What about other sources of sound: e.g., system sounds, youtube, CD, online music sources?
If it is truly a problem with the game then there is either a bug in the software or some related configuration setting is astray.
First check the game configuration to ensure that the correct audio output is selected/enabled.
Go into Playback devices via the little speaker icon (probably) lower right screen corner and work your way down through all the tabs and properties. There may be some Runescape setting therein that has been overlooked.
Run "Troubleshoot sound problems" - that may find something.