Explain Why Hs Equals 4 Times Standard Deviation
explain_hs_formula.RdEducational function explaining the physical and statistical basis for the relationship Hs = 4 * sigma.
Examples
cat(explain_hs_formula())
#>
#> SIGNIFICANT WAVE HEIGHT (Hs) = 4 * Standard Deviation
#> ======================================================
#>
#> STEP-BY-STEP CALCULATION
#> ------------------------
#> 1. MEASURE sea surface elevation eta(t) over 17.5 minutes
#> - Raw data: elevation in meters at each time point
#> - Example: eta = [0.5, -0.3, 1.2, -0.8, ...] meters
#>
#> 2. REMOVE THE MEAN (de-trend to eliminate tidal/ambient level)
#> - eta_prime(t) = eta(t) - mean(eta)
#> - This centers the data around zero
#> - Now we have pure wave-induced fluctuations
#>
#> 3. CALCULATE STANDARD DEVIATION of de-meaned elevations
#> - sigma = std(eta_prime)
#> - Units: meters
#> - This measures the typical amplitude of wave fluctuations
#>
#> 4. MULTIPLY BY 4
#> - Hs = 4 * sigma
#> - Units: meters
#>
#> WHY MULTIPLY BY 4? (The Statistical Derivation)
#> -----------------------------------------------
#> This comes from Longuet-Higgins (1952) wave spectrum theory:
#>
#> For a narrow-banded Gaussian random sea:
#> - Individual wave heights follow a RAYLEIGH DISTRIBUTION
#> - The Rayleigh PDF is: f(H) = (H/sigma^2) * exp(-H^2 / 2*sigma^2)
#>
#> The average of the HIGHEST 1/3 of waves (called H_1/3) can be computed
#> by integrating the Rayleigh PDF from the 67th percentile upward:
#>
#> H_1/3 = integral from H_67% to infinity of H * f(H) dH
#> = 4.004 * sigma
#> ~ 4 * sigma
#>
#> The factor 4.004 is a MATHEMATICAL CONSTANT from the Rayleigh distribution,
#> not an arbitrary choice. It rounds to 4 for practical use.
#>
#> PHYSICAL MEANING
#> ----------------
#> - Hs approximates what a trained observer visually estimates
#> - Observers naturally focus on the larger, memorable waves
#> - They unconsciously average the highest third
#> - Longuet-Higgins proved this matches H_1/3 = 4*sigma
#>
#> REFERENCES
#> ----------
#> 1. Longuet-Higgins, M.S. (1952). On the statistical distribution of the
#> heights of sea waves. J. Marine Research, 11(3), 245-266.
#> [Original derivation of Hs = 4*sigma]
#>
#> 2. Holthuijsen, L.H. (2007). Waves in Oceanic and Coastal Waters.
#> Cambridge University Press. Chapter 4.
#> [Modern textbook treatment]
#>
#> 3. WMO (2018). Guide to Wave Analysis and Forecasting. WMO-No. 702.
#> [Operational standards]
#>
#> RELATIONSHIP TO RMS WAVE HEIGHT
#> -------------------------------
#> RMS wave height: H_rms = sqrt(mean(H^2)) = sqrt(2) * sigma ~ 1.41 * sigma
#>
#> Therefore:
#> Hs = 4 * sigma = (4/sqrt(2)) * (sqrt(2) * sigma) = 2.83 * H_rms
#>
#> Or: H_rms ~ 0.35 * Hs