91Ó°ÊÓ

Group velocity is the velocity of envelop in which the wave is contained, it can be found by finding the differential of angular frequency.

Wavelength can be calculated by

$\mathrm{λ}=\frac{2\mathrm{Ï€}}{\sqrt{\frac{\mathrm{ÒÏ}g}{\mathrm{γ}}\left(\frac{2\sqrt{3}}{3}-1\right)}}$

Where, gacceleration due to gravity, yis the surface tension andp is the density of water

$\mathrm{Ó\neg }=\sqrt{gk+(\mathrm{γ}/\mathrm{ÒÏ})k^3}$

The group velocity ( $v_{group}$) is found by taking the derivative of the angular frequency with respect to wave number

$\begin{array}{rcl}{v}_{group}& =& \frac{d\mathrm{Ó\neg }}{dk}\\ & =& \frac{1}{2}{\left(gk+(\mathrm{γ}/\mathrm{ÒÏ})k^3\right)}^{-\frac{1}{2}}\left(g+\left(3\mathrm{γ}{k}^2/\mathrm{ÒÏ}\right)\right)\\ & =& \frac{g+\frac{3\mathrm{γ}{k}^2}{\mathrm{ÒÏ}}}{2\sqrt{gk+\left(\mathrm{γ}{k}^2/\mathrm{ÒÏ}\right)}}\\ & & \end{array}$

This graph has Group Velocity on its y-axis and Wave Number on its x-axis.

As clear from the formula that the Group Velocity approaches infinity when k approaches zero or infinity, it can also be seen in the above-mentioned graph.

The extreme values of the group velocity will occur when the derivative of the group velocity with respect to is equal to 0

$\begin{array}{rcl}\frac{d{v}_{group}}{{\displaystyle dk}}& =& 0\\ \frac{\frac{6\mathrm{γ}}{\mathrm{ÒÏ}}k\left(2\sqrt{gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}}\right)-2\left(g+\frac{3\mathrm{γ}}{\mathrm{ÒÏ}}{k}^2\right)\frac{1}{2}{\left(gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}\right)}^{-1/2}\left(g+3\mathrm{γ}{k}^3/\mathrm{ÒÏ}\right)}{4\left(gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}\right)}& =& 0\\ \frac{3\mathrm{γ}k}{\mathrm{ÒÏ}\sqrt{gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}}}-\frac{{\left(g+\frac{3\mathrm{γ}}{\mathrm{ÒÏ}}{k}^2\right)}^2}{4{\left(gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}\right)}^{3/2}}& =& 0\\ \frac{12\mathrm{γ}k\left(gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}\right)}{\mathrm{ÒÏ}}& =& {\left(g+\frac{3\mathrm{γ}{k}^2}{\mathrm{ÒÏ}}\right)}^2\\ \frac{12\mathrm{γ}k\left(gk+\mathrm{γ}{k}^3/\mathrm{ÒÏ}\right)}{\mathrm{ÒÏ}}& =& g^2+6\frac{g\mathrm{γ}{k}^2}{\mathrm{ÒÏ}}+9\frac{{\mathrm{γ}}^2{k}^2}{{\mathrm{ÒÏ}}^4}\\ \frac{12\mathrm{γ}{k}^2}{\mathrm{ÒÏ}}+\frac{12{\mathrm{γ}}^2{k}^4}{{\mathrm{ÒÏ}}^2}& =& g^2+6\frac{g\mathrm{γ}{k}^2}{\mathrm{ÒÏ}}+9\frac{{\mathrm{γ}}^2{k}^2}{{\mathrm{ÒÏ}}^2}\\ 3\frac{{\mathrm{γ}}^2{k}^4}{{\mathrm{ÒÏ}}^2}+6\frac{g\mathrm{γ}{k}^2}{\mathrm{ÒÏ}}-g^2& =& 0\end{array}$

Use the above-mentioned quadratic equation to produce two solutions for the potential values of $k^2$

$$\begin{gathered} k^2=\frac{\frac{-6gy}{\mathrm{ÒÏ}}Â\pm \sqrt{\frac{36g^2{\mathrm{γ}}^2}{{\mathrm{ÒÏ}}^2}-4\left(\frac{3y^2}{{\mathrm{ÒÏ}}^2}\right)\left(-g^2\right)}}{\frac{6{\mathrm{γ}}^2}{{\mathrm{ÒÏ}}^2}} \\ {k}^2=-\frac{\mathrm{ÒÏ}g}{\mathrm{γ}}Â\pm \frac{{\mathrm{ÒÏ}}^2}{6{\mathrm{γ}}^2}\sqrt{\frac{12g^2{\mathrm{γ}}^2(3+1)}{{\mathrm{ÒÏ}}^2}} \\ {k}^2=-\frac{\mathrm{ÒÏ}g}{\mathrm{γ}}Â\pm \frac{2\mathrm{ÒÏ}g}{3\mathrm{γ}}\sqrt{3} \\ {k}^2=\frac{\mathrm{ÒÏ}g}{\mathrm{γ}}\left(-1Â\pm \frac{2\sqrt{3}}{3}\right) \\ k=\sqrt{\frac{\mathrm{ÒÏ}g}{\mathrm{γ}}\left(\frac{2\sqrt{3}}{3}-1\right)} \end{gathered}$$

Now use the value of k to calculate the minimum group velocity

$\begin{array}{rcl}{v}_{group}& =& \frac{g+\frac{\frac{\mathrm{ÒÏ}g}{r}\left(\frac{2\sqrt{3}}{3}-1\right)}{\mathrm{ÒÏ}}}{2\sqrt{g{\left(\frac{pg}{\mathrm{γ}}\left(\frac{2\sqrt{3}}{3}-1\right)\right)}^{1/2}+\left(\mathrm{γ}{\left(\frac{\mathrm{ÒÏ}g}{\mathrm{γ}}\left(\frac{2\sqrt{3}}{3}-1\right)\right)}^{3/2}I\mathrm{ÒÏ}\right)}}\\ & =& \frac{g+g\left(2\sqrt{3}-3\right)}{2\sqrt{{\left(\frac{\mathrm{ÒÏ}{g}^3}{\mathrm{γ}}\left(\frac{2\sqrt{3}}{3}-1\right)\right)}^{1/2}\left(1+\left(\frac{2\sqrt{3}}{3}-1\right)\right)}}\\ & =& \frac{\sqrt{3}g-g}{\sqrt{{\left(\frac{\mathrm{ÒÏ}{g}^3}{\mathrm{γ}}\left(\frac{2\sqrt{3}}{3}-1\right)\right)}^{1/2}\left(\frac{2\sqrt{3}}{3}\right)}}\\ & & \end{array}$

Finally, substitute the given values of the density of water $\left(\mathrm{ÒÏ}\right)$, acceleration of gravity () and the surface tension of water$\left(\mathrm{γ}\right)$ then

$\begin{array}{rcl}{v}_{group}& =& \frac{\sqrt{3}\left(9.8 {\text{m/s}}^{\text{2}}\right)-\left(9.8 {\text{m/s}}^{\text{2}}\right)}{\sqrt{{\left(\frac{\left({10}^6 {\text{g/m}}^{\text{3}}\right){\left(9.8 {\text{m/s}}^{\text{2}}\right)}^3}{72 {\text{g/s}}^{\text{2}}}\left(\frac{2\sqrt{3}}{3}-1\right)\right)}^{1/2}\left(\frac{2\sqrt{3}}{3}\right)}}\\ & =& 0.177 \text{m/s}\\ & & \end{array}$

The wavelength corresponding to the slowest group velocity can be calculate to the wave number corresponding to the slowest group velocity

$\begin{array}{rcl}\mathrm{λ}& =& \frac{2\mathrm{Ï€}}{\sqrt{\frac{\mathrm{ÒÏ}g}{r}\left(\frac{2\sqrt{3}}{3}-1\right)}}\\ & =& \frac{2\mathrm{Ï€}}{\sqrt{\frac{\left({10}^6 {\text{g/m}}^{\text{3}}\right)\left(9.8 {\text{m/s}}^{\text{2}}\right)}{72 {\text{g/s}}^{\text{2}}}\left(\frac{2\sqrt{3}}{3}-1\right)}}\\ & =& 0.0433 \text{m}\\ & & \end{array}$

Hence,

The group velocity is$0.177 \text{m/s}$

The wavelength is$0.0433 \text{m}$