CACE Wave Equation and Standing Waves. (a) Prove by direct...

CACE Wave Equation and Standing Waves. (a) Prove by direct substitution that $y(x, t)=\left(A_{\mathrm{SW}} \sin k x\right) \sin \omega t$ is a solution of the wave equation, Eq. $(15.12),$ for $v=\omega / k$ . (b) Explain why the relationship $v=\omega / k$ for traveling waves also applies to standing waves.

Key Concepts

Wave Equation

The wave equation is a fundamental second?order partial differential equation that describes how waves propagate through a medium. It relates the second spatial derivative of a wave function to its second temporal derivative, often taking the form ?²y/?x² = (1/v²) ?²y/?t², where v is the speed at which the wave travels.

Standing Waves

Standing waves arise from the superposition of two traveling waves moving in opposite directions. This interference produces fixed nodes and antinodes, such that the spatial and temporal parts of the wave can be separated, revealing behavior where the wave does not appear to transport energy in the same way as traveling waves.

Dispersion Relation

The dispersion relation links the angular frequency (?) and the wave number (k) through the wave speed (v), typically expressed as v = ?/k for non-dispersive media. This relation is crucial because it connects time-dependent and space-dependent characteristics of a wave, and it holds true for both traveling and standing waves when the media do not exhibit dispersion.

Direct Substitution Method

Direct substitution is a verification technique where a proposed function is substituted into a differential equation to check whether it satisfies the equation. In the context of wave phenomena, this method confirms that the function’s spatial and temporal derivatives align to satisfy the wave equation, thereby validating the proposed solution.

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