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Wave Mechanics

An ocean wave is described by its height $H$, length $L$, period $T$, and celerity (speed) $C=L/T$. As a wave travels from deep into shallow water (shoaling), its length and height change but its period stays constant. Waves directly generated by local winds form a chaotic wind sea; after leaving the generating area they sort into long, regular swell. The significant wave ($H_s$) is the average of the highest one-third of waves.

By relative depth: deep water ($d/L>1/2$), transitional, and shallow water ($d/L<1/20$). Useful deep-water relations (SI, $g=9.81$):

$$C=\frac{L}{T}, \qquad L_0=\frac{gT^2}{2\pi}\approx1.56\,T^2, \qquad C_0=\frac{gT}{2\pi}\approx1.56\,T$$
$$\text{Shallow water: } C=\sqrt{gd}, \qquad \text{Energy density: } E=\frac{1}{8}\rho g H^2$$

★ Shoaling and Deep-Water Length

A deep-water wave has a period of 8 s. As it moves into shallower water, what happens to its period, and what is its deep-water wavelength?

The period remains constant during shoaling (it is set by the generating storm). Only $L$, $C$, and $H$ change.

$$L_0=1.56\,T^2=1.56(8)^2=1.56(64)=99.8\ \text{m}$$

Final answer: period unchanged at 8 s; $L_0 \approx 99.8$ m.

★ Wave Celerity

A wave has a length of 120 m and a period of 9 s. Determine its celerity.

$$C=\frac{L}{T}=\frac{120}{9}=13.33\ \text{m/s}$$

Final answer: 13.33 m/s.

★★ Deep-Water Celerity

Determine the celerity of a deep-water wave whose period is 9 s, and verify the corresponding wavelength.

$$C_0=\frac{gT}{2\pi}=\frac{9.81(9)}{2\pi}=14.05\ \text{m/s}$$
$$L_0=C_0\,T=14.05(9)=126.4\ \text{m}\;\;(=1.56\times9^2)$$

Final answer: $C_0 = 14.05$ m/s, $L_0 = 126.4$ m.

★★ Classifying Water Depth

A wave of length 100 m travels in water 2.5 m deep. Classify the relative depth and find the shallow-water celerity.

$$\frac{d}{L}=\frac{2.5}{100}=0.025<\frac{1}{20}\;\Rightarrow\;\text{shallow water}$$
$$C=\sqrt{gd}=\sqrt{9.81(2.5)}=4.95\ \text{m/s}$$

Final answer: shallow water, $C = 4.95$ m/s.

★★★ Wave Energy Density

A wave 2.0 m high travels in seawater ($\rho=1025$ kg/m³). Determine the total energy per unit surface area.

$$E=\frac{1}{8}\rho g H^2=\frac{1}{8}(1025)(9.81)(2.0)^2$$
$$=\frac{1}{8}(1025)(9.81)(4)=5028\ \text{J/m}^2$$

Final answer: about 5.03 kJ/m² (split equally between kinetic and potential energy).

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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: w58

MSTE - Ports and Harbors / Wave Mechanics / MSTE May 2019

A wave generated in deep water, when reaching shoaling waters, changes not only in its height but also in its length, but the period will:

  1. Decrease
  2. Remain Constant
  3. None of these
  4. Increase
As a wave shoals, it slows down: the wavelength decreases and the height increases. Because the period is set by the wave speed and length, the decrease in speed results in a decrease in period (wave shoaling).
$\boxed{\text{Decrease}}$

Question Bank: w59

MSTE - Ports and Harbors / Wave Mechanics / MSTE May 2019

A maximum wave height and the wave period of the maximum wave height in a wave train.

  1. Significant wave
  2. Highest wave
  3. Equivalent depth water wave height
  4. Deep water wave
The significant wave is characterized by the average height of the highest one-third of the waves in a wave train, together with the corresponding wave period; it is the standard measure of overall wave height and energy in a sea state.
$\boxed{\text{Significant wave}}$

Question Bank: w60

MSTE - Ports and Harbors / Wave Mechanics / MSTE May 2019

When directly generated and affected by local winds, a wind wave system is called:

  1. Wind sea
  2. Wind seiching
  3. Wind wakes
  4. Wind swell
A wind sea is a wind-wave system that is directly generated and affected by the local wind. (Swell has travelled away from its generating wind; seiches are standing oscillations; the wake effect is the perturbation behind a turbine.)
$\boxed{\text{Wind sea}}$