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My WiFi router is located inside a wooden box (1 cm, 0.4 inch thick) for aesthetic reasons. How much does the wood weaken the signal? What is the physical explanation to the answer?

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This depends on the type of wood. Putting unconfirmed jokes aside, you can verify this yourself using a program like inSSIDer (linux version here) and compare the signal strength and signal to noise ratio's before and after putting it in the box.

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Wood will not weaken the signal as it is not electrically conductive.

If the box were made of metal and sealed and grounded, all your signal would be blocked. Such a device is called a Faraday cage. If the metal box were ungrounded and had an open side, the signal would be redirected in and out of the open side, depending on the exact dimensions of the box and its shape. Under the right conditions you can get the signal to amplify in one direction. This is the principle upon which satellite dishes work.

If the wooden box were assembled with metal screws, nails, or staples, these would have an effect on the radiation pattern. Testing the exact signal attenuation would require an FCC Class-C testing facility. This is a wooden building constructed with no nails or screws, located away from radiation sources like TV and Radio transmission towers. The DUT is turned on and the radiation pattern recorded on a spectrum analyzer while rotating the DUT.

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