# Macagno formula

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Macagno formula is analytic estimate of coefficient of reflection of monochromatic wave at the surface of a liquid from a П–shaped inhomogeniety. [1]

The reflection coefficient $$R$$ is approximated with expression

$$\displaystyle R= \left( 1+ \left( kw \frac{ \sinh(kh) }{ 2 \cosh(kh\!-\!kd) }\right)^2 \right)^{-1/2}$$

where $$k$$ is wavenumber,

$$h$$ is water depth

$$w$$ is width of the obstacle

$$d$$ is draft (height) of the obstacle.

Using dimensions variables $$H=kh$$, $$W=kw$$, $$d=kD$$,

the estimate can be written as follows:

$$\displaystyle R= \left( 1+ \left( W \frac{ \sinh(H) }{ 2 \cosh(H\!-\!D) }\right)^2 \right)^{-1/2}$$

The formula looks strange, as it predicts reduction of the reflection coefficient to zero at large values of $$W$$.

## References

1. https://ascelibrary.org/doi/abs/10.1061/(ASCE)WW.1943-5460.0000153 Piero Ruol; Luca Martinelli; and Paolo Pezzutt. Formula to Predict Transmission for π-Type Floating Breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering. Volume 139 Issue 1 - January 2013. The aim of this paper is to define a simple and useful formula to predict wave transmission for a common type of floating breakwater (FB), supplied with two lateral vertical plates protruding downward, named п-type FB. Eight different models, with mass varying from 16 to 76 kg, anchored with chains, have been tested in the wave flume of the Maritime Laboratory of Padova University, under irregular wave conditions. Water elevation in front and behind the structure has been measured with two arrays of four wave gauges. Our starting point for the prediction of wave transmission was the classical relationship established by Macagno in 1954. Flis relationship was derived for a box-type fixed breakwater assuming irrotational flow. Consequently, he significantly underestimated transmission for short waves and large drafts. This paper proposes an empirical modification of his relationship to properly fit the experimental results and a standardized plotting system of the transmission coefficient, based on a simple nondimensional variable. This variable is the ratio between the peak period of the incident wave and an approximation of the natural period of the heave oscillation. A fairly good accuracy of the prediction is found analyzing the data in the literature relative to variously moored rr-type FBs, tested in small-scale wave tanks under regular and irregular wave conditions. DOI: 10.1061/ (ASCE)WW.1943-5460.0000153. © 2013 American Society of Civil Engineers.