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Why is the wavelength of circular cyclotron radiation different than that emitted by a non-relativistic electron passing through an undulator?

From what I understand, the wavelength of circular cyclotron radiation (for electrons) is dependent only on the strength of the cyclotron’s magnetic field.

$$\lambda = \frac{2\pi m_e c^2}{e B}$$

I’ve read that the radiation emitted by a non-relativistic electron ($\gamma = 1$) passing through an undulator is dependent only on the undulator’s period and the undulator parameter K.

$$\lambda = \frac{\lambda_u}{2\gamma^2}\left(1+\frac{K^2}{2}\right)$$ $$K = \frac{e B \lambda_u}{2\pi m_e c}$$

Isn’t a sine wave just a bunch of alternating circular arcs? I feel like the wavelength of radiation emitted should be either (a) dependent only on the magnetic field strength (like circular motion) or (b) related to its frequency of oscillation (and therefore its velocity/undulator period length)?

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