Summary: A quadratic line complex is a three-parameter family of lines in projective space \(\mathbb{P}^3\) specified by a single quadratic relation in the Plücker coordinates. Fixing a point \(\mathbf{p}\) in \(\mathbb{P}^3\) and taking all lines of the complex passing through \(\mathbf{p}\) we obtain a quadratic cone with vertex at \(\mathbf{p}\). This family of cones supplies \(\mathbb{P}^3\) with a conformal structure, which can be represented in the form \(f_{ij}(\mathbf{p})\,dp^idp^j\) in a system of affine coordinates \(\mathbf{p}=(p^1, p^2, p^3)\). With this conformal structure we associate a three-dimensional second-order quasilinear wave equation \[ \sum _{i, j}f_{ij}(u_{x_1}, u_{x_2}, u_{x_3}) u_{x_ix_j}=0, \] whose coefficients can be obtained from \(f_{ij}(\mathbf{p})\) by setting \(p^1=u_{x_1}, \;p^2=u_{x_2}, \;p^3=u_{x_3}\). We show that any partial differential equations (PDE) arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classification of linearly degenerate wave equations into 11 types, labelled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the structure \(f_{ij}(\mathbf{p}) \, dp^idp^j\) is conformally flat, as well as Segre types for which the corresponding PDE is integrable.