Question

A damped harmonic oscillator is given by $$ f(t)=\left\\{\begin{array}{cl} A e^{-a t} e^{i \omega_{0} t}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$ a. Find \(\hat{f}(\omega)\) and b. the frequency distribution \(|\hat{f}(\omega)|^{2}\). c. Sketch the frequency distribution.

Short answer

The Fourier transform of the given function is \( \hat{f}(\omega) = \frac{A}{a-i(\omega-\omega_{0})} \). The frequency distribution \(|\hat{f}(\omega)|^{2}\) is \( \frac{A^{2}}{(a^{2}+(\omega-\omega_{0})^{2})} \).

Step-by-step solution

Compute the Fourier Transform

Fourier Transform of a function \( f(t) \) is given by:\[\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt\]Applying this to the given function \( f(t) = A e ^{-at} e^{i \omega_{0} t} \) for \( t >= 0 \), we get:\[\hat{f}(\omega) = \int_{0}^{\infty} A e ^{-at} e^{i \omega_{0} t} e^{-i \omega t} dt\]Simplifying:\[\hat{f}(\omega) = A \int_{0}^{\infty} e ^{-at} e^{i t (\omega_{0} - \omega)} dt\]

Evaluate the integral

This is an integral of exponential function which can be evaluated. After integrating, we obtain:\[\hat{f}(\omega) = \frac{A}{a-i(\omega-\omega_{0})}\]

Calculate the frequency distribution

The frequency distribution is given by \(|\hat{f}(\omega)|^{2}\). Given that \( \hat{f}(\omega) = \frac{A}{a-i(\omega-\omega_{0})} \), we find the square magnitude as:\[|\hat{f}(\omega)|^{2} = \hat{f}(\omega) \cdot \hat{f}^{*}(\omega)\]where \( * \) denotes complex conjugation. On calculation, we get:\[|\hat{f}(\omega)|^{2} = \frac{A^{2}}{(a^{2}+(\omega-\omega_{0})^{2})}\]

Key concepts

Fourier Transform

The Fourier Transform is a mathematical tool that transforms a function from its original domain (often time or space) into the frequency domain. The Fourier Transform of a function is integral to many areas of physics and engineering, especially in signal processing where it helps analyze the frequencies present in a signal.

For a damped harmonic oscillator, represented by a time-dependent function, the Fourier Transform allows us to calculate the complex amplitudes of the exponential components at different frequencies. It's defined as \[ \hat{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt \.
\] In the given exercise, the function only exists for \( t \geq 0 \), simplifying the limits of the integral. The result is a frequency domain representation of the damped oscillator, highlighting how the amplitude of oscillations decays over time and how the oscillations are spread out over a range of frequencies, not just a single natural frequency \( \omega_0 \).

Frequency Distribution

Frequency distribution in the context of the Fourier Transform refers to the distribution of power or amplitude over a range of frequencies. For the damped harmonic oscillator, this gives us the spectrum of frequencies that make up the damped oscillation. The magnitude squared of the Fourier Transform, \( |\hat{f}(\omega)|^{2} \), is particularly important as it represents the energy density per frequency interval.

By calculating \( |\hat{f}(\omega)|^{2} \), we can visualize how the oscillation's energy is distributed across different frequencies. In the exercise, this frequency distribution is obtained by multiplying the Fourier Transform by its complex conjugate, resulting in \[ |\hat{f}(\omega)|^{2} = \frac{A^{2}}{(a^{2}+(\omega-\omega_{0})^{2})} \]. This describes a Lorentzian distribution centered at the natural frequency \( \omega_{0} \) with a width at half maximum determined by the damping factor \( a \). It quantitatively describes how the damped oscillator gradually loses energy over time across its frequency components.

Complex Exponential Functions

Complex exponential functions play a pivotal role in the analysis of oscillatory systems, such as the damped harmonic oscillator. When dealing with the harmonic oscillator, we encounter expressions of the form \( e^{i \omega_{0} t} \), which represents oscillations at a specific natural frequency \( \omega_{0} \). Euler's formula tells us that \( e^{i \theta} = \cos(\theta) + i \sin(\theta) \), reflecting the oscillatory nature both in sine and cosine components.

The use of complex exponentials is particularly useful when applied to the Fourier Transform. In the given problem, the function \( f(t) = A e^{-at} e^{i \omega_{0} t} \) uses the complex exponential to encode both the oscillatory behavior and the exponential decay due to damping. Through the Fourier Transform, these complex exponentials are analyzed to determine how different frequency components, both real and imaginary, contribute to the overall behavior of the damped system. The manipulations of these functions lead to a clearer understanding of the system's characteristics in the frequency domain.