On the integrals of a partial differential equation

The ordinary one-dimensional wave equation has special integrals of the form which satisfy the first-order equations respectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight lines in an xt -plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.