Deep-Water Wave Problem

Problem Statement

An ocean swell propagates in deep water where the water depth is $h = 300\ \text{m}$ and the wavelength is $L = 100\ \text{m}$. Assume gravitational acceleration $g = 9.81\ \text{m/s}^{2}$.

Determine:

Whether the wave satisfies the deep-water condition.
The wave speed.
The physical significance of the result.

Solution

Step 1: Deep-Water Condition

Deep-water waves satisfy:

$$ \frac{h}{L} \ge 0.5 $$ $$ \frac{h}{L} = \frac{300}{100} = 3.0 $$

Since $3.0 \gg 0.5$, the wave clearly lies in the deep-water regime.

Step 2: Wave Speed

For deep-water waves, the phase speed is given by:

$$ c = \sqrt{\frac{gL}{2\pi}} $$

Substituting the values:

$$ c = \sqrt{\frac{(9.81)(100)}{2\pi}} $$ $$ c = \sqrt{156.1} $$ $$ \boxed{c \approx 12.5\ \text{m/s}} $$

Step 3: Physical Interpretation

The wave travels at approximately 12.5 m/s.
The wave speed depends only on the wavelength.
Water depth no longer affects the speed.
Deep-water waves are dispersive: longer waves move faster.

Optional Check: General Dispersion Relation

$$ c = \sqrt{\frac{gL}{2\pi}\tanh\left(\frac{2\pi h}{L}\right)} $$ $$ \frac{2\pi h}{L} = \frac{2\pi(300)}{100} \approx 18.85 $$ $$ \tanh(18.85) \approx 1 $$ $$ c \approx \sqrt{\frac{gL}{2\pi}} $$