Wave Equation & Harmonics: Traveling waves satisfy $$\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$$ with harmonic profile $$y(x,t) = A\sin(kx - \omega t + \phi)$$, where wave number $$k = \frac{2\pi}{\lambda}$$, angular frequency $$\omega = 2\pi f$$, and speed $$v = \lambda f$$.
Superposition & Standing Waves: Counter-propagating waves form standing waves $$y(x,t) = [2A\sin(kx)]\cos(\omega t)$$ featuring stationary nodes ($$y = 0$$) and antinodes ($$y = 2A$$).
Doppler Frequency Shift: For relative motion between source and observer in a medium with wave speed $$v$$: $$f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$$.
Operating Instructions: Use the tab switcher above to explore Sound wave harmonics, 2D Wave interference patterns, Standing wave nodes, and Doppler pitch shifts.