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?
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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