Standing Waves

PDEs · Axiom Academy

Understanding wave superposition, nodes, antinodes, and normal modes in vibrating systems 1. Superposition of Traveling Waves A standing wave forms when two traveling waves of equal amplitude and frequency move in opposite directions. Consider two waves: When these waves superpose, they interfere constructively at some points and destructively at others, creating the standing wave pattern. Superposition means the total displacement is the sum of individual displacements: u(x,t) = u₁(x,t) + u₂(x,t) 2. Mathematical Form of Standing Waves Adding the two traveling waves using the sum-to-product identity, we obtain: This is the standard form of a standing wave. Notice how the spatial and temporal parts separate - the amplitude A sin(kx) is modulated by cos(ωt). This means every point oscillates in phase, but with different amplitudes. Standing waves have characteristic points that define their structure: Points where the amplitude is always zero. These occur when sin(kx) = 0, giving x = 0, λ/2, λ, 3λ/2, ... Points where the amplitude is maximum. These occur when |sin(kx)| = 1, giving x = λ/4, 3λ/4, 5λ/4, ... 4. Normal Modes for a String with Fixed Ends For a string of length L with both ends fixed (like a guitar string), the boundary conditions require u(0,t) = 0 and u(L,t) = 0. This means nodes must exist at both ends. The allowed wavelengths must satisfy: where n = 1, 2, 3, ... is the mode number. Each value of n represents a different normal mode or harmonic.

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