= Solution
The <wave equation> is second order in time. Even one <normal mode> obeys a second-order <ordinary differential equation>, so its initial displacement and initial velocity are independent data. Physically they specify stored elastic deformation and the initial motion.
The <heat equation> is first order in time; its initial <temperature> already fixes the instantaneous time derivative through the spatial <Laplacian>. It needs one initial field. In either case spatial <boundary conditions> remain necessary and are not substitutes for initial data. Classical fixed-end string data must satisfy the appropriate endpoint compatibility, including $f(0)=f(L)=0$ and, when endpoint velocities exist, $g(0)=g(L)=0$.
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