  We study generalized solutions of the nonlinear wave equation
$$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ with periodic conditions in $t$ and
homogeneous Dirichlet conditions in $s$, under the assumption that the ratio of
the period to the length of the interval is two. When $p\equiv 0$ and $\lambda$
is a nonzero eigenvalue of the wave operator, we give a proof of the existence
of two families of curves (which may coincide) in the Fu\v{c}ik spectrum
intersecting at $(\lambda,\lambda)$. This result is known for some classes of
self-adjoint operators (which does not cover the situation we consider here),
but in a smaller region than ours. Our approach is based on a dual variational
formulation and is also applicable to other operators, such as the Laplacian.
In addition, we prove an existence result for the nonhomogeneous situation,
when the pair $(a,b)$ is not `between' the Fu\v{c}ik curves passing through
$(\lambda,\lambda)\neq(0,0)$ and $p$ is a continuous function, sublinear at
infinity.
