Someone argues: "The diffusion equation needs one initial condition because it is…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy
Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions
Problem
Someone argues: "The diffusion equation ∂tT=D∂x2T needs one initial condition because it is first order in time. The wave equation ∂t2y=c2∂x2y is second order in time, so by the same reasoning it also needs just one initial condition, namely the initial shape y(x,0)." Find the error in this reasoning, state the correct number and identity of the initial conditions the wave equation needs, and use a plucked string (released from rest) and a struck string (starting straight but with a given velocity profile) to show that shape alone is not enough.