Chapter 12, Problem 1P

To determine

The speed of the waves will travel towards upwards and downward stream and Froude number

Explanation of Solution

Given:

Velocity of river flow $\left(v_r\right)$ is $1.5\text{m}/\text{s}$.

Depth of water flow in the river $\left(y\right)$ is 2 m.

Calculation:

Calculate the speed of surface wave $\left(c\right)$.

  $\begin{array}{c}c=\sqrt{gy}\\ =\sqrt{9.81\text{m}/{\text{s}}^{\text{2}}\times 2\text{m}}\\ =\sqrt{19.62}\\ =4.429\text{m}/\text{s}\end{array}$

Calculate the velocity in upstream stream direction ${\left(v_w\right)}_u$.

  ${\left(v_w\right)}_u=v_{w/r}-v_r$

Here, the velocity of waves independent of the river is $v_{w/r}$.

  $\begin{array}{c}{\left(v_w\right)}_u=4.429\text{m}/\text{s}-1.5\text{m}/\text{s}\\ =2.93\text{m}/\text{s}\end{array}$

Thus, the velocity of wave in the upstream direction is $2.93\text{m}/\text{s}$.

Calculate the velocity in downstream stream direction ${\left(v_w\right)}_d$.

  $\begin{array}{c}{\left(v_w\right)}_d=v_{w/r}+v_r\\ =4.429\text{m}/\text{s}+2\text{m}/\text{s}\\ =5.93\text{m}/\text{s}\end{array}$

Thus, the velocity of wave in the downstream direction is $5.93\text{m}/\text{s}$.

Calculate the Froude number $\left(\text{Fr}\right)$.

  $\begin{array}{c}\text{Fr}=\frac{v_r}{c}\\ =\frac{1.5\text{m}/\text{s}}{4.429\text{m}/\text{s}}\\ =0.339\end{array}$

Thus, the Froude number of the flow is $0.339$.