One of important problems of the theory of antennas is the research of boundary effects in finite antenna arrays. Two methods are used to determine characteristics of such arrays. The first one is based on applying direct numerical methods to solve the corresponding boundary value problems of electrodynamics. This method can be applied to arrays containing a small number of elements. The second method is based on using an infinite array as a model of the finite array. This method allows determining the characteristics of radiating elements situated in the central part of the array containing a large number of such elements. Also, combination methods are used that consider the antenna array to be a part of an infinite array where nonexcited radiating elements are loaded with matched load. The above methods have an essential drawback: they do not reveal the physics of boundary effects and do not allow using the mathematical apparatus effectively. The attempt to use formal mathematical approach to researching properties of radiating elements in the boundary area by analyzing the resulting set of algebraic equations does not succeed because of the complexity of such a problem.

This work suggests another method of analyzing characteristics of finite antenna arrays, free of drawbacks listed above. The method is effective because it is based on the adequate model of physical processes that take place in the boundary areas of finite arrays. The main point of this model is considering currents in the radiating elements of a finite antenna array to be a superposition of unperturbed currents of an infinite array that contains the finite antenna array as a part, and a boundary wave that propagates from the boundary area to the center of the array. In this case, changes in amplitude and phase of currents of radiating elements in the boundary area is the result of interference of unperturbed currents of the infinite array and the currents of the boundary wave. As the research shows, the boundary wave has quite stable characteristics. For example, in linear arrays only the amplitude of the boundary wave is changed while scanning, but the speed of the wave in the array and the changes in the amplitude from the boundaries to the center remain almost unchanged. This allows formulating a qualitative representation of changes in current distribution in the array while scanning. Understanding of the physics of boundary effects allows using the mathematical apparatus most effectively.

The theory of finite antenna arrays, based on the concept of boundary waves can be derived using various methods. At first, it is necessary to show that boundary waves do exist. It is enough to consider the simplest model of a semiinfinite antenna array, which allows excluding the interaction of the opposite boundaries. As an antenna array that satisfies this condition, the semiinfinite flat slotted array of narrow infinite parallel slots, connected to feeder lines exciting the reacting elements, through matching circuit is chosen. The matching circuit consists of an ideal transformer and a reactance. The characteristics of these elements are chosen to match the slot radiator in the infinite antenna array. As an operator equation the energy balance equation is used

where S — closed surface containing the radiating elements and the reactive elements of the matching circuit; P — complex power, accumulated in the reactive elements. Let , , , n be voltage, current strength, impedance of the reactive element and transformation coefficient of the ideal transformer respectively. Then the equation (1) can be rendered as follows:

where — magnetic-field strength of the p-th slot with unit voltage on the k-th slot.

After representing the current and voltage in the semiinfinite antenna array as a sum of the corresponding elements in the infinite array and the boundary wave

and inserting (3) into (2), we receive the set of equations for determining the voltage and current of the boundary wave at the input of the matching circuits of the radiating elements and the equations for determining the current and voltage in the infinite antenna array:

where the v value is expressed by the array step, wave length and phase angle .

If the matching of the array is performed for the phasing direction, defined by the parameter, the characteristics of the elements of the matching circuit are determined by the following relations:

where — impedance of the feeder line. The formal solution of the set of equations (4) can be found using the Viner-Hopf method [1,2]. However the implementation of factorization procedure in this case involves the numerical integration methods. Thus, it is worthwhile considering the possibility of using other methods.

The analytical solution can be found in the extreme case when the slot step tends to zero. It can be shown that in this case the set of equations (4) is transformed into the Fredholm integral equation of the second kind with weak polar nucleus and semiinfinite integration limits

where — voltage at the input of the radiating elements of the infinite array.

The theory of such equations is developed in [3]. Using this theory it is possible to solve the equation (8). In particular, when the arrays is phased in the direction of the tangent to the screen and the voltage in the array is equal to the voltage of the boundary wave, the integral equation above is transformed into the shore effect problem, solved in [3]. The analysis of the solution to this problem directly characterizes the boundary wave in a small step array.