(a) Does a function of the form u(t)=a cosγt-b cosωt still exhibit beats when γ≈ω, but a ≠b ? Us

(a) Does a function of the form u(t)=a cosγt-b cosωt still...

(a) Does a function of the form $u(t)=a \cos \gamma t-b \cos \omega t$ still exhibit beats when $\gamma \approx \omega$, but $a \neq b$ ? Use a computer to graph some particular cases and discuss what you observe. (b) Explain to what extent the conclusions based on (10.103) do not depend upon the choice of initial conditions (10.102).

Key Concepts

Beats Phenomenon

The beats phenomenon occurs when two sinusoidal functions with nearly equal frequencies are superposed, resulting in a waveform whose amplitude is modulated at a frequency equal to the difference between the two frequencies. Even if the amplitudes of the two oscillations are different, the interference leads to periodic variations in amplitude – a key characteristic of beats observed in many physical systems like sound and electrical signals.

Superposition of Harmonic Oscillations

In linear systems, the response to multiple inputs is the sum of the responses to individual inputs. When two cosine functions with nearly equal frequencies are added, their combination can be expressed using trigonometric sum-to-product identities. This mathematical treatment reveals an underlying modulation (or beat pattern) that is present regardless of whether the amplitudes are equal or not.

Influence of Initial Conditions in Linear Systems

In linear differential equations, the behavior of the system’s steady-state solution is often independent of the specific initial conditions. Although initial conditions affect transient responses, the dominant characteristics of phenomena like beats—which are determined by the system's intrinsic frequencies and damping (if present)—remain consistent. This demonstrates that conclusions derived from the analysis of the inherent oscillatory behavior are robust against variations in initial conditions.

Transcript

00:01 A signal is described by the function d t equals to c e to the power t by toe.

00:08 We have to calculate the fourier transform for this function that is a omega and sketch and interpret the results and for b part how are d t and a omega are affected by the change in door.

00:37 First for transform, the formula we will be using is ak is 1 by 2 pi integration from minus infinity to plus infinity, f of x, e to the power minus ik x dx.

00:52 Here the function is a omega equals to 1 by 2 pi from minus infinity to plus infinity, c e to the power minus t by toe e to the power minus i omega t this is equals to c by two pi c taking outside of the integration from minus infinity to plus infinity e to the power minus t by toe e to the power minus omega -t can be written as cos of omega -t minus i -s -rine omega -t.

01:40 Opening the bracket and multiplying by the exponent, e by 2 -py, the integration from minus infinity, e to the power minus t by toe, cos omega -t minus of integration from minus infinity to plus infinity, e to the power minus t by t sine omega t.

02:07 This is the imaginary part since e to the power minus t by tau is an even function due to symmetric about the y -axis and sine is an odd function the product of them will be an odd function and when integrating between the equal and opposite limits the result will be zero so the remaining equation we have is the integration of minus infinity to plus infinity e to the power minus t by toe, cos of omega t.

02:49 Here cost is an even function and e to the power minus t by tau is also an even function.

02:55 Their product becomes even.

02:57 And when integrating between the equal and opposite units, the result will be equal to twice the integral and integrating between the zero and the upper limit, e to the power minus t by toe, cos omega, t, d t.

03:24 So, on integrating, we have, since that integral doesn't have a closed form solution, we can take it as a equals to 1 by 2 and b equals to omega...

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