Wave spectra: frequency, energy and direction

A spectrum describes how surface variability is distributed across frequencies and, when directional information exists, across directions. It is another view of the sea state rather than a picture of waves arranged in space.

A frequency graph’s horizontal axis represents oscillations per second; its vertical axis may represent variance density. A peak’s height on paper is not a wave height in metres.

A frequency of 0.10 Hz means 0.10 cycles per second, or one cycle in 10 s. At 0.20 Hz, the period is 5 s.

T=1/f requires positive frequency and the same time reference. Transforming a frequency axis into a period axis changes interval spacing: relabelling numbers is insufficient.

Transforming a density also requires respecting the axis variable and its units.

In a non-directional elevation spectrum, S(f) may use m²/Hz. Multiplying a band’s density by its width in Hz gives a contribution in m².

Adding contributions approximates total variance m0. A tall narrow band can therefore contribute less than a low broad one.

Reading only the highest point confuses density per unit frequency with the integrated quantity in a band.

If two components are treated as uncorrelated in a statistical model, their variances can add. Their Hm0 heights do not follow the same addition rule because they depend on a square root.

In a separate example, components with Hm0 of 2 m and 4 m have m0 of 0.25 m² and 1 m². Combined Hm0 is 4√1.25≈4.472 m, not 6 m.

The correlation assumption matters: this neither adds two observed crests nor calculates an instantaneous wave.

Components with different periods can arrive from different directions. A directional spectrum retains that distribution; a “mean direction” arrow compresses it.

Direction associated with the peak band need not equal a mean across the whole spectrum. FROM and TO conventions also differ: here, maritime direction FROM 315° identifies a north-west origin; corresponding propagation is TO 135° under the same true reference.

A buoy-estimated spectrum depends on a finite record, sensor response and processing. It contains neither the entire surface nor the exact sequence of future waves.

Our two-band arithmetic is transparent integration for learning units and summaries, not a forecast algorithm. Applied interpretation of crossing seas, current or vessel response needs more context than this statistical representation.