If m is already equal to 1 and the signal still cannot propagate, the distance between the walls must be increased.įinally, Equation (10-11) may be substituted into Equation (10-10) to give the very important universal equation for the guide wavelength, which does not depend on either waveguide geometry or the actual mode (value of m) used. If this signal must be propagated, a mode of propagation with a larger cutoff wavelength should be used, that is, m should be made smaller. Furthermore, when a Parallel Plane Waveguide fails to propagate a signal, it is because its free-space wavelength is too great. It follows from Equation (1040) that the wavelength of a signal propagating in a waveguide is always greater than its free-space wavelength. When m is made unity, the signal is said to be propagated in the dominant mode, which is the method of propagation that yields the longest cutoff wavelength of the guide. This means that the longest free-space wavelength that a signal may have and still be capable of propagating in a Parallel Plane Waveguide, is just less than twice the wall separation. The largest value of cutoff wavelength is 2a, when m = 1. From Equation (10-10), the cutoff wavelength is that value of λ for which λ p becomes infinite, under which circumstance the denominator of Equation (10-10) becomes zero, giving This implies that any larger free-space wavelength certainly cannot propagate, but that all smaller ones can. The free-space wavelength at which this takes place is called the cutoff wavelength and is defined as the smallest free-space wavelength that is just unable to propagate in the waveguide under given conditions. We then haveįrom Equation (10-10), it is easy to see that as the free-space wavelength is increased, there comes a point beyond which the wave can no longer propagate in a waveguide with fixed a and m. It is now possible to use Equation (10-9) to eliminate λ n from Equation (10-3), giving a more useful expression for λ p, the wavelength of the traveling wave which propagates down the waveguide.
The previous statements are now seen in their proper perspective: Equation (10-9) shows that for a given wall separation, the angle of incidence is determined by the free-space wavelength of the signal, the integer m and the distance between the walls. Substituting for λ n from Equation (10-4) gives M = number of half-wavelengths of electric intensity to be established between the walls (an integer)
Λ n = wavelength in a direction normal to both walls If a second wall is added to the first at a distance a from it, then it must be placed at a point where the electric intensity due to the first wall is zero, i.e., at an integral number of half-wavelengths away. Another important difference is that instead of saying that “the second wall is placed at a distance that is a multiple of half-wavelengths,” we should say that “the signal arranges itself so that the distance between the walls becomes an integral number of half-wavelengths, if this is possible.” The arrangement is accomplished by a change in the angle of incidence, which is possible so long as this angle is not required to be “more perpendicular than 90 °.” Before we begin a mathematical investigation, it is important to point out that the second wall might have been placed (as indicated) so that a′ = 2λ n/2, or a″ = λ n/2, without upsetting the pattern created by the first wall.
A major difference from the behavior of transmission lines is that in Parallel Plane Waveguide the wavelength normal to the walls is not the same as in free space, and thus a = 3λ n/2 here, as indicated.