Evaluation and validation of the spectral linear wave theory and ‘traditional’ formulae for pulsating wave loads for unimodal and bimodal seas

Comparison to Goda and measurements


  • Henry Tuin Arcadis the Netherlands BV
  • Hessel Voortman Hessel Voortman Engineering consultancy BV
  • Bas Hofland Delft University of Technology
  • Ermano de Almeida Arcadis the Netherlands BV




Spectral Linear wave theory, Linear Wave Theory, New Wave, wave spectrum, wave force spectrum, bimodal spectrum, significant wave force


For the design of vertical hydraulic structures pulsating wave forces need to be calculated. The total wave force is a result of every wave component (long waves and short waves) within a wave field. The common formulae are derived for regular or unimodal narrow sea states and use one characteristic wave height and period. Broad-banded spectra like bimodal sea states are present at many locations. Moreover, new hydraulic structures like Panamax or post-Panamax locks do have a large vertical surface exposed to pulsating wave loads. Swell components within the wave spectrum are disproportionally contributing to the total wave force compared to short waves. This depth effect for broad-banded or bimodal wave spectra is not considered by the traditional wave formulae which could result in significant underestimations of wave forces on hydraulic structures.

This paper aims to determine the wave loads of irregular non-breaking wave fields under any wave spectrum: narrow banded, broad-banded, or bimodal. Spectral linear wave theory (LWT) is used to transform any wave spectrum to a wave force spectrum. The wave force or wave pressure at any level can directly be evaluated from the wave force spectrum or wave pressure spectrum for any shape of the wave spectrum considered within this research. Spectral LWT is compared to the outcome of wave flume experiments with bimodal seas and other wave force formulae, like the Goda formula and quasi-regular LWT and the NewWave theory.

This paper gives a description and evaluation of the spectral LWT applied for bimodal wave spectra and a comparison of the accuracy and validity of other wave force formulae. The peak forces and peak pressures distribution obtained by spectral wave theory compare well to the measurements. It appears that the use of a spectral LWT to obtain characteristic extreme forces improves the accuracy of the extreme load more than the use of a second order wave model with a quasi-regular assumption (i.e. where the spectral shape is not considered). For the typical conditions that occur at hydraulic structures (horizontal bed, intermediate to deep water, non-breaking, and uni- and bimodal seas) the often-used Goda formula can both under of overestimate the peak loads. Goda is well applicable for conditions with (breaking) waves narrow wave spectra and values of kph <0.5.


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How to Cite

Tuin, H., Voortman, H., Hofland, B., & de Almeida, E. (2022). Evaluation and validation of the spectral linear wave theory and ‘traditional’ formulae for pulsating wave loads for unimodal and bimodal seas: Comparison to Goda and measurements. Journal of Coastal and Hydraulic Structures, 2. https://doi.org/10.48438/jchs.2022.0021