Wave Reflection

PDEs · Axiom Academy

Understanding how waves behave at boundaries and interfaces 1. Reflection from a Fixed Boundary When a wave on a string reaches a fixed endpoint (clamped boundary), the displacement must remain zero at that point. To satisfy this boundary condition, the wave reflects with a phase shift of π, creating an inverted reflection . The incident wave u i (x,t) = A sin(kx - ωt) traveling in the +x direction reflects as u r (x,t) = -A sin(-kx - ωt), with the negative sign indicating inversion. 2. Reflection from a Free Boundary At a free endpoint (where the string can move freely but has zero tension gradient), the boundary condition requires zero slope: ∂u/∂x = 0. This produces a non-inverted reflection with no phase shift. The reflected wave has the same sign as the incident wave, doubling the amplitude at the free end. This is why the amplitude at a free boundary is maximum. 3. Method of Images for Semi-Infinite Strings The method of images is a powerful technique for solving wave problems on semi-infinite domains (x ≥ 0). We extend the solution to the entire real line by placing an appropriate "image" source on x < 0. For a fixed boundary at x = 0: place an inverted image source at -x 0 when the actual source is at x 0 . For a free boundary : place an upright image source at -x 0 . The superposition of actual and image waves automatically satisfies the boundary condition. 4. Standing Waves from Incident and Reflected Waves

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