This problem is designed to illustrate the superposition principle and the concepts of modulated and modulating functions in a wave packet. Consider two wave functions Ψ1(y,t) = 5y cos 7t and Ψ2(y,t) = -5y cos 9t, where y and t are in meters and seconds, respectively. Show that their superposition

Question

This problem is designed to illustrate the superposition principle and the concepts of modulated and modulating functions in a wave packet. Consider two wave functions [latex]\psi _1(y,t)=5y\cos 7t and \psi _2(y,t)=-5y\cos 9t,[/latex] where y and t are in meters and seconds, respectively. Show that their superposition generates a wave packet. Plot it and identify the modulated and modulating functions.

Accepted Answer

Using the relation [latex]\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta ,[/latex] we can write the superposition of [latex]\psi _1(y,t) and \psi _2(y,t)[/latex] as follows:

[latex]\psi (y,t)=\psi _1(y,t)+\psi _2(y,t)=5y\cos 7t-5y\cos 9t[/latex]

[latex]=5y(\cos 8t\cos t+\sin 8t\sin t)-5y(\cos 8t\cos t-\sin 8t\sin t)[/latex]

[latex]=10y\sin t\sin 8t.[/latex] (1.215)

The periods of [latex]10y \sin t and \sin(8t)[/latex] are given by [latex]2\pi and 2\pi/8,[/latex] respectively. Since the period of [latex]10y \sin t [/latex] is larger than that of [latex]\sin 8t, 10y \sin t[/latex] must be the modulating function and [latex]\sin 8t[/latex] the modulated function. As depicted in Figure 1.19, we see that [latex]\sin 8t[/latex] is modulated by [latex]10y \sin t.[/latex]

Question 1.10.14: This problem is designed to illustrate the superposition pri...

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