# Surf similarity parameter

 Definition of Surf similarity parameter: The surf similarity parameter compares the wave surface slope to the bed slope in the surf zone and represents important features of the hydrodynamics of the surf zone. This is the common definition for Surf similarity parameter, other definitions can be discussed in the article

## Notes

Characteristic features that play a major role in the wave hydrodynamics of the breaker zone are the seabed slope $m$, the still water depth $h$, the significant incident wave height $H_s$ and the wave period $T$, where all quantities refer to a location just seaward of the breakpoint. Battjes (1974[1]) has shown, however, that important processes depend mainly on a single parameter $\xi$, generally called surf similarity parameter or Iribarren parameter:

$\xi = \Large\frac{m}{\sqrt{H_s/L}}\normalsize , \qquad (A1)$

where $L=\Large\frac{g T^2}{2 \pi}\normalsize$ is the wavelength at the seaward boundary of the breaker zone and $H_s/L$ the wave steepness at this location. Using the expression for the shallow-water wave celerity ($c \approx \sqrt{gh}$) and the condition of surf zone saturation (all waves are broken and decay in proportion to depth, $H_s/h =\gamma$, see breaker index), Eq. (A1) can be expressed as

$\xi \approx \Large \sqrt{\frac{\gamma}{2 \pi}}\, \frac{m}{H_s /L}\normalsize . \qquad (A2)$

The surf similarity parameter thus represents approximately the ratio of the seabed steepness and the steepness of the breaking wave.

Experiments show that $\xi$ is also related to the breaker type[1]:

• for $\xi \gt 2$ a breaker is collapsing or surging,
• for $0.4 \lt \xi \lt 2$ a breaker is plunging;
• for $\xi \lt 0.4$ a breaker is spilling,

where the values of $H_s$ and $L$ are to be taken at the breakpoint.

It further appears that:

• The breaker index $\gamma$ depends on $\xi$; an empirical formula is[1][2] $\quad \gamma \sim 1.06+0.14 \ln \xi$. See also Breaker index.
• The wave reflection coefficient $C_r$ depends on $\xi$; an empirical formula is [1][3]$\quad C_r \sim 0.1 \xi^2 \quad$ for $\xi \lt 2.5$.
• The wave run-up $R$ is approximately proportional to $\xi$ according to the empirical formula[4] $\quad R \sim H_s \xi \quad$ for $0.1 \lt \xi \lt 2.3$, see Wave run-up.
• The value $\gamma \approx 0.3$ (corresponding to $4 H_s/L =5m$) defines the boundary between two different wave height transformation regimes: over-dissipative conditions for $\gamma \lt 0.3$ (more dissipation than required for depth-limited waves) and under-dissipative conditions for $\gamma \gt 0.3$ (energy dissipation insufficient for depth-limited conditions)[5].

## Related articles

Breaker index
Shoreface profile
Shallow-water wave theory
Swash zone dynamics
Tsunami
Stability of rubble mound breakwaters and shore revetments

## References

1. Battjes, J.A. 1974. Surf similarity. Proceedings 14th International Conference on Coastal Engineering, pp. 466–480
2. Camenen, B. and Larson, M. 2007. Predictive Formulas for Breaker Depth Index and Breaker Type. Journal of Coastal Research 23: 1028–1041
3. CIRIA/CUR/CETMEF 2007. The Rock Manual. The use of rock in hydraulic engineering (2nd ed.). C683. London: CIRIA
4. Hunt, I.A., 1959. Design of sea-walls and breakwaters. Transactions of the American Society of Civil Engineers, 126: 542-570
5. Power, H.E. 2013. Surf zone states and energy dissipation regimes. Coastal Engineering Journal 55, 1350003