Mike, W4EF wrote:
"---standing waves on the resonant antenna, and a "law" that says what
we can predict that taper with a simple formula---."

It`s as simple as ON4UN`s Fig 9-22 center loading diagram. The resonant
element is 90-degrees long. Any missimg wire length must be provided by
the loading.

Best regards, Richard Harrison, KB5WZI