You will be presented with the full text of a science research paper. Please write an abstract for this paper. Your response should include the abstract and no additional text.

Paper text:
consider a layered 2d acoustic waveguide , which is homogenous in the @xmath0-direction and has a sandwich structure in the @xmath1-direction .
the media constituting the waveguide can be elastic , liquid or gaseous .
waves in such a waveguide are described by a dispersion diagram in the @xmath2 coordinates ( @xmath3 is the temporal circular frequency , @xmath4 is the wavenumber in the @xmath0-direction ) .
usually the dispersion diagram is a graph consisting of several branches ( curves ) .
each point of the dispersion diagram is characterized by two important parameters , namely by phase velocity @xmath5 and group velocity @xmath6 ( the derivative @xmath7 is the slope of corresponding branch of the dispersion diagram ) .
it is well known that wave pulses propagate in the waveguide with corresponding group velocities .
it is well known also that in the experiment typically one can observe the _
first arriving signal _
( fas ) propagating with the velocity of the fastest medium in the structure of the waveguide . in the case of a single elastic isotropic medium
the velocity of the fas is close to the velocity of the longitudinal waves . in some cases
the velocity of the fas is bigger than any of the group velocities provided by the dispersion diagram .
the fas pulse decays with propagation distance unlike the pulses corresponding to usual modal pulses ( the latter are called _ guided waves _ in the medical - related literature ) . thus , fas is a transient process in a waveguide .
it can negligibly small very far from the source , however at moderate distances it can be an important phenomenon .
the most known applications of fas are related to medical acoustics ( see e.g. @xcite ) .
a long bone can be considered as a tubular waveguide constituted of three media : a thin outer layer of dense ( cortical ) bone , a sponge bone underneath , and a liquid marrow core in the center .
the fastest wave that can theoretically propagate in such media ( taken separately as infinite spaces ) is the longitudinal wave in the cortical bone .
the sponge bone and the marrow bear much slower waves .
however , when a standard analysis of a waveguide is performed ( say , by the finite element method ) the dispersion diagram contains no branch having group velocity close to the longitudinal cortical velocity .
there exist different approaches to describe fas .
the most comprehensive approach has been proposed by miklowitz and randles in @xcite .
this approach considers an analytical continuation of the dispersion diagram . after passing some branch points , it is possible to find a single branch of the dispersion diagram whose group velocity and decay correspond to fas .
another analytical approach is an application of the classical cagniard 
dehoop technique to invert the fourier transform in the time domain @xcite
. an approximate approach to fas is described in @xcite , where this type of waves is treated as a head wave .
also fas can be approximately described as a leaky mode . for this , the slow media composing the waveguide are declared as elastic half - spaces , while the fastest medium remains to be a layer .
such an approach enables one to compute ( at least in the simplest cases ) the velocity and the decay of fas with an acceptable accuracy .
the aim of the current paper is to present a model of fas based on an elementary analysis of real dispersion diagrams of a layered waveguide
. it is known @xcite that a wave process in a layered waveguide can be treated as an interaction between the modes of different types and velocities .
thus , the dispersion diagram has a `` terrace - like '' structure formed by overlapping of different sets of branches .
since typically no crossing of the branches can happen ( except the branches corresponding to non - interacting waves ) , there occur quasi - crossings at which the type of the mode is changing . a typical fragment of a dispersion diagram is shown in fig .
[ fig01 ] ( this is a dispersion diagram of a model two - media acoustic waveguide studied in the paper ) . the visible line of the smallest slope ( although this line can be composed of segments corresponding to different branches ) is a _ pseudo - branch _ ) relating to the fas .    in the current paper
this idea is developed into an analytical model .
an approximation of a fragment of the dispersion diagram by a sum of a tangent function and a linear function is constructed .
an analytical continuation of the approximation to the domain of positive @xmath8 $ ] is performed .
the tangent function tends to an imaginary constant there . using this trick ,
the estimation of the decay of fas is obtained .    the idea to study the analytical continuation of the dispersion diagram is inspired by the miklowitz 
randles method @xcite . in the paper
we briefly describe this method and its connection with the description of the fas as a leaky wave .
the structure of the paper is as follows . in section
ii a sample problem is formulated . in section  iii a numerical modeling of pulse propagation in the two - media problem
is performed .
the presence of fas and its exponential decay are established . in section
iv the approach by miklowitz 
randles is applied to the waveguide .
the branch of the dispersion diagram responsible for fas is found . in section
v the properties of fas are compared to those of a corresponding leaky wave . in section
vi the signal in the waveguide is represented in the form of an approximation of the phase based on tangent function .
the approximation is analyzed and the parameters of fas are estimated .
the results obtained by miklowitz  randles approach , leaky wave approach and the analysis of the real dispersion diagram are compared .
consider a waveguide in the @xmath9-plane occupying the strip @xmath10 ( see fig .
[ fig03 ] ) .
the layer @xmath11 is filled with a medium having density and speed of sound equal to @xmath12 , @xmath13 , respectively .
the layer @xmath14 is filled with a medium with parameters @xmath15 , @xmath16 .
the wave equations in the media are as follows : @xmath17 where @xmath18 , @xmath19 are the field variables ( say , acoustical potentials ) , notation @xmath20 stands for the second time derivative .
the boundary conditions are as follows .
the surface @xmath21 is rigid ( neumann ) : @xmath22 on the interface @xmath23 the pressure and the normal velocity are continuous : @xmath24 the surface @xmath25 is also rigid , but a point source is located at the point @xmath26 : @xmath27 where @xmath28 is the dirac delta - function , @xmath29 is the time shape of the probe pulse .
the observation point is located at @xmath30 , i.  e. the function @xmath31 is recorded .
the problem of finding the signal on the receiver is quite standard and it can be solved easily .
namely , perform the 2d fourier transform of @xmath32 in the domain of time and frequency : @xmath33 get a 1d problem for @xmath34 as functions of @xmath1 .
these functions should obey the equations @xmath35 the following boundary conditions should be valid : @xmath36 @xmath37 where @xmath38    the solution of ( [ eq0209 ] ) , ( [ eq0210 ] ) , ( [ eq0211 ] ) can be found : @xmath39 @xmath40 @xmath41 the field on the receiver can be obtained by inverting the fourier transformation : @xmath42    formula ( [ eq0215 ] ) can not be used directly , since zeros of the denominator belong to the plane of integration .
the limiting absorption principle is used to change the contour of integration .
namely , for @xmath43 we assume that the velocities @xmath44 have vanishing _ negative _ imaginary parts , while for @xmath45 the velocities @xmath44 have vanishing _ positive _ imaginary parts . due to this , for each @xmath3 the zeros of @xmath46 in the complex @xmath4-plane become displaced from the real axis .
the dispersion diagram represents the zeros of @xmath47 ( for real @xmath44 ) , i.  e.  the dispersion equation is @xmath48 the roots of ( [ eq0216 ] ) are curves in the @xmath49 plane , each point of which corresponds to a wave freely propagating in the waveguide and having @xmath0- and @xmath50-dependence of the form @xmath51 .
the following parameters have been selected for a numerical demonstration of fas presence : @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 .
dimensionless values are used in the computations , since no particular physical medium is under the investigation .
the values plotted in the graphs are , therefore , also dimensionless . the dispersion diagram for this waveguide
is shown in fig .
[ fig01 ] .
the fastest pseudo - branch is shown in the figure as a dashed line .
the pseudo - branch is composed of parts of real branches of the diagram .
one can see that for the selected parameters the pseudo - branch is quite loose , i.  e.  the gaps between its parts are quite wide .    for the demonstration we are using the probe pulse @xmath29 having the spectrum corresponding to the pseudo - branch .
namely , we are using the region @xmath57 . in this region
the group velocities of the guided waves are smaller than  @xmath58 . in the demonstration we are going to show the presence of a pulse whose velocity is approximately equal to the inverse of the slope of the dashed line . this velocity is close to to  4 .
thus , the velocity of the fas is considerably higher than that of any of the guided waves .
the shape of the pulse @xmath29 is shown in fig .
[ fig03 ] , left .
the spectrum of this pulse is shown in fig .
[ fig03 ] , right .
one can see that @xmath29 is a radio pulse centered around @xmath59 .
the central circular frequency is about @xmath60 .
the results of the computations made by formula ( [ eq0215 ] ) for @xmath61 are shown in fig .
[ fig04 ] , fig .
[ fig05 ] , fig .
[ fig06 ] , respectively .
the field at the receiver , i.  e. @xmath62 , is plotted .
one can see that in all graphs there exists a small pulse , which can be interpreted as fas .
parameters of fas approximately determined from these graphs are given in table  [ tab01 ] .
tof is the `` time of flight '' , i.  e.  the travel time of fas .
one can see that the velocity of the pulse is more than 4 , and the amplitude decay is close to exponential .
such a behavior is typical for fas .
the aim of the rest of the paper will be to estimate the group velocity of the fas pulse @xmath63 and the decay parameter @xmath64 .
the decay parameter is the coefficient in the exponential dependence of the amplitude vs @xmath65 : @xmath66    .parameters of the fas [ cols="^,^,^",options="header " , ]     the estimation of @xmath67 is , obviously , @xmath68 this parameter is an estimation of @xmath69 .
the estimation of @xmath70 is @xmath71 finally , the estimation of @xmath72 is as follows : @xmath73 since the values of @xmath72 given by ( [ eq1005 ] ) are different for different points @xmath74 , we come to conclusion that @xmath72 in ( [ eq1001 ] ) is a ( slow ) function of @xmath3 . by ( [ eq1005 ] )
we obtain an estimation of @xmath72 at the points @xmath75 , and after that we can , say , interpolate @xmath72 between these points . as we mentioned above
, function @xmath72 is an estimation of the decay parameter @xmath64 .    using the data extracted from fig .
[ figa06 ] and fig .
[ fig01 ] collected in table  [ tab02 ] , we can estimate the position of the pseudo - branch and the values of the attenuation parameters .
the points related to the peaks of the group velocity are plotted in fig  [ figa05 ] as small circles .
one can see that the points obtained from the the real dispersion diagram are in reasonably good agreement with the dispersion curve of the leaky wave for real @xmath3 .
it is shown that fas pulses correspond to terrace - like structures of the dispersion diagrams .
these structures are called pseudo - branches in the paper . a typical form of the pseudo - branch is shown in fig .  [ fig01 ] .
the pulse propagation velocity for fas is the inverse of the slope of the dashed curve in fig .
[ fig01 ] , and it can be larger than any of the group velocities available in the considered part of the dispersion diagram .    in the paper
we describe three different approaches to finding the velocity and the decay of fas pulses .
they are the miklowitz 
randles approach based on the analytical continuation of the dispersion diagram , the leaky wave approach and the approach based on the analysis of the pseudo - branch of the real dispersion diagram .
only the third approach is original . in the third approach
one should study the local peaks of the the group velocity , i.  e.  find the positions of the peaks and the values of the group velocity .
these parameters provide data describing the fas .
our computations prove consistency of all three approaches .
m.  muller , p.  moilanen , e.  bossy , p.  nicholson , v.  kilappa , j.  timonen , m.  talmant , s.  cheng , and p.  laugier ( 2005 ) `` comparison of three ultrasonic axial transmission methods for bone assessment , '' ultrs
biol . * 31 * , 633642 .
j.  grondin , q.  grimal , k.  engelke , and p.  laugier ( 2010 ) `` potential of first arriving signal to assess cortical bone geomtery at the hip with qus : a model based study , '' ultras .   med
.  biol . *
36 * , 656666 .
the first arriving signal ( fas ) in a layered waveguide is investigated . 
 it is well known that the velocity of such a signal is close to the velocity of the fastest medium in the waveguide , and it may be bigger than the fastest group velocity given by the dispersion diagram of the waveguide . 
 usually the fas pulse decays with the propagation distance . 
 a model layered waveguide is studied in the paper . 
 it is shown that the fas is associated with the pseudo - branch structure of the dispersion diagram . 
 the velocity of fas is determined by the slope of the pseudo - branch . 
 the decay is exponential and it depends on the structure of the pseudo - branch .