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Duhamel worked on partial differential equations and applied his methods to the theory of heat and to acoustics. His acoustical studies involved vibrating strings and the vibration of air in cylindrical and conical pipes. His techniques in the theory of heat were mathematically similar to Fresnel's work in optics.
His theory of the transmission of heat in crystal structures was based on work of Fourier and Poisson. 'Duhamel's principle' in partial differential equations arose from his work on the distribution of heat in a solid with a variable boundary temperature.
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