I’ve experienced this issue many times. A new email arrives, or some other such event that ought to trigger and audible alert from my Mac, but I get nothing but silence.
Until Apple decides to fix this long-standing problem, the Mac Observer has one possible solution. They suggest unplugging any external speakers that use the headphone jack, and then plug them back in. In some cases, it works. But not reliably. My Mac has failed to produce an alert noise after waking from sleep, and even without any external speakers connected.
Of course, it could be settings. In Mail>Preferences>General, check the box for “Play sounds for other Mail actions.” Then in System Preferences>Sound>Special Effects, turn the Volume for “Alerts” all the way up.
There also appears to be problems with a stuck internal microswitch. You can see if that’s a problem by looking for a red LED in the headphone port. Try plugging and unplugging your speakers or headphones about half a dozen times to fix it.
Apple support boards have been filled with people talking about these problems for years. There’s a variety of fixes, but they don’t seem to stick. Apple, can you hear us? Perhaps Cupertino’s alert sounds are muted, too.