In fluid dynamics and coastal engineering, the Miche criterion, Miche formula or Miche breaking index is a theoretical upper bound on the steepness of a non-breaking, periodic wave in finite water depth. It gives the maximum wave height that can persist at a given depth and wavelength. Waves that exceed this bound are unstable and must break. The criterion was derived by French engineer Robert Miche in 1944 at the École nationale des ponts et chaussées in a study of wave motion over constant and decreasing depth.
Formulation Miche's work provides an upper limit for wave breaking, with observations in deep sea locations indicating that breaking criteria can be independent of steepness. Miche shows that, theoretically, the maximum height of a fixed-form, periodic wave is controlled by the fact that the particle velocity at the wave crest u x {\displaystyle u_{x}} cannot be larger than the celerity of the wave, c {\displaystyle c} , resulting in the following:
H max ≈ 0.14 L tanh ( 2 π D L ) {\displaystyle H_{\max }\approx 0.14L\tanh \left({\frac {2\pi D}{L}}\right)}
In deep water, this makes the steepness of an individual wave smax = Hmax/L ≈ 0.14. In his 1944 paper, Miche expressed the limiting steepness in two equivalent forms: Steepness form:
H b L ≤ 0.142 tanh ( 2 π h b L ) {\displaystyle {\frac {H_{b}}{L}}\;\leq \;0.142\,\tanh \!\left({\frac {2\pi h_{b}}{L}}\right)}
Wavenumber form:
k H b ≤ 0.88 tanh ( k h b ) , k = 2 π L {\displaystyle k\,H_{b}\;\leq \;0.88\,\tanh(kh_{b}),\quad k={\frac {2\pi }{L}}}
where H b {\displaystyle H_{b}} is the wave height at incipient breaking, L {\displaystyle L} the wavelength, h b {\displaystyle h_{b}} the local water depth, and k {\displaystyle k} is the wavenumber.
Limits Two limits follow directly from the criterion:
Deep water ( k h ≫ 1 {\displaystyle kh\gg 1} ): H / L ≤ 0.142 {\displaystyle H/L\leq 0.142} . Shallow water ( k h ≪ 1 {\displaystyle kh\ll 1} ): H / h ≤ 0.88 {\displaystyle H/h\leq 0.88} (often called the breaker index).
Interpretation and use Miche's result gives a necessary condition for non-breaking waves, and an upper theoretical limit for wave breaking. If the inequality is violated at a point, a steady periodic wave cannot exist and breaking must occur. In practice the criterion is used to:
check numerical or physical model results for wave heights in shallow areas; estimate an upper bound for local wave run-up and loads on coastal structures when explicit breaking dissipation is not modelled; define a cap for random-sea parameters by applying the bound to a representative height such as significant wave height H s {\displaystyle H_{s}} (or the wave energy parameter H m 0 {\displaystyle H_{m0}} ) as a conservative proxy. For random waves on natural slopes, empirical breaker indices used in design are often somewhat lower than the shallow water upper bound of 0.88, however Miche's relation provides a theoretical ceiling.
History Miche developed the criterion while studying the limiting form of wave crests at the point of breaking, including effects of finite depth and possible rotational components. His work focused on periodic waves in constant depth, wave transformation over regularly decreasing depth, and the geometry and kinematics of limiting (breaking) waves near the shore. The first part of Miche's 1944 paper focused on application of breaking wave limits to coastal engineering structures such as breakwaters, as well as patterns of standing waves.
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