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The element thermal conductivity matrix is reduced to that for a conventional (nonspectral) finite element when the frequency tends to zero. The accuracy of a pitch is what musicians call intonation. The spectral finite element method is formulated for 1d Maxwell–Cattaneo heat conduction based on the SATP wave solutions. Especially at the end of wave 5, a trade can catch all wave A-B-C and profit hundreds of pips. When harmonic patterns show up at the end of wave 3 or 5, we have a perfect setup to trade. Only the SATP waves survive when the equation turns parabolic. The harmonic pattern might only appear at the end of wave 5 but not wave 3 or vice versa. 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. 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 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 harmonic waves can be added and subtracted. The phase velocities of both waves are equal to the energy velocities and less than the group velocities. Harmonic Waves: The equation of waves which repeats after a certain interval this repetitive motion is known as a harmonic wave. 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 telegraph equation τ ∂ 2 u / ∂ t 2 + ∂ u / ∂ t = τ c 2 ∂ 2 u / ∂ x 2 arises in studies of waves in dissipative media with a damping coefficient 1 / τ, or from a Maxwell–Cattaneo type heat conduction with a relaxation time τ.