- #1

- 726

- 27

Hey! Two questions:

1) How exactly does the proof go for the formula for the group-velocity:

[tex] v_g = \frac{d \omega}{dk}[/tex]

Is it something along the line of using the superposition principle to add all the (arbitrarily numerous) wave-functions making up the wave-packet together, then using trigonometric identities to discover that the wave-packet is like a "beat-phenomenon" moving with a velocity equal to the group velocity through the medium? Ie, it's just function of the type

[tex] A \sin(\omega_0 t -k_0 x) \cos(\omega_{beat} t -k_{beat} x)[/tex]

? And then finding out that [tex] v_g = \frac{\omega_{beat}}{k_{beat}} = \frac{d \omega}{dk} [/tex]?

2) How do you actually calculate the group velocity? What wave-lengths and frequencies do you insert into the formula? I mean, there can be hundreds of different waves making up the wave-packet, all with different frequencies and wavelengths. Do you just use the averages of their wavenumbers, or something?

3) What exactly does the formula for the phase-velocity in a wave-packet mean?

[tex] v_p = \frac{\omega}{k}[/tex]

Is this the general formula for the velocity of each wave of the packet? Or is it the velocity of the sine-term in the equation above, assuming the waves making up the wave-packet have nearly identical frequencies and wavelengths? So:

[tex] v_p = \frac{\omega_0}{k_0}[/tex]

?

1) How exactly does the proof go for the formula for the group-velocity:

[tex] v_g = \frac{d \omega}{dk}[/tex]

Is it something along the line of using the superposition principle to add all the (arbitrarily numerous) wave-functions making up the wave-packet together, then using trigonometric identities to discover that the wave-packet is like a "beat-phenomenon" moving with a velocity equal to the group velocity through the medium? Ie, it's just function of the type

[tex] A \sin(\omega_0 t -k_0 x) \cos(\omega_{beat} t -k_{beat} x)[/tex]

? And then finding out that [tex] v_g = \frac{\omega_{beat}}{k_{beat}} = \frac{d \omega}{dk} [/tex]?

2) How do you actually calculate the group velocity? What wave-lengths and frequencies do you insert into the formula? I mean, there can be hundreds of different waves making up the wave-packet, all with different frequencies and wavelengths. Do you just use the averages of their wavenumbers, or something?

3) What exactly does the formula for the phase-velocity in a wave-packet mean?

[tex] v_p = \frac{\omega}{k}[/tex]

Is this the general formula for the velocity of each wave of the packet? Or is it the velocity of the sine-term in the equation above, assuming the waves making up the wave-packet have nearly identical frequencies and wavelengths? So:

[tex] v_p = \frac{\omega_0}{k_0}[/tex]

?

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