T-38 flight crew reported a rumbling and high frequency vibrations were felt in cruise. After landing; ground crew noticed the left stab was sheared off aft of the axis of rotation.

Narrative

During cruise operations at approximately FL370; a distinct rumbling and high frequency vibration in the aircraft was felt by both crew members. Flight surfaces; engine parameters; hydraulics and other indications were checked with nothing off nominal noted. Flap and ailerons were confirmed normal visually via direct view from the RCP and flap; ailerons and stab were checked via the mirror from the RCP with nothing noted. To troubleshoot the engines; right engine throttle was retarded from MIL to midrange and returned to MIL slowly. During the movement back to MIL; an additional; distinct and quick onset rumbling and vibration of lower frequency was felt; to include a wing dip and overall aircraft uncommanded motion. This felt more like standard heavy turbulence and lasted for about 1-2 seconds. After the increased vibrations; the right engine temperature went to 700 as it reached MIL then settled into a normal temperature. No further vibrations of high or low frequency were felt after the increased vibration stopped. The same throttle retard and return to MIL was done on the left engine with nothing abnormal noted. Crew decided to RTB and quickly reached final altitude of FL380 and requested this change with ATC. The right throttle was put in IDLE due to suspected right engine issues. All indications in IDLE were normal. Later after a switch to next controller; assistance was offered and we declined need for assistance. Upon reporting situation to Operations; we were advised to attempt operation of the right engine to MIL. During our descent from FL380 to FL220; we put the right engine in MIL. Engine performed normally and no vibrations were noted; so rest of flight was conducted two-engine as normal. Rest of flight was uneventful and no further vibrations or rumblings were felt. One exception is that a pop that seemed similar to but louder than a usual ECS system pop was heard at an altitude of ~10k ft on the arrival.The clearance given from approximately 17 NM east of ZZZ to ZZZ1 was direct ZZZZZ then the ZZZ STAR at FL330. Later cleared to FL250 and eventually 14k-12k by ZZZZZ1. Descent between FL250 and 12k was rapid and speed brakes were used. We were taken to a heading off the STAR soon after crossing ZZZZZ1 and soon after; given a speed restriction of 230 KIAS. We proceeded via vectors to the RNAV XXL. We were given a speed restriction of 170 KIAS before being cleared for the approach at/near ZZZZZ2. Configured to gear down/flaps 60 at this time with nothing abnormal noted. Configured to flaps full at approximately 250 feet. Touchdown speed was approximately 145 KIAS and we exited XXL on Charlie taxiway after normal aerobraking and braking. Nothing was noted during Flap return to 0 in after landing checklist.Upon shutdown; lineman noticed the left Stab was sheared off aft of the axis of rotation.