Brown, a headmaster in Paisley, Walter was educated at Allan Glen's School and then studied at the University of Glasgow (BSc Hons Mathematics and Physics 1907; and BSc Pure Science 1910).

He was soon promoted to become Professor in Pure and Applied Mathematics, a post he held from 1918 to 1946.

A member of the Royal Naval Volunteer Reserve, Brown was captured when Hong Kong surrendered to the Japanese, and he was held as a Prisoner of War at Stanley Camp (1941–45).

He organised study groups in the internment camp, and helped attend to the medical needs of the prisoners.

He travelled extensively and died in Marandellas in Rhodesia (now Zimbabwe) in April 1957.