How is wavelength defined for standing waves? I cannot understand why the first harmonic is half a wavelength rather than a full wavelength. I see how wavelength is defined for progressive waves but I cannot apply this to standing waves.
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A standing wave is the sum of two waves with wave numbers $k_1$ and $k_2$, with the condition that:
$$ k_1 = -k_2 $$
so we'll say:
$$k = ||k_1|| = ||k_2|| $$
The wavelength is then:
$$ \lambda = \frac{2\pi}k$$
A full wavelength has 3 nodes (inclusive) and 2 anti-nodes. You only need 2 nodes to get a standing wave on a string, so the length is $\lambda/2$.
JEB
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