Question

7. Based up what we showed in a previous homework set, the displacement of a damped harmonic oscillator that is initially displaced to $x = A$ is given by $x(t) = egin{cases} 0 & t < 0\ Ae^{-t/ au} sin omega_0 t & t ge 0 end{cases}$ Find the Fourier transform of this function, $ ilde{x}(omega)$. Sketch $| ilde{x}(omega)|^2$ – this essentially shows how energy is distributed over different frequencies in this system.

          7. Based up what we showed in a previous homework set, the displacement of a damped harmonic oscillator that is initially displaced to $x = A$ is given by
$x(t) = egin{cases} 0 & t < 0\ Ae^{-t/	au} sin omega_0 t & t ge 0 end{cases}$
Find the Fourier transform of this function, $	ilde{x}(omega)$. Sketch $|	ilde{x}(omega)|^2$ – this essentially shows how energy is distributed over different frequencies in this system.

7. Based up what we showed in a previous homework set, the displacement of a damped harmonic oscillator that is initially displaced to x = A is given by
x(t) = egincases 0 t < 0 Ae^-t/ au sin omega0 t t ge 0 endcases
Find the Fourier transform of this function, ildex(omega). Sketch | ildex(omega)|^2 – this essentially shows how energy is distributed over different frequencies in this system.

Added by Nicholas B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shaiju T

04/25/2022

Step 1

The Fourier transform of a function f(t) is given by the integral from -∞ to ∞ of f(t)e^(-iωt) dt. So, we need to compute the integral from -∞ to ∞ of Ae^(-t/τ) sin(ωt + φ)e^(-iωt) dt. This integral is a bit tricky to compute directly, so we'll use the Euler's Show more…

Show all steps

Thanks for your feedback!

Based on what we showed in the previous homework set, the displacement of a damped harmonic oscillator that is initially displaced to x(0) = A is given by: x(t) = Ae^(-t/̄̄) sin(̄̄t + ̄̄) Find the Fourier transform of this function, Z(̄̄). Sketch |Z(̄̄)|²; this essentially shows how energy is distributed over different frequencies in this system.