  In this paper we study weakly hyperbolic second order equations with time
dependent irregular coefficients. This means to assume that the coefficients
are less regular than H\"older. The characteristic roots are also allowed to
have multiplicities. For such equations, we describe the notion of a `very weak
solution' adapted to the type of solutions that exist for regular coefficients.
The construction is based on considering Friedrichs-type mollifiers of
coefficients and corresponding classical solutions, and their quantitative
behaviour in the regularising parameter. We show that even for distributional
coefficients, the Cauchy problem does have a very weak solution, and that this
notion leads to classical or ultradistributional solutions under conditions
when such solutions also exist. In concrete applications, the dependence on the
regularising parameter can be traced explicitly.
