# What is the physical idea behind the mathematical definition of the dimensionless “surf similarity parameter”?

My question refers to a parameter called "iribarren number", whose value corresponds qualitatively to the type of breaking of water waves (spilling, plunging, collapsing and surging) on a uniformly sloping beach. This parameter is defined as:

$$\epsilon = \frac{tan \alpha}{\sqrt{H/L_0}}$$

so the numerator is the slope of the bottom of the coast and the denumerator is the square root of the wave steepness. This formula implies for example that for the surging type of breaker (reflective beaches), increasing the beach slope by a factor of two should be compensated by factor 4 increase in wave steepness in order to get the same relative amount of wave energy reflection.

I have some intuition about why both the wave steepness and beach slope enter the definition; obviously when beach slope is 0 the wave is undisturbed, and when the slope angle is 90 degrees the wave undergoes (almost) complete reflection. Between this two extreme cases there is a "spectrum" of behaviours; there is an optimal slope angle for the formation of plunging breakers which surfers look for. Also i have some intuition for the importance of wave steepness.

But i'm completely unable to formulate this intuitions mathematically from fundamental physical priniciples. Especially i'd like to know why the iribarren numbers is proportional to slope beach but is inversely proportional only to the square root of the wave steepness.

So i'd like to know if anyone can give a motivating explanation for the definition of the "surf similarity parameter".