with the firm foam amplifies the 1 -- 3 kHz components of the
transmitted signal. When the copper is backed by the soft foam, there is amplification up to around 2 kHz and
then some attenuation for frequencies greater than this.
5. Discussion. In this paper we examined the transmission behavior of a two­layer jacket on the chest
using a well­established method of describing the transmission of waves through layered media. The method is
often applied in acoustics and is closely related to methods used in optics and even in transmission line theory.
We provided rigorous and self­consistent derivations of two recurrent techniques (the matrix technique and

34 Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su)
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
n
c
y
,
k
H
z
Layer1=7.5mmSoftFoam( f
1
=0.323)
Layer2=0.5mmcopper( f
2
=0.001)
f
E
x
a
c
t
.
=
H
z
,
1
3
9
2
=
2
.
d
B
8
6
-
T ransmittanceacrossa2­layerjacket,DesignE
2
6
maxmax
f
f
n
f
n
(
)
4
f
n
,
-
1
2
m
-
2
k
1
-
=
H
z
1
4
0 4
l
o
g
,
d
B
Normalizedtransmittance
(
)
f
f
0.63dB
1.09dB
0
1
2
3
4
5
7
8
­5
­4
­3
­2
­1
0
1
3
F
r
e
q
u
e
n
c
y
,
k
H
z
Layer1=7.5mmair( f
1
=0.284)
Layer2=0.5mmcopper( f
2
=0.001)
f
E
x
a
c
t
.
=
H
z
,
3
1
2
=
2
.
d
B
5
9
-
T ransmittanceacrossa2­layerjacket,DesignF
2
6
maxmax
f
f
n
f
n
(
)
4
f
n
,
-
1
2
m
-
2
k
1
-
=
H
z
3
1 2
l
o
g
,
d
B
Normalizedtransmittance
(
)
f
f
-
1.93
dB
-
3.9dB
Fig 9. As in the Figure 8, but a softer foam is used
here
Fig 10. As in Figure 8, but air is used instead of foam
behind the copper
the admittance technique) for calculating reflection and transmission properties of elastic structures.
We find that it is difficult to design a jacket of moderate weight and bulk that effectively blocks the
transmission of incident blast energy over frequencies of 1 to 3 kHz. In fact, it can be shown that, in the absence
of dissipation, a soft two­layer jacket (where soft means that the impedance of the outer layer is less than that
of the chest and the impedance of the inner layer is intermediate to that of the chest and the air) is guaranteed
to amplify some range of low frequencies and that attenuation is possible only for frequencies above this range.
The same holds true for some combinations of ``hard'' jackets (meaning the impedance of the outer layer is
greater than that of the chest), but it is harder to specify what these combinations are.
If we consider the peak amplification to occur at a natural frequency f n , the task then is to push f n
to lower frequencies. We derived an approximation for the transmission behavior of the jacket at frequencies
f ? fn and also an approximation for f n that turns out to be identical to the natural frequency predicted by
a lumped­parameter model. Neither approximation is particularly good for most of our jacket designs, but the
insight we gain from them allows us to systematically vary the design in order to push f n to a lower frequency.
We find that, given the constraints we apply and the materials we consider in this paper, the only designs that
block the transmission of energy over our entire target frequency range are those with an inner layer of air.
Our interest was in blocking the transmission of energy for frequencies of 1 to 3 kHz. We have not considered
the ramifications of amplifying the transmission of the low frequencies, but they could be quite serious. Note,
however, that because we are using a continuum model, the predicted transmission remains finite in the absence
of dissipation. In contrast, a lumped­parameter model would predict an infinite response at resonance unless
dissipation were added.
Unfortunately, we find that --- in the absence of anisotropy or any non­linear behaviors --- amplification
of low frequencies is unavoidable for any ``soft'' two­layer jacket and for many two­layer jackets with a hard
outer shell (and very possibly for all such jackets). Our models can readily include dissipation by using complex
moduli to describe dissipative materials. Frequency­dependent material behavior can be modeled by allowing
the moduli to be functions of frequency. Anisotropic behaviors can also be handled: in fact, these behaviors are
important in non­destructive testing and ultrasonic imaging.
Though we systematically optimized the design of a two­layer jacket, optimization becomes more difficult as
more layers are added or if nonlinear behaviors are modeled. For these cases, we find that optimization methods
implemented computationally become valuable.
We caution that, in reality, the chest probably should not be modeled as a half­space. Better approximations
would consider the chest wall to be a layer with the appropriate thickness and might also include the lungs and
even the opposite wall of the torso.
6. Acknowledgement. This work was supported by the U.S. Civilian Research & Development Foundation

Numerical Methods and Programming, 2006, Vol. 7 (http://num­meth.srcc.msu.su) 35
under award RE1--5007--MO--03.
Appendix. The coefficients ff and fi in Eq. (28) can be derived. They are
ff = \Gamma4ú 2
i
c 1 c 2 Z 1 Z 2 (Z 0 + Z i )
\Delta 2
\Gamma
c 2
1 h 2
2 Z 2
1 (Z 2
0 \Gamma Z 2
2 )(Z 2
2 \Gamma Z 2
i ) + 2c 1 c 2 h 1 h 2 Z 1 Z 2 (Z 2
0 \Gamma Z 2
1 )(Z 2
2 \Gamma Z 2
i )
+ c 2
2 h 2
1 Z 2
2 (Z 2
0 \Gamma Z 2
1 )(Z 2
1 \Gamma Z 2
i )
\Delta
(43)
and
fi = 4ú 4
3
\Gamma
c 2
1 c 2
2 Z 1 Z 2 (Z 0 + Z i )
\Delta 2
i
c 4
2 h 4
1 Z 2
2
\Gamma
\Gamma4Z 4
1 + 3Z 2
i Z 2
1 \Gamma 2Z 0 Z i Z 2
1 + Z 2
0 (3Z 2
1 \Gamma 4Z 2
i )
\Delta
+ 16c 1 c 3
2 h 3
1 h 2 Z 1 Z 2 (Z 2
0 \Gamma Z 2
1 )(Z 2
2 \Gamma Z 2
i ) + 12c 2
1 c 2
2 h 2
1 h 2
2 (Z 2
0 \Gamma Z 2
1 )(Z 2
1 + Z 2
2 )(Z 2
2 \Gamma Z 2
i )
\Gamma 16c 3
1 c 2 h 1 h 3
2 Z 1 Z 2 (Z 2
1 \Gamma Z 2
0 )(Z 2
2 \Gamma Z 2
i ) + c 4
1 h 4
2 Z 2
1
\Gamma
\Gamma4Z 4
2 + 3Z 2
i Z 2
2 \Gamma 2Z 0 Z i Z 2
2 + Z 2
0 (3Z 2
2 \Gamma 4Z 2
i )
\Delta j
:
(44)
For the case of a blast wave in the air impinging on the chest covered by a two­layer jacket, it is appropriate
to assume that
Z 0 AE Z i ; Z 1 ? Z i ; Z 2 AE Z i : (45)
In other words, the characteristic impedances of the output media (the chest) and the outer layer j = 2 are
significantly greater than the characteristic impedance of the input media (air), while the inner layer j = 1
must be made of something with impedance no less than that of the air (that is because we really do not have
a practical material with a lower impedance than air.)
In the body of the paper, we assume that ff ! 0 and fi ? 0. By inspection of Eq. (43) with the assumptions
of Eqs. (45), we see that ff ! 0 is guaranteed if Z 2 ! Z 0 . In other words, when the garment is ``soft.''
While the condition Z 2 ! Z 0 is sufficient to ensure that ff ! 0, it is not necessary, and we find that ff ! 0
for many choices of Z 2 ? Z 0 .
References
1. Bedford A. and Drumheller D.S. Introduction to elastic wave propagation. N.Y.: Wiley, 1994.
2. Benzinger T. Physiological effects of blast in air and water. Vol. II of German Aviation Medicine: World War II.
Chapter XIV­B. Washington: U.S. Government Printing Office, 1950. 1225--1259.
3. Brekhovskikh L.M. Waves in layered media. N.Y.: Academic Press, 1980.
4. Cooper G.J. Protection of the lung from blast overpressure by thoracic stress wave decouplers // J. of Trauma, Injury,
Infection, and Critical Care. 1996. 40, N 3. 105--110.
5. Cooper G.J., Townend D.J., Cater S.R., and Pearce B.P. The role of stress waves in thoracic visceral injury from
blast loading: modification of stress transmission by foams and high­density materials // J. of Biomechanics. 1991.
24, N 5. 273--285.
6. Furman S. and Tikhonravov A.V. Basics of optics of multilayer systems. Gif­sur­Yvette (France): Editions Frontieres,
1992 (see also http://www.optilayer.com).
7. Johnson D.L., Yelverton J.T., Hicks W., and Merickel B. Blast overpressure studies with animals and man: Nonau­
ditory damage risk assessment for simulated 155 mm self­propelled Howitzer blast. Technical Report ADA329424.
EG and G Management Systems, Inc. Albuquerque (NM, USA), 1997.
8. Phillips Y.Y. Primary blast injuries // Annals of Emergency Medicine. 1986. 15, N 12. 1446--1450.
9. Richmond D.R., Yelverton J.T., Fletcher E.R., Phillips Y.Y., Jaeger J.J., and Young A.J. Damage­risk criteria for
personnel exposed to repeated blasts // Proc. of the 20th Department of Defense Explosives Safety Seminar. Norfolk
(VA, USA), 1982. 1489--1512.
10.Stuhmiller J.H., Ho K.H.­H., Vander Vorst M.J., Dodd K.T., Fitzpatrick T., and Mayorga M.A. A model of blast
overpressure injury to the lung // J. of Biomechanics. 1996. 29, N 2. 227--234.
11.Winfree N.A. and Kang J.H. The effects of layered garments on the profile of a blast wave entering the thorax //
Proc. of the 72nd Shock and Vibration Symposium. San Destin (FL, USA), 2001.
Received 10 December 2005