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A linear partial differential equation is a partial differential equation in which the unknown function and its partial derivatives occur linearly, with coefficients independent of the unknown. Its homogeneous solutions obey the superposition principle.

By the superposition principle, interchange $x$ and $y$ in the solution from part (c) and add the two solutions:The first term supplies the required data on the positive $x$-axis and vanishes on the positive $y$-axis; the second does the reverse. Explicitly,It is harmonic in the quarter-plane, has the stated boundary values, and decays in both unbounded directions.

The linearized equations are homogeneous and have constant coefficients in $x$ and $t$. Consequently, spatial Fourier modes evolve independently, and the superposition principle reconstructs arbitrary sufficiently regular initial disturbances from them. It is therefore enough to seek the normal modesHere the real $k$ is the wavenumber, $\sigma$ is the possibly complex temporal exponent, and taking the real part produces a physical displacement.