crest. We then place one first-order difference on $\varphi$ and one on $K$ to deploy an interpolation argument between the global $C^{\alpha}$-regularity and the $C^{1/2}$-regularity exactly at the top, to obtain global $C^{1/2}({\mathbb R})$-regularity. The highest wave is qualitatively depicted in Figure \[fig:highest\]. We conjecture that it is everywhere convex and satisfies $$\varphi = {\textstyle\frac{\mu}{2}} - {\textstyle \sqrt{\frac{\pi}{8}}}|x|^{1/2} + o(x) \quad\text{ as }\quad x \to 0,$$ but a proof of these facts has so far evaded us. If such a formula holds, then one can show that the constant in front of $|x|^{1/2}$ must indeed take the value $\sqrt{\pi /8}$. The same constant would furthermore scale with $K$, so that a different normalisation of the inverse Fourier transform would lead to a corresponding scaling in the limiting constant.

Some comments on related recent work on the Whitham equation not mentioned above. The equation features the same kind of Benjamin–Feir instability as the full Euler equations [@MR3298879; @MR3226084], although its uni-directional character excludes other (small-amplitude, high-frequency) instabilities seen in the Euler equations [@DT15]. It is locally well-posed in $H^{3/2+}$, in both the periodic setting and on the line [@MR3375167], but a large-time existence theory is so far lacking for equations with 