Has it been proven that being exposed to WiFi waves is harmful?
WIFi is non-ionising radiation and so has similar issues to other radiation using similar frequencies such as mobile telephones and microwave ovens. These produce heating effects. WiFi is not focused, so any impact should be very small and perhaps not measurable.
I am not aware of any health studies specifically on WiFi. There have been studies on mobile phones which has shown that while the phone is in use and held next to the head, there is small but measurable heating effect on human tissue. My guess is that it has less impact than standing at right angles to the Sun so one side of the head gets warmer faster than the other. Even then, these studies have produced no evidence that this has any health impact, positive or negative:
The Sun does emit ionising radiation (ultra violet) and that has significant health effects such as sunburn, pigmentation changes and Vitamin D production. WiFi's impact, if anything, is nothing like this.
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I will take a moment to expand a bit. Currently Henry has pointed out the only affect that non-ionizing radiation can have, heating.
People commonly perceive this "heating" effect as a threat which will not amount to more then a hundredth of a degree at most, and usually significantly less. If minor heating is causing cancer, should we not be working as a society to wipe out anything that could possibly give a fever? A fever which will give 6 or 8 degrees of heating would dwarf any exposure to low power RF fields as there are orders of magnitude of difference and your body allows this heating.
Your blood is an amazing conductor of heat. When working with RF on radar systems (where our output power was continuous and very very high), there was danger to a human standing within 3 feet of our antenna. This danger is significantly increased if you are very overweight and we are irradiating fat. Adipose has poor circulation and will not naturally cool itself, as opposed to muscle and many other tissues which have good circulation and will naturally cool if they start to heat. On that note, the output power we were dealing with amounted to more than 100,000 times the power a cell phone will emit.
Other Forms of Electromagnetic Radiation
Most people do not realize that at a much higher frequency EM radiation is what we call light. These photons are much higher power then anything at the microwave range and pose a significantly increased risk for heating. Really, I think we should start with a sun-shield around the earth before we start on the evils of WiFi.
Thought I would add some information, I am sure everyone will enjoy the absence of links. The bulk of this is first hand experience from someone in the field, please forgive me for sharing without googling and finding someone else whom has published the facts also.
I am unaware of any research done on this specific question. The closest I've come across was this Dutch study, which states that there may be a connection between Wi-Fi transmissions and tree sickness. The research in itself was not sufficient to make that conclusion.
There are people who report a negative feeling when exposed to Wi-Fi, but those claims don't seem to face a proper double-blind testing, see for example:
Personally, based on my understanding of physics, I doubt that Wi-Fi radiation is harmful. The explanations that fallow are not 100% accurate or bullet-prof, but they should give the general idea.
Wi-Fi transmits at the 2.4 GHz frequency range; that's close to you microwave oven in the kitchen - and it has very much the same effect, that is it heats you up a little bit. Now, while that may be in theory a harmful effect, it should be noted that the change in temperature is much less than due to, say, a having a little warmer weather (I'm cheating a little bit here).
But maybe we should look at it in a different way. This is a nice table from xkcd:
While admittedly not the most solid source of information, I do believe it can give the general notion. Look at the lowest dose of radiation that any research ever found to be linked with cancer (100 mSv), and compare how much radiation you are exposed to on your daily life.
The difference is quite large. So even with the (rather small) addition of radiation from Wi-Fi, you should be fine.
I did find one study about WiFi and trees. Conclusion translated to English from Dutch:
Also per this study by the California Department of Health Studies -- be advised, this is about EMFs from power lines, and I'm not clear how much stronger / weaker those are compared to radio frequency EMFs from cell phones and wifi:
As you can see from the "degree of certainty" from these professionals, the jury is very much out on this one.
In my opinion in the absence of even mildly compelling proof of danger, I don't think there is harm from reasonable, typical exposure to WiFi.