A localized displacement in a three-mass chain

Three identical masses \(m\) move horizontally without friction. Adjacent masses are joined by identical springs of constant \(k\), and the two end masses are also joined to fixed walls by springs of the same constant. Let \(\mathbf x=(x_1,x_2,x_3)^T\) measure displacement from equilibrium. Derive the matrix equation \(m\ddot{\mathbf x}+K\mathbf x=0\). Find all three exact normal-mode frequencies and a Euclidean-orthonormal eigenvector for each. Connect your result with the fixed-end formula \(q_j^{(p)}\propto\sin(jp\pi/4)\), but verify the vectors directly rather than quoting the formula alone. The chain is released from rest with \[ \mathbf x(0)=(a,0,0)^T,\qquad \dot{\mathbf x}(0)=\mathbf0. \] Use modal projection to write every component \(x_j(t)\) explicitly. Check both initial conditions, identify which modes have a stationary middle mass, and prove from the modal coordinates that total energy is constant. State why the modal projection is Euclidean here but would become mass-weighted for unequal masses.

The stiffness matrix is \(k\) times the tridiagonal matrix with diagonal entries two and neighboring entries minus one.