Correct Answer - Option 3 : Group velocity = phase velocity
In a dispersive medium, the group velocity is less than the phase velocity only.
When there is no dispersion, i.e. for a non dispersive medium, both the velocities are equal.
Derivation:
Phase velocity is defined as:
\({V_p} = \frac{\omega }{\beta }\)
β is the phase constant defined as:
\(\beta = \sqrt {{\omega ^2}\mu \epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} \)
\({V_p} = \frac{\omega }{{\sqrt {{\omega ^2}\mu \epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} }}\)
\({V_p} = \frac{1}{{\sqrt {\mu \epsilon - {{\left( {\frac{{m\pi }}{{a\omega }}} \right)}^2}} }}\)
\({V_p} = \frac{{\frac{1}{{\sqrt {\mu \epsilon } }}}}{{\sqrt {1 - {{\left( {\frac{{m\pi }}{{a\omega \sqrt {\mu \epsilon} }}} \right)}^2}} }}\)
Using \(c = \frac{1}{{\sqrt {\mu C} }}\) where c = speed of light, the above expression becomes:
\({V_p} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{m\pi C}}{{a\omega }}} \right)}^2}} }};\)
Also \({\omega _c} = \frac{{m\pi c}}{a}\)
\({V_p} = \frac{C}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }}\)
Using \(\sin \theta = \frac{{{\omega _c}}}{\omega }\), we get:
\({V_p} = \frac{c}{{\sqrt {1 - {{\sin }^2}\theta } }}\)
\({V_p} = \frac{c}{{\cos \theta }};\)
Since -1 ≤ cos θ ≤ 1
∴ Vp > c
Group velocity is given by:
\({V_g} = \frac{{d\omega }}{{d\beta }}\)
\(\beta = \sqrt {{\omega ^2}\mu \epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} \)
\(\frac{{d\beta }}{{d\omega }} = \frac{{2\omega \mu\epsilon }}{{2\sqrt {{\omega ^2}\mu\epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} }}\)
\(\frac{{d\beta }}{{d\omega }} = \frac{{\sqrt {\mu\epsilon } }}{{\sqrt {1 - {{\left( {\frac{{m\pi }}{{a\omega \sqrt {\mu\epsilon } }}} \right)}^2}} }}\)
\(\frac{{d\beta }}{{d\omega }} = \frac{1}{{C\sqrt {1 - {{\left( {\frac{{{\omega _C}}}{\omega }} \right)}^2}} }}\)
\({V_g} = c\sqrt {1 - {{\left( {\frac{{{\omega _C}}}{\omega }} \right)}^2}} \)
Vg = c cos θ
Vg < c
Conclusion:
The phase velocity is always greater than the speed of light and group velocity is always less than the speed of light. Hence, the group velocity is less than the phase velocity
For a Non-dispersive medium, the one-dimensional wave defined as:
U(x, t) = A0 sin (ωt – kx + ϕ) has a phase angle (θ) of ωt – kx + ϕ
In general, the phase is constant,
i.e. \(\frac{{d\theta }}{{dt}} = \frac{\omega }{k} = {v_p}\;\left( {phase\;velocity} \right)\)
Group velocity is defined as:
\({V_{group}} = \frac{{{\omega _2} - {\omega _1}}}{{{k_2} - {k_1}}} = \frac{{d\omega }}{{dk}}\).
Dispersion is when the distinct phase velocities of the components of the envelope cause the wave packet to “Spread out” over time.
When there is no dispersion derivative term is 0 and
Vp = Vg
