Natural Science Forum / Earth Science / Oceanography / January 2004

My local yacht club needs an instrument to measure significant wave height.

I've investigated commercially available wave height measurement
instruments and found they are way too expensive for the yacht club.

Consequently I've built an ultrasonic wave height instrument which seems
to measure instantaneous distance from the water surface usefully
accurately. However, I think I've a problem deriving significant wave
height from this stream of instantaneous height data. I'm using a
formula from the US National Data Buoy Center site, but the readings
from the instrument using this formula seem about twice what I'd expect
from looking at the waves.

On the National Data Buoy Center page
http://www.ndbc.noaa.gov/wavecalc.shtml they state:

<start quote>
Significant wave height, Hs, is approximately equal to the average of
the highest one-third of the waves. Hs is calculated using:

Hs = 4.0 * sqrt(m0)

where m0 is the variance of the wave displacement time series acquired
during the wave acquisition period.
<end quote>

For a sine wave of peak amplitude 1 (and peak-to-trough amplitude 2),
sqrt(m0) is .707, so Hs would be 2.828.

It seems odd to me that Hs calculated using this formula is some 1.4
times the peak-to-trough amplitude.

To me, the formula would make sense if the 4.0 was within the sqrt
brackets, as this would make Hs twice the root mean square of the wave
form. As an electronics-type chap, this would seem to me sensible, as
RMS is much used in electronic waveform energy-related formulae.

I emailed the National Data Buoy Center site a couple of weeks ago but
have received no reply.

I'd be grateful for comment on whether the formula they give is the
"generally accepted" formula and my logic is faulty, or whether there
may be an error in it.

I'm aware there are other formulae for SWH which involve spectral
distribution, but the maths looks too complex both for the simple
microcontroller in the instrument and for my failed-engmaths3 brain.