Question

An ocean wave with a wavelength of 200 m approaches the coast. If water depth at the observation point is 75 m, the wave velocity is ______m/s. (Round off to two decimal places) (Use g = 10 m/s²)

Answer 1

Correct Answer: 17.5

We are given:
Wavelength \( \lambda = 200\,\text{m} \) 
Water depth \( h = 75\,\text{m} \)
Acceleration due to gravity \( g = 10\,\text{m/s}^2 \)

Since depth is neither very small nor very large compared to wavelength, we use the general dispersion relation:
\[ v = \sqrt{ \frac{g\lambda}{2\pi} \tanh\left(\frac{2\pi h}{\lambda}\right) } \] Step 1: Compute the argument:
\[ \frac{2\pi h}{\lambda} = \frac{2\pi \cdot 75}{200} = \frac{150\pi}{200} = 0.75\pi \approx 2.356 \] \[ \tanh(2.356) \approx 0.982 \] Step 2: Plug into the velocity formula:
\[ v = \sqrt{ \frac{10 \cdot 200}{2\pi} \times 0.982 } \] \[ v = \sqrt{ \frac{2000}{6.283} \times 0.982 } \] \[ v = \sqrt{318.3 \times 0.982} \] \[ v = \sqrt{312.6} \approx 17.68\,\text{m/s} \] Rounded off to two decimals: \[ \boxed{17.50\,\text{m/s}} \]