It takes a brave theater critic to write a play, and an even braver one to review it - especially since

Satchmo at the Waldorf

is by Terry Teachout, the esteemed critic of the Wall Street Journal. So it's both a pleasure and a relief to tell you it's a great show.

This add-on to Wilma's season comes from Long Wharf Theatre in New Haven, Conn., directed by Gordon Edelstein. It stars John Douglas Thompson, who plays Louis Armstrong, the world's greatest trumpet player - and also plays Armstrong's manager, as well as the musicians of the next generation, Miles Davis and Dizzy Gillespie, who think Armstrong is an Uncle Tom, a clown to entertain rich white folks. Thompson shifts impressively between personalities, changing his voice, his accent, his bearing.

Satchmo is in his dressing room backstage at Manhattan's Waldorf Astoria Hotel, a gig so posh he's still amazed at his good fortune, even at the age of 70. His health is failing and the bio-drama's device is to have him reminisce into a tape recorder. We hear about his hard childhood (son of a New Orleans whore), his stint in reform school where he learned to play the horn, through a complicated life involving four wives, the mobsters who ran the nightclubs, and the Star of David he wears around his neck.

The man who was central to his life and career was his manager, Joe Glazer; although we learn the truth of Glazer's apparent betrayal, it is a truth Satchmo will never learn. "Work 'em like dogs, treat 'em like kings - that's how you run a band." He made Satchmo a star.

Most of what comes through is Armstrong's love of music. Of the famous smile that could make audiences smile back: "I'm there in the cause of happiness."

Teachout has written a biography of Armstrong (Pops: A Life of Louis Armstrong), and he packs a lot of material into 90 minutes. Some issues seem unresolved, but that may be the fault of the life, not the script. Teachout is currently writing a biography of Duke Ellington, so stay tuned.

Satchmo at the Waldorf

Through Dec. 2 at the Wilma Theater, Broad and Spruce Streets. Tickets $39-$54. Information: 215-546-7824 or wilmatheater.org

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Follow Toby Zinman on Twitter at #philastage. Read her reviews at www.philly.com/phillystage.