  We investigate in this paper the existence of the leading profile of a WKB
expansion for quasilinear initial boundary value problems with a highly
oscillating forcing boundary term. The framework is weakly nonlinear, as the
boundary term is of order O($\epsilon$) where the frequencies are of order
O(1/$\epsilon$). We consider here multiple phases on the boundary, generating a
countable infinite number of phases inside the domain, and we therefore use an
almost periodic functional framework. The major difficulties of this work are
the lack of symmetry in the leading profile equation and the occurrence of
infinitely many resonances (opposite to the simple phase case studied earlier)
The leading profile is constructed as the solution of a quasilinear problem,
which is solved using a priori estimates without loss of derivatives. The
assumptions of this work are illustrated with the example of isentropic Euler
equations in space dimension two.
