Re: [Phys-L] Halfway point ???

["Donald Polvani" <dgpolvani@verizon.net>]

> On 04/03/2015 06:54 PM, Bob Sciamanda wrote:

> The Log of the geometric mean of two quantities is the halfway point 
> between the Logs of the two quantities (using any logarithmic base).

In addition to the arithmetic and geometric means, the harmonic mean is
sometimes useful in situations requiring an average of reciprocals.  For
example, in acoustic transponder navigation (and in the days when computing
power wasn't so prevalent), it was a computational simplification to have an
average sound speed determined from the harmonic mean of the sound speed
depth profile in order to determine slant ranges from travel time
measurements.  One could determine this average speed from ray tracing
calculations, but this takes more computing power than doing a simple
average.  The harmonic mean (x_HM) is defined as:

1/x_HM = (1/x_1 + 1/x_2)/2  or  x_HM = 1/(1/x_1 + 1/x_2)/2

So, the reciprocal of the harmonic mean is the arithmetic average of the
reciprocal quantities. 

Don