\gamma, \qquad \mu \mapsto \mu + 2 \gamma, \qquad \lambda \mapsto \lambda + \gamma(1-\mu-\gamma),$$ maps solutions of $-\mu \varphi + L\varphi +\varphi^2 = \lambda$ to solutions of a new equation of the same form. The equation can be rigorously justified as a model for shallow water waves travelling rightward with a permanent form and a constant, nondimensionalised, wave speed $\mu$ [@MR3060183], and can, consistently, be obtained from the Euler equations via an exponential scaling [@MR3390078]. We shall deal with somewhat generally. *With a solution of the steady Whitham equation we denote a real-valued, continuous and bounded function $\varphi$ that satisfies almost everywhere.* As Whitham himself conjectured in [@MR1699025 p. 479] (here, the notation has been changed to match that of ),

> …it seems reasonable to assume that in fact a critical height is reached when $\varphi = \frac{\mu}{2}$. If $K(x)$ behaves like $|x|^p$ as $x \to 0$ and $\varphi(x)$ behaves like $\frac{\mu}{2} - |x|^q$, a local argument in suggests that $2q - 1 = p + q$; hence $ q = p +1$. According to this, the crest would be cusped with $\varphi \sim \frac{\mu}{2} - |x|^{1/2}$ for $K$.

The simplicity in Whitham’s formal argument is striking, even the 