Measuring voltage at two points A and B along a line carrying
a sinusoidal signal, and measuring the phase delay between A and B, delay_{AB
} , we may calculate the mean propagation velocity between
A and B.

The time delay between A and B is directly proportional
to the distance between A and B and inversely proportional to the velocity.

The sinusoidal signal will vary in frequency, and it will be convenient to relate delay and velocity to phase shift (angular shift, in radians).

Phase shift may also be written

.

If velocity is independent of frequency, then phase shift
is proportional to frequency. This is the S-K case.

If velocity increases as the square root of frequency,
as it does in the K-R case, phase shift will be proportional to the square
root of frequency.

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Let's work these phase shift-frequency relations out for
the very low G cases set out in http://www.wisc.edu/rshowalt/vel_explan.

For the K-R case, velocity in the low G limit reduces
to

.

In the KR case, phase shift is proportional to the square
root of frequency.

For the S-K case, velocity is independent of frequency
in the low G limit and reduces to:

Phase shift is directly proportional to frequency in this case.

.

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The theoretical predictions of S-K and K-R differ, but
the measurement inference of velocity from phase shift will in any case
be as follows: