Prove that for any sine function sin(kx + φ) of wavelength shorter than 2a, where a is the atomic spacing, there is a sine function with a wavelength longer than 2a that has the same values at the points x = a, 2a, 3a, and so on. (Note: It is probably easier to work with wave number than with wavelength. We seek to show that for eveiy wave number greater than π/a there is an equivalent one less than π/a.)
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