
Anything repeated ad nauseum is annoying, even Grace Slick's classic White Rabbit. Source: Wikipedia
This morning I received the following email query (sender's name withheld to protect the guilty):
"I know what I'm asking is probably against FCC regulations and all, but I have this really obnoxious old man neighbor who blasts sappy classical music all day long. I am wondering what it takes to broadcast white noise or even better John Denver easy listening hits, the twenty feet up to his radio."
First, recognize that the phrase "sappy classical music" is an oxymoron. The word "classical" means the music (or anything else it is applied to) has been time tested to be of superior quality. The word "sappy" implies that an objective test would determine it to be of inferior quality. You can't have it both ways.
On the other hand, anything repeated ad nauseam is annoying. For example, I just went through a period of having Grace Slick's White Rabbit running through my head - the whole thing, start-to-finish, in order, endlessly repeated.
It took two weeks (because I really, really like that piece) for the thing to become intolerable, but it did. In the end, I had trouble getting to sleep because of it, dreamed of it when I finally got to sleep, and woke up to it in the morning. I finally had to spend a night in a karaoke bar to get rid of it.
But the answer to your question is: yes, it is possible to overwhelm your neighbor's radio receiver (assuming he's using a radio, rather than playing recorded music). It's called "jamming" and, as you surmise, is highly illegal. The trick is to first determine the frequency he's listening to, the modulation method carrying the signal, and the radio signal's field strength at his antenna's location.
If the radio station uses amplitude modulation (unlikely) or frequency modulation (very likely), you can (but should not) interfere with that signal. If it's a digital broadcast you have a snowball's chance in the hot place of effectively jamming it, though. In any case, you have an approximately 100% chance of getting caught.
To do so, obtain a white-noise source by amplifying the thermal noise naturally generated by, say, a 1,000 ohm resistor until it has a useful amplitude (1-10 Volts). That will require a voltage amplifier with a gain of a few thousand. The John Denver idea would likely be less effective, but would be easier to accomplish - just use the output from your CD player.
Then, build (you cannot buy without having a broadcast station license, which you would not get) a radio transmitter that will broadcast at the radio station's frequency using the same modulation method the radio station uses. The transmitter must put out enough power to create an RF field at least 2X greater than that of the radio station at the receiver's antenna. Pipe the noise source into the transmitter's modulation input, and turn the thing on. Finally, wait for the knock on your door from the FCC's Spectrum Enforcement Division. It should take a day or so for the complaints (you'll also interfere with everyone else's radio in the neighborhood, as well as your own) to reach the FCC, and then about 20 minutes for them to trace the interference to your door.
You probably don't have the requisite skills to do the job. The easiest way to get them is to go back to college and get a degree in electrical engineering with a specialty in radio frequency design. At that point, you'll have a very lucrative career beckoning, and won't want to throw it away by antisocial behavior involving RF electronics. Instead, you'll take the best offer, make lots of money, and move to a very nice suburban home where your neighbors will be better behaved.
In the short run, it would be better to knock on the guy's door and ask him to either turn the music down, close his windows, or get some headphones. If that fails, or if you don't want to be confrontational, call the local police and ask them to do it for you. That, after all, is what you pay them for. Disturbing the peace is illegal, too.

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