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- Using sin a + sin b = 2 cos ((a - b) / 2) sin ((a + b) / 2):
A sin(k x - ω

This solution, with ω_{1}t) + A sin(k x - ω_{2}t + φ)= 2A cos((ω

_{1}- ω_{2}) t / 2 + φ / 2) sin(k x - (ω_{1}+ ω_{2}) t / 2 + φ / 2)_{2}= - ω_{1}(a superposition of a left mover and a right mover of equal frequency), is a**standing wave**:2A cos (ω

The value of the wave number is constrained by the_{1}t + φ / 2) sin (k x + φ / 2)**boundary conditions**. For a standing wave of length L:**closed-closed**: k = n π / L (n*half*wavelengths fit in L)**closed-open**: k = n π / (2 L), n odd (n*quarter*wavelengths fit in L)**open-open**: k = n π / L, φ = π

Think of the second factor as describing a "static standing wave", and the first factor as its time-dependent amplitude.

- At a fixed position (here with k x = π / 2 and φ = 0), with ω
_{1}independent of ω_{2}, we have**beats**2A cos((ω

with_{1}- ω_{2}) t / 2 ) cos((ω_{1}+ ω_{2}) t / 2)**beat frequency**ω_{1}- ω_{2}:Here, the first factor describes the "envelope" and the second describes the oscillation inside; the picture shows one

*half*of an envelope cycle, which is heard as a single beat. - This applet will demonstrate beats.
This one will demonstrate standing waves (the
**mode**is "n" in the boundary constraints). This applet demonstrates the constructive and destructive interference in traveling waves.For

**constructive interference**, the phase difference must be an even multiple of π.

For**destructive interference**, the phase difference must be an odd multiple of π.

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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