The authors study the hyperbolic equations with delays \(\sigma\) and \(\tau\) of the form \[ u_{xy}\pm f(x,y,u(x-\sigma,y-\tau))=0, \] with characteristic initial data \(u(x,y)=\phi (x,y)\) if \(-\sigma \leq x<\infty\), -\(\tau\leq y\leq 0\) and \(u(x,y)=\psi (x,y)\) if -\(\sigma\leq x\leq 0\), and \(-\tau \leq y<\infty\). Here f, \(\phi\), \(\psi\), are continuous, f is positive and \(f(x,y,0)=0\). The paper contains a study of the effects of the delays on the oscillatory nature of the solutions.