http://forums.androidcentral.com/verizon-galaxy-s-iii/204407-turn-off-audible-notification-new-email.html

If this is for your gmail account do the following:

  • Open gmail
  • Hit the menu button and select settings
  • Tap the account you want to turn notifications off for
  • Tap on Ringtone & Vibrate (“Sound & vibrate” after Gmail updates”)
  • Tap on Ringtone and select Silent, then tap OK
  • Hit the back button until it takes you back to your inbox. No more audible notifications

For the standard email client do the following:

  • Open Email
  • Hit the menu button and select Settings
  • Tap the account you want to turn notifications off for
  • Scroll down to Notification settings and tap on Select ringtone
  • Select Silent and tap OK
  • Hit the back button until it takes you back to the inbox. No more audible notifications