Answer to Question #162381 in Optics for Hans Vans

Question

Calculate the wave and group velocities of water waves at (a) A = 2 cm, (b) A =

8.0 cm, and (c) A = 20.0 cm. The wave velocity of short waves such as these are

given by

where A is the wavelength in meters, T is the surface tension in newtons per meter,

which at room temperature is 0.073N/m, g isthe acceleration due to gravity, 9.80m/s2

,

and d is the density of the liquid in kilograms per cubic meter.

Answer 1

$wave\:velocity:\:v=\sqrt{\frac{\lambda }{2\pi }\left(g+\frac{4\pi T^2}{\lambda ^2\rho }\right)}$

$T=0.073\frac{N}{m}\:$

$\rho =100\frac{kg}{m^3}$

$g=9.8\frac{m}{s^2}$

$\lambda =0.02m\:v=\sqrt{\left(\frac{0.02}{2\times3.14}\right)\left(9.8+\frac{4\times 3.14\times\left(0.073\right)^2}{\left(0.02\right)^2\times1000}\right)}=0.178\frac{m}{s}$

$\lambda =0.08m\:v=\sqrt{\left(\frac{0.08}{2x3.14}\right)\left(9.8+\frac{4x3.14x\left(0.073\right)^2}{\left(0.08\right)^2x1000}\right)}=0.354\frac{m}{s}$

$\lambda =0.2m\:v=\sqrt{\left(\frac{0.2}{2x3.14}\right)\left(9.8+\frac{4x3.14x\left(0.073\right)^2}{\left(0.2\right)^2x1000}\right)}=0.559\frac{m}{s}$