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The loss factor and the wave number are replaced by an expression depended by the damped frequency. Could the last expression be a valid solution for the underdamped wave equation when the initial frequency is known?

\begin{align} \omega_d = \sqrt{\omega_0^2-(lc)^2} \\ \frac{2 \pi}{\lambda_d} = \omega_d /c \\ y[z,t] = A e^{-\sqrt{\omega_0^2-\omega_d^2} z/c} \cos(2 \pi \frac{z}{\lambda_d} - \omega_d t) \\ y[z,t] = A e^{-\sqrt{\omega_0^2-\omega_d^2} z/c} \cos(\omega_d z/c - \omega_d t) \end{align}

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