B737NG FO reported a passenger was injured during a wake turbulence encounter with an opposite direction B737 at FL340 in Jacksonville Center airspace.

Narrative

While traveling southbound from ZZZ to ZZZ1 at FL340; [we] encountered the wake turbulence of an opposite direction Boeing 737 traveling at FL350 (the aircraft type was reported by ATC after a flight crew inquiry). [We] requested a climb to FL380 per the flight plan and were cleared by ATC to climb from FL340 to FL380 after the opposite direction traffic had passed by on the right side (west of our track; distance unknown). The light level outside was dusk and the aircraft above and to the right was not producing contrails. Winds were out of the west at 100 kts. No turbulence had been encountered at cruise and the fasten seat belt sign was off. Shortly after the CA (PF) (Captain Pilot Flying) initiated the climb; the aircraft rolled to the left approximately 30 degrees and shook violently from the encounter. The PF disengaged the autopilot; rolled back to wings level; and re-engaged the autopilot to continue the climb. No 'bank angle; bank angle' aural alert occurred. The FO (PM) (First Officer Pilot Monitoring) turned on the fasten seat belt sign and instructed the Flight Attendants over the PA: 'Flight Attendants; take your seats.' (The FO did not use standard phraseology for a turbulence encounter due to the startle effect). The encounter lasted for less than 5 seconds. No deviation from the cleared routing or altitude was noted as the CA was quickly able to regain control of the aircraft. The CA transferred aircraft control and radios to the FO while the CA had the flight attendants check in. Upon check in; the CA learned that a passenger waiting to use the aft lavatory fell into the cover for the escape slide at 2R and twisted her right knee/leg. The CA contacted Dispatch for a discrete frequency and then Medlink to discuss treatment options. In the course of determining the extent of the passenger's injuries; the CA offered diversion to the passenger which the passenger declined. Medlink recommended a cold compress and three acetaminophen tablets from the medical kit. The flight attendants had already applied the cold compress but the passenger refused the acetaminophen. The CA and FO determined the turbulence would have been classified as severe. Since the aircraft was flying normally; there was no concern of structural damage from the short duration event. Nothing other than normal Gs were sensed by the flight deck crew. No turbulence from the event was displayed on [display]. We discussed whether we should request priority handling or declare an emergency. The CA and FO determined the best course of action; based on Medlink's assessment of the injuries and recommendations; was to continue to the destination and have EMTs meet the aircraft at the gate. Normal checklists were completed for the descent and arrival into ZZZ1. We were already flying at a high cost index so we were flying at the fastest speed for the arrival into ZZZ1. ATIS was calling for arrivals to XXL. We requested and were granted to fly an approach to Runway XXR after a brief explanation to ATC. Runway XXR had a shorter taxi distance to the ramp and fewer turns to our gate. After an uneventful landing; the CA taxied as fast as practical to arrive at the gate quickly. EMTs were standing by at the gate. We left the seatbelt sign on until the injured passenger was transported off the aircraft using a straight back wheelchair.