Revisit Problem 14.21, but with the current phase allowed to vary with frequency (this will automatically occur if the phase difference is established by a simple time delay between the feed currents). Now, the current phase difference will be ξ' = ξ f/ f₀, where f₀ is the original (design) frequency. Under this condition, radiation will maximize in the ϕ = 0 direction regardless of frequency (show this). Backward radiation (along ϕ = π) will develop, however, as the frequency is tuned away from f₀. Derive an expression for the front-to-back ratio, defined as the ratio of the radiation intensities at ϕ = 0 and ϕ = π, expressed in decibels. Express this result as a function of the frequency ratio f/ f₀. Evaluate the front-to-back ratio for

(a) f = 1.5 f₀;

(b) f = 2 f₀ and;

(c) f = 0.75 f₀.