Boundary Conditions for Waves

PDEs · Axiom Academy

LESSON Boundary Conditions for Waves Understanding how boundaries shape wave behavior and determine natural frequencies 1. Fixed (Dirichlet) Boundary Conditions The most common boundary condition is fixed ends , where the displacement is zero at the boundaries. This models strings that are clamped at both ends, like a guitar or violin string. These conditions force the wave to have nodes (points of zero displacement) at both ends. Watch how the wave must vanish at x = 0 and x = L: 2. Free (Neumann) Boundary Conditions Free ends occur when there's no constraint on displacement, but the slope (derivative) vanishes at boundaries. This models situations where ends can move freely but experience no external force. Free boundaries create antinodes (points of maximum displacement) at the ends. The wave slope must be horizontal at x = 0 and x = L: Real systems often have different conditions at each end . For example, a string fixed at one end but free at the other, or attached to a spring or damper. Mixed conditions create asymmetric modes. Watch how the wave behaves with one fixed end (left) and one free end (right): When a domain extends infinitely in one direction (x > 0), we need a boundary condition at x = 0 plus a boundedness condition as x → ∞. This models waves on a semi-infinite string or vibrations in a half-space. Watch how waves can travel into the infinite domain: 5. Effect on Normal Modes and Frequencies

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