Good evening Art, I have a few minutes to put some thoughts on email
regarding efficiency per unit length of an antenna, or antenna elements.
I hope I don't glaze your eyes :-)

Efficiency by definition is work delivered to a load divided by the
total work available.

The work [power] delivered to the load is the work available minus
losses all divided by work available. Your car engine may have 100
horsepower available but losses [heat] in the transmission, drive shaft,
differential, oils, fluids, bearings, tires and wheels may limit
horsepower delivered to the road to 30 horsepower. In this model then
the efficiency is 30%.

Note: bandwidth is not a factor. It is total work[power] delivered
divided by total work[power] available.

For an antenna the following definition is applicable by similarity:

Efficiency = Radiated Power/Total power. By definition an antenna is a
linear device and the Principle of Reciprocity is applicable. That is,
it has the same efficiency either transmitting or receiving.

Now, total power = I^2*Reffective

Reffective = Radiation Resistance [Rradiation] + Loss Resistance.

Radiated Power = I^2*Rradiation.

Where I = Io*cos[wt + b]. Where wt = frequency, b = phase shift along
antenna element.

By definition radiation resistance is that determined by integrating the
total radiated power over the surface of the sphere containing that
power. [1] For a half wave dipole that converges to the value at a
current maximum. So, radiation resistance for a 1/2 wave infinitely thin
dipole is 73 ohms at the current maximum.[1]

As one moves away from the current maximum the radiation resistance
increases as 1/cosine(angle from the maximum)^2 i.e. 1/cos^2[theta].[2]

Now, in a uniform cross section antenna the Loss Resistance per unit
length is constant.

So, the Radiation Resistance at the ends of the antenna is infinite.
[Cosine 90 degrees = 0] That means the efficiency at the ends of a 1/2
wavelength dipole is 100%. Isn't that a surprise?? [It's the same for a
dipole or a Yagi!!!!]

The Radiation Resistance at the 45 degree point from the current maximum
of a thin 1/2 wavelength dipole is 73 ohms/(cos^2(45 degrees)) = 146 Ohms.

The conclusion is that the efficiency of an antenna element varies along
it's length and can vary between maximum of infinity at the ends and
have a minimum value, that depends on the length of the antenna, at a
current maximum.

The total antenna efficiency is measured in the radiated pattern by
integrating the power density per square steradian [or square degrees]
over the full surface of a sphere divided by the power into the antenna.

So, a Yagi with 8.14 dBi (6 dBd + 2.14 dBi) gain has concentrated it's
radiated power into 15.38% of the three dimensional space defined by the
surface of a sphere. [See Note 1.] Now if the Yagi is 95 % efficient and
a dipole is 95% efficient the 6 dBd value remains constant.

_____________
Note 1 [I'm using degrees instead of steradians to simplify
understanding. The Science/math is the same]

The diameter of the earth is 360 degrees as measured from E-W and N-S.
The surface is then proportional to 360^2 = 129600 square degrees. [I
will be dividing the constant Pi in a subsequent step so it is deleted
here.]

A 6 dBd Yagi is also +2.14 dBi giving a net gain of 8.14 dBi. This
normalizes the value to a sphere.

So, 8.14 dBi = 10*Log(129600/X)
X = 19938.4 square degrees.

The Yagi has concentrated it's pattern into a piece of the surface of
the sphere that represents 15.38% of the total surface [19938.4/129600].
This is Gain NOT efficiency.

_____________
Reference [1]: Antennas, Kraus, McGraw-Hill 1950, Chapter 5, section
5-6, pages 143-146.

Reference [2]: Antennas, Kraus, McGraw-Hill 1950, Chapter 5, section
5-7, pages 147-148.