Telegraph equation: two types of harmonic waves, a discontinuity wave, and a spectral finite element
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The telegraph equation \(\tau \partial ^2 u/\partial t^2 + \partial u/ \partial t = \tau c^2 \partial ^2 u / \partial x^2\) arises in studies of waves in dissipative media with a damping coefficient \(1/\tau \), or from a Maxwell–Cattaneo type heat conduction with a relaxation time \(\tau \). To elucidate basic properties of this equation, two harmonic wave solutions are compared: (1) temporally attenuated and spatially periodic (TASP) and (2) spatially attenuated and temporally periodic (SATP). The phase velocities of both waves are equal to the energy velocities and less than the group velocities. The phase velocities of the two waves are different, and less than c, but both naturally lead to a speed c for the propagation of discontinuities. The two harmonic wave solutions are suitable for different initial-boundary value problems: TASP for those with space periodicity and SATP for those with time periodicity. The asymptotic behaviors of the harmonic wave solutions when the telegraph equation transitions into a nondissipative wave equation or into a parabolic diffusion equation are presented. Only the SATP waves survive when the equation turns parabolic. The spectral finite element method is formulated for 1d Maxwell–Cattaneo heat conduction based on the SATP wave solutions. The element thermal conductivity matrix is reduced to that for a conventional (nonspectral) finite element when the frequency tends to zero.
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This work was partially supported by the NSF under Grants CMMI-1462749 and IIP-1362146.