Interaction of elastic waves with ice layer in shelf zone

. A theoretical model for the propagation of elastic waves of arbitrary wave sizes from 0.5 to 20 units in an ice layer has been developed. The calculation was based on Green’s function theory for Helmholtz equ ation. Special “directed” Green’s functions were introduced. They make it possible to anayze wave fields in closes volumes limited by different-angle impedances. The developed calculation algorithms allow one to anayze fields on medium-powered computers for 15 minutes. The suggested methods are capable of estimating elastic wave interactions with different impedances in bays, lakes and other volumes with limited wave sizes.

2 0 ( ) ( ) 4 ( ), r k r q r   +  = − (1) where Ф is the oscillating velocity potential; q(r0) is the density of source distribution in domain r0; r is the distance from coordinate origin to observation point М; k is the wave number.
Equation (1) is solved by directed Green's functions. Function GHl(M, M0) has a sectorshaped pattern in the defined angle interval ∆φ min max 1, when , 0, in the rest angle interval.
where GHl is the directed Green's function. Green's function of a point-source radiator has the form where l is an angle sector where Green's function equals a unit; L is the number of sectors overlapping the angle 4π. Ice surface is approximated by a plane set as long as a plane wave is Helmholtz equation solution in Cartesian coordinate system (Fig. 1).
The spherical wave in the form of plane waves superposition. Helmholtz equation in the Cartesian system has the form where ; is the wave number;   (5) is the basic one to make further numerical experiments.
Hereinafter, the coefficient of wave reflection from interface of a plane layer is denoted as V. Transmission (transparency) coefficient is W.
Green's functions for position mediums: medium 1, 3 ( Fig.2) is assumed in the form where R = r -r0, Rpr = rpr -r0 are the observation point coordinates relatively a source.

Fig. 2.
Layer medium geometry taking into account the ice.
As we introduce the expression in general form which allows us to take into account the reflection from a layer and its transparency, we basically form a generalized mathematical model which we shall further modify for concrete tasks considered within the framework of 3 this paper. We turn to the expressions for reflection coefficient V and transmission coefficient W.
Address to the theory of wave reflection from the interface of homogeneous half-spaces, a plane layer and a layer system [8,9]. As long as this theory is widely known, it is unreasonable to describe it. We consider only the basic principle, interactions and resulting formulas which are applied in this paper to make numerical experiments.
If we follow the problem, formulated in the previous section, the generalized physical model has the form. The radiator position medium, the layer and the medium, where a wave should arrive, are assigned the numbers 3, 2, 1, respectively. The angles of incidence, reflection and passage are denoted by θ3, θ2, θ1.
To find the coefficient of reflection from a layer, we need to find the impedance of the layer under consideration. The coefficient of reflection from layer 2 though impedance at the boundary 3-2 (ZIN) has the form As the result of rereflection at the boundaries, two waves with different propagation directions, symmetrical to plane 2, are formed inside the layer. Owing to that, the pressure inside layer 2 has the form ) , where А and В are undetermined constants.
It is known that in case of harmonic wave propagation in a homogeneous medium, particle oscillating velocity takes the form grad , where p = p(x, y, z, t) is the pressure; ω is the frequency; ρ is the medium density. Expression for z-component of velocity v2 is obtained by the substitution of (8) into (9): 1 , Be Ae e iz we denote Substituting (8) and (10) into (15), we obtain the expression for the layer input impedance Now, substituting expressions (16) into (7), we obtain a formula for the coefficient of reflection from layer 2 .
From the described expression of pressure continuity when passing the boundary z = 0, the following expression is fair: .
The condition of sound pressure continuity at the boundary z = d can be written as If d layer thickness tends to zero, the transmission coefficient is given by the formula In free space, Green's function at the frequency f=1 kHz behaves as follows:  frequency is f=1кГц; ultrasound wave propagation velocity in water is 1500 m/s; ultrasound wave propagation velocity in air is 331 m/s; ultrasound wave propagation velocity in ice is 3980 m/s; distance between the radiator and the hydrophone is 1000 m. Initial data of medium characteristics are: frequency is f=1кГц; ultrasound wave propagation velocity in water is 1500 m/s; ultrasound wave propagation velocity in air is 331 m/s; ultrasound wave propagation velocity in ice is 3980 m/s; distance between the radiator and the hydrophone is 1000 m. Initial data of medium characteristics are: frequency is f=3кГц; ultrasound wave propagation velocity in water is 1500 m/s; ultrasound wave propagation velocity in air is 331 m/s; ultrasound wave propagation velocity in ice is 3980 m/s; distance between the radiator and the hydrophone is 1000 m.

Conclusions
1. Applying an ice layer as a path for wave propagation, we can extend the communication system distance by about 1.5-2 times since the sound wave P(x) decreases according to the cylindrical law (Fig. 6,7,8)). 2. The time for calculation of one variant of a field does not exceed one minute on medium-powered computers. 3. The developed theory may be applied in online mode for theoretical analysis of sound fields accompanying earthquakes.