This paper deals with quasilinear hyperbolic systems of the form \[ \frac{\partial u_i}{\partial t}+ \lambda_i (u_1,\dots, u_{2g+1}) \frac{\partial u_i}{\partial x}=0, \qquad x,t,u_i\in \mathbb{R},\tag{1} \] with ordering \(u_1> u_2>\dots> u_{2g+1}\). For a given \(g\), system (1) is called \(g\)-phase Whitham equations. The author shows that, if the monotonic initial data \(x=f(u)\) satisfy the condition \[ \frac{d^{2N+1}} {du^{2N+1}} f(u):= f^{(2N+1)}(u)< 0, \qquad 1\leq N\in \mathbb{N}, \] for all real \(u\) except for a number of isolated points, then the genus of the global solution of the Whitham equations is at most \(N\). See the announcement in Applied and industrial mathematics, Venice-2, 1998, 235-244 (2000; Zbl 1001.37064).