For a superposition of two since waves of equal amplitude in a dispersive media, we find that the group velocity is given exactly by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]v_g = \frac{\omega_2-\omega_1}{k_2-k_1}[/tex]

and approximately by [itex]d\omega / dk|_{k=k_0}[/itex].

How do we show that this approximation holds for any type of waves (not just sine), and for a superposition of any number of them?

Here's the developement that allowed me to conclude to the statement above. You may not read this.

In the 2 sine wave superposition problem (beats), we have two progressive waves of equal amplitude, [itex]y_1(x,t) = Asin(k_1x - \omega_1t)[/itex] and [itex]y_2(x,t) = Asin(k_2x - \omega_2t)[/itex] and their superposition is

[tex]y(x,t)=2Acos\left(\frac{k_2-k_1}{2}x-\frac{\omega_2-\omega_1}{2}t\right) sin\left(\frac{k_2+k_1}{2}x-\frac{\omega_2+\omega_1}{2}t\right)[/tex]

The speed [itex]v_g[/itex] of the modulation envelope is found by setting [itex]x=x(t)=v_g t[/itex] in [itex]cos(\frac{k_2-k_1}{2}x-\frac{\omega_2-\omega_1}{2}t)[/itex] and arguing that since x(t) moves at the same speed as the cos wave, the phase is constant. I.e.,

[tex]\frac{k_2-k_1}{2}v_g t -\frac{\omega_2-\omega_1}{2}t = \mbox{cst}[/tex]

and taking the derivative wrt t gives

[tex]v_g = \frac{\omega_2-\omega_1}{k_2-k_1}[/tex]

This gives [itex]v_g[/itex] exactly. Here, the distribution of amplitude wrt k is discrete rather than continuous as in the case of the wave packet, but extending the definition of [itex]k_0[/itex] to the discrete case we have,

[tex]k_0 = \frac{|k_2+k_1|}{2}[/tex]

So evidently, as [itex]k_1[/itex] approaches [itex]k_2[/itex], by definition of derivative, [itex]v_g[/itex] approaches [itex]d\omega / dk|_{k=k_0}[/itex], but the equality [itex]v_g = d\omega / dk|_{k=k_0}[/itex] is not exact! For nonlinear dispersion relations, this is only an approximation.

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# Homework Help: Group velocity for a discrete superposition of waves

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