  Here we present inelastic neutron scattering measurements from the
frustrated, quantum spin-1/2 chain material linarite, PbCuSO_4(OH)_2. Time of
flight data, taken at 0.5K and zero applied magnetic field reveals low-energy
dispersive spin wave excitations below 1.5meV both parallel and perpendicular
to the Cu-chain direction. From this we confirm that the interchain couplings
within linarite are around 10% of the nearest neighbour intrachain
interactions. We analyse the data within both linear spin-wave theory and
density matrix renormalisation group theories and establish the main magnetic
exchange interactions and the simplest realistic Hamiltonian for this material.
