Maximum gain of a line source antenna if the distribution fuction is a finite fourier series

If the highest permitted harmonics of the distribution function (this is the quantity of interest in practical application) and the supergain ratio are given, what is the maximum attainable gain? In this paper this optimum problem is solved through the introduction of
n
dimensional vectors and tensors and the result is obtained in closed form. For the case of two harmonics, numerical results are given. It is concluded that a small improvement in the specific gain is accompanied by a large change in the supergain ratio. The examination of the maximum of the specific gain, without auxiliary condition, leads to the conclusion that the addition of every harmonic increases the value of specific gain, but that the improvement is negligibly small until the number of harmonics becomes greater than the size of the aperture in wavelength. Using the method developed in this paper it is possible for any finite number of harmonics to calculate the optimum distribution function, but the computations become very lengthy as the number of harmonics increases.