We have encountered the same problem at some point; you search for a video and by the time you finally find the exact video you were looking for, it has been removed due to some copyright infringement of the audio. Youtube realizes that this is a problem and is now trying to do something about it.
Youtube now contains an audio library of royalty free tracks to choose from. Currently there are only 150, but I'm sure it is growing daily. These songs are available for any use, not just Youtube videos. They can be downloaded forever to your computer as an Mp3 file.
Although this is an awesome tool for anyone in the media industry, it still has major flaws. 150 is a relatively small number, even for a free audio library. The songs can only be played with a start and stop button, with no option for skipping to another part of the song.
Obviously a paid service would better streamline the process of selecting an audio file, but lets be honest, who pays for music anymore? This is an awesome resource that I am very excited about using in the future, and I'm sure Youtube will be providing frequent updates to the library and video process.