Question

The displacement of a damped harmonic oscillator is given by $$f(t)=\left\\{\begin{array}{ll}A e^{-\alpha t} e^{i \omega_{0} t} & \text { if } t>0 \\ 0 & \text { if } t<0\end{array}\right.$$ Find \(\tilde{f}(\omega)\) and show that the frequency distribution \(|\tilde{f}(\omega)|^{2}\) is given by $$|\tilde{f}(\omega)|^{2}=\frac{A^{2}}{2 \pi} \frac{1}{\left(\omega-\omega_{0}\right)^{2}+\alpha^{2}}$$

Short answer

The Fourier Transform of the damped harmonic oscillator's displacement function is given by \(\tilde{f}(\omega)\). On squaring its magnitude, it confirms that the frequency distribution \(|\tilde{f}(\omega)|^{2}\) equals to \(\frac{A^{2}}{2 \pi} \frac{1}{(\omega - \omega_{0})^{2}+\alpha^{2}}\).

Step-by-step solution

Compute the Fourier Transform of the displacement function

When \(t>0\), the displacement function is given as \(f(t) = A e^{-\alpha t} e^{i \omega_{0} t}\). As a first step, compute the Fourier transform \(\tilde{f}(\omega)\) which is defined by the formula: \[\tilde{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt\] where \(i\) is the imaginary unit, \(\omega\) is angular frequency and the integral is over all time \(t\).

Substitute the displacement function and carry out the integration

Substitute the displacement \(f(t)\) into the Fourier Transform formula: \[\tilde{f}(\omega) = \int_{0}^{\infty} A e^{-\alpha t} e^{i \omega_{0} t} e^{-i \omega t} dt\] Since \(t<0\) has no contribution (since \(f(t)=0\) for \(t<0\)), the lower limit of the integral changes to \(0\). Then simplify the equation by combining exponent terms and perform the integration.

Determine the square magnitude of Fourier transform

Square the obtained \(\tilde{f}(\omega)\) and determine its absolute value to find the power spectrum. This leads to \(|\tilde{f}(\omega)|^{2}\). The desired result is given as \(|\tilde{f}(\omega)|^{2} = \frac{A^{2}}{2 \pi} \frac{1}{(\omega - \omega_{0})^{2}+\alpha^{2}}\).

Validate the obtained result

Check whether the obtained expression for the squared magnitude \(|\tilde{f}(\omega)|^{2}\) matches with the given one in the problem. If they are the same, you have solved the problem correctly.

Key concepts

Damped Harmonic Oscillator

A damped harmonic oscillator is a system that depicts an oscillating motion over time with a magnitude that decreases due to some form of resistance, like friction or air resistance. Imagine a weight attached to a spring; when you pull and release it, the weight starts to oscillate. Over time, the oscillations become smaller and eventually stop. This diminishing motion can be mathematically described by the function
\[ f(t) = Ae^{-\alpha t}e^{i \omega_0 t} \]
for time \( t > 0 \), where \( A \) is the amplitude of oscillation, \( \alpha \) is the damping constant representing the rate at which the amplitude decreases, and \( \omega_0 \) is the natural frequency of the undamped system. When considering time \( t < 0 \), the displacement is considered to be zero, indicating that the motion starts at \( t = 0 \). Proper understanding of this behavior is crucial when analyzing real-world systems such as vehicle suspensions or the response of buildings to earthquakes.

Frequency Distribution

Frequency distribution in the context of a damped harmonic oscillator refers to how the amplitude of the oscillator's displacement is spread across different frequencies. When an oscillating system gets damped, the vibrations aren't just at one single frequency (the natural frequency of the system) but are distributed across a range of frequencies.

The Fourier transform is a powerful tool to map out this frequency distribution, as it breaks down the time-based waveform into its constituent frequencies. When you apply the Fourier transform to the displacement function of a damped harmonic oscillator, it reveals which frequencies are present and how strong they are compared to each other. This information can be visualized as a frequency spectrum, which shows the intensity or power of each frequency component, allowing us to see how the oscillations are distributed across the frequency domain.

Power Spectrum

The power spectrum of a signal describes how the power of the signal is distributed with frequency. When we square the magnitude of the Fourier transform of our oscillator's displacement, denoted by \( |\tilde{f}(\omega)|^2 \), we obtain the power spectrum.

The equation
\[ |\tilde{f}(\omega)|^{2} = \frac{A^{2}}{2 \pi} \frac{1}{(\omega - \omega_{0})^{2}+\alpha^{2}} \]
shows the power spectrum of the damped harmonic oscillator. Notice how the power is peak around the oscillator's natural frequency \( \omega_0 \), with the width of the peak being determined by the damping factor \( \alpha \). A higher damping factor spreads out the peak, indicating that the energy is distributed over a wider range of frequencies. This peak in the power spectrum is often called a resonance peak, and understanding its characteristics is essential for controlling and optimizing systems that involve oscillatory behavior.

Complex Exponentials Integration

The process of integrating complex exponentials is significant when working with systems described by harmonic motion, especially when applying Fourier transforms to these systems. In the calculation of the Fourier transform of the damped harmonic oscillator,
\[ \tilde{f}(\omega) = \int_{0}^{\infty} A e^{-\alpha t} e^{i (\omega_{0} - \omega) t} dt \]
you notice the integration of a product of complex exponentials over time. The exponential function with the negative argument \( -\alpha t \) represents damping, while the \( e^{i (\omega_{0} - \omega) t} \) term ties into the oscillatory motion.

Mathematically, integrating this product yields the frequency-domain representation of the damped oscillator. This integral has particular interest because it forms the basis for analyzing a wide range of physical, engineering, and signal processing problems where understanding the behavior of systems in the frequency domain is crucial.