I found the new Google Hangouts sounds annoying and they don't provide a way to disable them. The only thing you can do is disable notifications entirely, per contact even, not globally.
How can you just mute the sound alerts for new messages?
Google now offers an option to mute sounds:
This setting applies to all Hangouts notifications from Gmail, Google+ and the Chrome extension on the same desktop computer.
Turn sound off for notification sounds
- Click menu ▼ at the top of your Hangouts list.
- Check the box next to "Mute."
Edit 2013/12/16: they added an option to mute all sounds, indefinitely. Finally! See this answer
As of today (2013/05/21), you can't (even though it's requested), but I found a way to prevent the sound to be loaded and therefore played.
In short: I redirect the link to the alert audio file (which you can find at the bottom of the answer) to a non-existent one.
This is how:
Text to enter:
To see if it works, try this url, it shouldn't load
Use your favourite adblock plugin to block the URL to the mp3, which at the time of writing is:
In Chrome this appears to only work in web interfaces, and not in the browser extension.