Short answer: Coral doesn't need the light, it's the zooxanthellae that is symbiotic with some corals that relies on light to grow; it adapts to the frequency spectrum available at the depth the particular coral grows, full spectrum at shallow depths and at over ~30m the zooxanthellae relies on blue light because other frequencies are absorbed by water.
Some species of coral, such as Lophelia pertusa, grow in water so deep that there is too little light for them to rely on zooxanthellae, instead their polyps gather food directly with their tentacles, feeding heterotrophically.
There are even instances of coral feeding on jellyfish, see: "Opportunistic feeding by the fungiid coral Fungia scruposa on the moon jellyfish Aurelia aurita" (May 21 2009), by A. Alamaru, O. Bronstein, G. Dishon and Y. Loya
The physics portion of your question is best answered on our Physics.SE site's question: "Why doesn't the frequency of light change during refraction?"
Light's wavelength and frequency are related to each other based on a mathematical formula, it's the intensity that is affected by passing through a medium; different materials will attenuate some frequencies more than others. Water doesn't transpose light's frequency or wavelength as much as it absorbs (filters) the frequencies of light affecting which color can penetrate deeper.
"What is a light wave?
Light is a disturbance of electric and magnetic fields that travels in the form of a wave. Imagine throwing a pebble into a still pond and watching the circular ripples moving outward. Like those ripples, each light wave has a series of high points known as crests, where the electric field is highest, and a series of low points known as troughs, where the electric field is lowest. The wavelength is the distance between two wave crests, which is the same as the distance between two troughs. The number of wave crests that pass through a given point in one second is called the frequency, measured in units of cycles per second called Hertz. The speed of the light wave equals the frequency times the wavelength.".
See this other Physics.SE question: "Does light color change when refracting?":
"The color will not change. What you're not taking into account is the speed of light in the medium. It's not the same $c$ en vacuo. The frequency stays the same. What changes is that speed of light in the refracting medium and as a result wavelength.".
Here is what sunlight (natural light) looks like underwater:
Despite that fact in shallow water enough full spectrum light penetrates sufficiently, and the zooxanthellae adapts to the spectrum available, no longer requiring blue light for growth.
See: "The spectral quality of light is a key driver of photosynthesis and photoadaptation in Stylophora pistillata colonies from different depths in the Red Sea" by T. Mass, D. I. Kline, M. Roopin, C. J. Veal, S. Cohen, D. Iluz, O. Levy in the Journal of Experimental Biology 2010 213: 4084-4091; doi: 10.1242/jeb.039891
"Depth zonation on coral reefs is largely driven by the amount of downwelling, photosynthetically active radiation (PAR) that is absorbed by the symbiotic algae (zooxanthellae) of corals. The minimum light requirements of zooxanthellae are related to both the total intensity of downwelling PAR and the spectral quality of the light. Here we used Stylophora pistillata colonies collected from shallow (3 m) and deep (40 m) water; colonies were placed in a respirometer under both ambient PAR irradiance and a filter that only transmits blue light. We found that the colonies exhibited a clear difference in their photosynthetic rates when illuminated under PAR and filtered blue light, with higher photosynthetic performance when deep colonies were exposed to blue light compared with full-spectrum PAR for the same light intensity and duration. By contrast, colonies from shallow water showed the opposite trend, with higher photosynthetic performances under full-spectrum PAR than under filtered blue light. These findings are supported by the absorption spectra of corals, with deeper colonies absorbing higher energy wavelengths than the shallow colonies, with different spectral signatures.".
"Fig. 4. Percent surface irradiance received of different wavelengths at 3 m (grey) and 40 m (black) depth in the waters adjacent to the Interuniversity Institute for Marine Sciences, Eilat, Israel, on 3 April 2008.".
So if you deep dive for your own coral you should be aware that it will thrive under blue light. If you obtain shallow water coral, or buy from the local aquarium supply store, white light is suitable.
Also from: "The spectral quality of light is a key driver of photosynthesis and photoadaptation in Stylophora pistillata colonies from different depths in the Red Sea", (full info above):
"Studies in the Caribbean have found that other corals that are found along a broad depth gradient switch from hosting clade A or B zooxanthellae in shallow water to clade C in deeper water, possibly owing to the ability of the different clades to adapt to different light levels (Rowan and Knowlton, 1995; Baker et al., 1997).".