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Let $c_0=\sqrt{g^{\prime }{h}_0/2}$, as printed in the PDF. The converted TeX misrenders this expression. The undisturbed reservoir has $u=0$, $c=c_0$. The expansion fan moving into it is a simple wave with $R_{+}=u+4c=4c_0$ constant. On the other characteristic family, a self-similar fan satisfiesSolving these two equations givesThe fan begins at $\xi =-c_0$, where it matches the initially resting reservoir.

At the front, write $k_F=\sqrt{2}F$. The nose closure is $u_f=k_F{c}_f$ and the invariant gives $u_f+4c_f=4c_0$. ThereforeA uniform region of this state connects the right edge of the fan, ${\xi }_n=u_f-c_f$, to the advancing front $x_f=u_{f}t$. Its width is $c_{f}t>0$. The complete lower-layer solution for $t>0$ isandAhead of the front there is no lower-layer current, so its velocity is not a physical variable there. Both transitions inside the current are continuous; the finite-height nose is closed by the ambient-front condition, not by an unsupported vacuum jump. These formulas give the front-regularized triangular-channel dam break for any specified positive constant $F$.