Born at Stockport, Lamb studied at Cambridge University. His earliest interest had been in classics, but soon after arriving in Cambridge he discovered his true vocation lay in mathematics. Among his teachers were George Stokes and the great mathematical physicist James Clerk Maxwell. Lamb's own interest in mathematics was very much shaped by contact with the ideas of these two men, and was almost entirely in applied mathematics. In 1875 he went to Australia to become the first professor of mathematics at the newly founded University of Adelaide. In 1885 he returned to England to take up the chair of mathematics at Manchester University, and held this post until his retirement in 1920. He was knighted in 1931.

Among the great variety of fields of applied mathematics to which Lamb made important contributions are electricity and magnetism, elasticity, acoustics, vibrations and wave motion, statics and dynamics, seismology, and the theory of tides and terrestrial magnetism. However, it is for his work in fluid mechanics that Lamb is most celebrated. He wrote a book on the subject, Hydrodynamics (1895), which immediately became a classic and by 1932 had gone through six editions. Lamb's work in geophysics was also important. He wrote a paper in 1904 on the propagation of waves over the surface of an elastic solid, in which he virtually laid the whole theoretical foundations for modern mathematical seismology.