Hennessey, who had been on the verge of a first-team return following cruciate ligament surgery, injured his knee during training on Monday.
The 25-year-old underwent a second operation on Thursday and has been ruled out for up to 11 months.
"It's a tragedy for Wolves and it's a tragedy for Wayne," Solbakken said. "He's had the operation and everything should be fine. There was nothing unusual in what they found. He will probably be out for nine, 10 or 11 months.
"The plan will be exactly the same as he did last time - to get back ahead of schedule, because he did everything right.
"To say he's upbeat isn't the right word because he doesn't want to go through all that injury period again, but he's a strong character and can bounce back."