The sea as a frequency mixer

Two waves meet on the sea. In some circumstances, their interaction helps set the ground vibrating at twice the waves’ frequency. That is a rather good invitation to a rabbit hole.

I started with a question about a connection to sound synthesis: could something in the sea behave a little like a ring modulator?

A quiet signal beneath the waves

Longuet-Higgins’s 1950 paper explains a mechanism in which opposing ocean waves produce a second-order pressure variation at twice their frequency. That component avoids the simple depth attenuation of ordinary travelling-wave pressure and can help generate the small ground motions known as secondary microseisms. The full theory also considers water compressibility and seabed response. Read the original paper (PDF).

The interesting part is the interaction. A useful way into the frequency arithmetic is to multiply two ideal wave patterns and see what survives when we average across space.

One small piece of maths

Take two sinusoidal waves with the same wavelength and period, travelling in opposite directions. Write their surface elevations as:

incident:  a cos(kx − ωt)
opposing:  b cos(kx + ωt)

Their product can be written:

(ab / 2) [cos(2kx) + cos(2ωt)]

Average over whole wavelengths and the spatial term cancels. What remains oscillates at twice the original frequency. In musical terms, frequency doubling takes us up an octave. With a 14-second input period, that surviving product has a 7-second period.

I also checked the spatial averaging numerically. Over eight wavelengths and twenty input periods, the calculation matches the analytic result to floating-point precision. The corresponding product of waves travelling in the same direction has a constant spatial average, so removing its time mean leaves essentially zero.

Bring the idea into hearing range

The illustrative wave frequencies are much too low to hear. For this short audio sketch, I multiplied both by 4,096—a shift of twelve octaves. The first tone is approximately 292.57 Hz; the next is 585.14 Hz. A final crossfade moves between them.

Where the analogy stops

The multiplication above is an algebraic toy, not a fluid-dynamics or seismic model. Its product is in square metres, not pascals. It omits depth, realistic wave spectra, bathymetry, and the Earth’s response. Ring modulation is a helpful mathematical connection; the analogy alone does not establish the physical mechanism.

There is also more than one way for ocean waves to excite seismic motion. Ardhuin, Gualtieri, and Stutzmann’s 2015 model separates the secondary interaction from a primary mechanism involving bottom topography. Their results attribute the longer 13–300-second seismic periods chiefly to the latter. The slower Earth “hum” should not be casually folded into the frequency-doubling story. Read their research paper.

And a real seafloor recording contains more than wave information: currents, instrument motion, and other environmental influences matter. Hilmo and Wilcock’s observational study is a useful reminder of that complexity. Read their 2020 paper.

Keep the workings

The small numerical experiment is available to inspect and repeat. It uses Python and NumPy; it contains no measured ocean or seismic data.

• Experiment source Python
• Calculated time series CSV
• Results and checks JSON

I like the connection because it puts a familiar piece of audio arithmetic beside a much larger physical question. It also leaves an obvious next question: what changes when the perfectly regular waves become finite, uneven wave packets?