Physic Labs
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
problems.proof.intuition

Like both ends of a guitar string pinned to the frame — they cannot move at all, even while the middle of the string vibrates violently.

y(0,t)=0,y(L,t)=0y(0,t) = 0,\quad y(L,t) = 0
problems.proof.analysis

Both ends of the string are clamped, so the displacement at x=0x=0 and x=Lx=L must be zero at every instant tt — this is precisely the definition of a node.