Can light have zero wavelength? As you increase the energy of a photon it's wavelength shortens. Is it theoretically posible for light to not have a wavelength? Like a still pond?
 A: The relationship between energy and wavelength:
$$E = h f = \frac{h c}{\lambda}$$
As $\lambda$ goes to zero, $E$ goes to infinity.
So "no".
A: In case of photon's wave nature they have definite wavelengths for definite energies. If wavelength become zero then its energy become infinite which is impossible.
Secondly, every wave must have wavelength which defines its motion. If wavelength become zero then wave become motionless.
A: How about doing it the other way? Quote "The earth is a magnet, and it is accelerating as it rotates the Sun, so it radiates EM waves with a period of 365 days and a wavelength of 1 light year." The frequency would be so low as to have no detectable wavelength. Like a still pond.
A: A still pond would be equivalent of infinite wavelength and the frequency of 0. The equivalent of 0 wavelength would be an infinitely tall tsunami wave in the pond. It is not possible to have wavelength at exactly 0, but you can get arbitrarily close to 0. Proof: any positive non-zero wavelength, no matter how small, can be made smaller via blueshift (approaching towards the source). At some point, though, you will be limited by the amount of energy needed to gather in order to create such radiation.
