Question 1

An ocean wave has a velocity of 20 m/s. (a) What is its period?
(b) What is its wavelength?
(c) If the energy per meter of wave front is $1 \mathrm { MJ }$ per period, what is the wave height?

Accepted Answer

(a) From equation (12.10) the velocity is
$$v = \frac { g T } { 2 \pi }$$ $$\text { So the period is given in terms of the velocity as }$$ $$T = \frac { 2 \pi v } { g } = \frac { 2 \times 3.14 \times ( 20 \mathrm {~m} / \mathrm { s } ) } { \left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right) } = 12.8 \mathrm {~s}$$
(b) The wavelength is
$$\lambda = \frac { g T ^ { 2 } } { 2 \pi } = \frac { \left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right) \times ( 12.8 \mathrm {~s} ) ^ { 2 } } { 2 \times 3.14 } = 256 \mathrm {~m}$$
(c) The energy per meter of wave front for one period is given in terms of the power as
$$E / l = P T / l$$ where
$$P / l = \left[ 1.96 \times 10 ^ { 3 } \mathrm {~W} / \left( \mathrm { m } ^ { 3 } \cdot \mathrm { s } \right) \right] H ^ { 2 } T$$
Solving for the wave height gives
$$H = \left[ \frac { E / l } { \left( 1.96 \times 10 ^ { 3 } \mathrm {~W} / \left( \mathrm { m } ^ { 3 } \cdot \mathrm { s } \right) \right) \times T ^ { 2 } } \right] ^ { 1 / 2 } = \left[ \frac { 10 ^ { 6 } \mathrm {~J} / \mathrm { m } } { \left( 1.96 \times 10 ^ { 3 } \mathrm {~W} / \left( \mathrm { m } ^ { 3 } \cdot \mathrm { s } \right) \right) \times ( 12.8 \mathrm {~s} ) ^ { 2 } } \right] ^ { 1 / 2 } = 1.76 \mathrm {~m}$$

Question 2

Hawaii has about 1200 km of coast line and a population of about 1.4 million.
(a) If Hawaii's per capita energy consumption is similar to the rest of the United States, calculate the total annual primary energy use in this state.
(b) Compare the results from part (a) with the total annual wave energy available in Hawaii. What fraction of the total wave energy would satisfy the primary energy needs?

Accepted Answer

(a) From Table 2.1 the per capita average power consumption in the United States is $11,730 \mathrm {~W}$. Per year this represents
$$( 11,730 \mathrm {~J} / \mathrm { s } ) \times \left( 3.15 \times 10 ^ { 7 } \mathrm {~s} / \mathrm { y } \right) = 3.69 \times 10 ^ { 11 } \mathrm {~J} / \mathrm { y }$$
For the state of Hawaii the total annual energy use is
$$\left( 1.4 \times 10 ^ { 6 } \right) \times \left( 3.69 \times 10 ^ { 11 } \mathrm {~J} \right) = 5.17 \times 10 ^ { 17 } \mathrm {~J}$$
(b) From the map the average wave power in Hawaii is $100 \mathrm {~kW} / \mathrm { m }$. For the entire state this represents
$$\left( 1.0 \times 10 ^ { 5 } \mathrm {~W} / \mathrm { m } \right) \times ( 1000 \mathrm {~m} / \mathrm { km } ) \times ( 1200 \mathrm {~km} ) = 1.2 \times 10 ^ { 11 } \mathrm {~W}$$ This represents an annual energy of
$$\left( 1.2 \times 10 ^ { 11 } \mathrm {~J} / \mathrm { s } \right) \times \left( 3.15 \times 10 ^ { 7 } \mathrm {~s} / \mathrm { y } \right) = 3.78 \times 10 ^ { 18 } \mathrm {~J} / \mathrm { y }$$
Compared to the annual energy use this is
$$\left( 3.78 \times 10 ^ { 18 } \mathrm {~J} / \mathrm { y } \right) / \left( 5.17 \times 10 ^ { 17 } \mathrm {~J} / \mathrm { y } \right) = 7.3 \text { times as much }$$

Question 3

The LIMPET wave generating facility in Scotland has a width of 21 m.
Estimate its efficiency.

Accepted Answer

From the map, the average wave power in Scotland is 70 kW/m. If the LIMPET device has a width of 21 m then the total average wave power will be $$( 70 \mathrm {~kW} / \mathrm { m } ) \times ( 21 \mathrm {~m} ) = 1470 \mathrm {~kW}$$
If the LIMPET device is rated at $250 \mathrm {~kW}$ then the rating compared to the incident wave power gives an efficiency of
$$100 \times ( 1470 \mathrm {~kW} ) / ( 250 \mathrm {~kW} ) = 17 \%$$

Question 4

The average wave power available in Hawaii (Figure 12.1) is 100 kW/m.
For waves with a period of 10 s, what is the average wave height?

Accepted Answer

The answer of The average wave power available in Hawaii...

Question 5

A tsunami in deep water has a period of $$5.7 \times 10 ^ { 2 } \mathrm {~s}$$ (a) What is its velocity?
(b) What is its wavelength?
(c) If the tsunami has a height of 2.5 m, what is its power per km of wave front?

Accepted Answer

The answer of A tsunami in deep water has a...

Question 6

An ocean wave has a period of 10 s and a height of 2 m in deep water. (a) What is its wavelength?
(b) As it approaches shore, its period decreases to $6 \mathrm {~s}$. What is its wavelength at this time?
(c) For the situation in part (b), if wave energy per wavelength is conserved, what is the new wave height?

Accepted Answer

The answer of An ocean wave has a period of...

Question 7

An ocean wave has a height of 2.5 m and a period of 10 s. What is the total
power available from 20 m of wave front for this wave?

Accepted Answer

The answer of An ocean wave has a height of...

Question 8

In deepwater, tsunamis have a relatively small height (typically 2 m) and a
very long period (typically 30 minutes). The amplitude becomes larger as they pile up when they reach shallow water. For a tsunami in deep water, calculate the energy per meter of wave front (in J/m). Compare this with the energy per meter of width of a 2-m wide,1500-kg automobile traveling at 120 km/h. This comparison  mphasizes the damage that can be caused by a tsunami.

Accepted Answer

The answer of In deepwater, tsunamis have a relatively small...

Question 9

A seiche is a standing wave that occurs due to resonant conditions on the
surface of lakes and enclosed ocean bays. It is the result of surface perturbations caused by phenomena such as seismic activities or weather conditions. The period of a seiche is approximated by the expression $$T = \frac { 2 L } { \sqrt { g h } }$$
where $L$ is the length of the resonant enclosed region, $g$ is the gravitational acceleration, and $h$ is the mean water depth. Lake Ontario is particularly susceptible to seiches. Calculate the period of a seiche along the length of Lake Ontario.

Accepted Answer

The answer of A seiche is a standing wave that...

Question 10

A small boat with two occupants and a 15 kW outboard motor (total mass = $500 \mathrm {~kg}$ ) moves up and down in a wave with a height of $1 \mathrm {~m}$ and a period of 10 seconds. Calculate the ratio of the average wave power lifting the boat from the trough of the wave to the crest of the wave to the power available from the motor.

Accepted Answer

The answer of A small boat with two occupants and...

Question 11

An ocean wave has a wavelength of 120 m and a period of 16 s.
(a) Calculate the wave's velocity.
(b) If the amplitude of the wave is 1.5 m, calculate the energy per km of wave front in one period.

Accepted Answer

The answer of An ocean wave has a wavelength of...

Question 12

An ocean wave has a velocity of 8.5 m/s. What are its period and
wavelength?

Accepted Answer

The answer of An ocean wave has a velocity of...

Question 13

If a tsunami in deep water has a period of 10 minutes, what would the wave height have to be to provide the same power in $1 \mathrm {~km}$ of wave front as the output of a $1000 \mathrm { MW } _ { \mathrm { e } }$ coal-fired generating station? This result emphasizes the fact that a tsunami in deep water is actually difficult to detect in comparison to normal ocean waves with wavelengths of hundreds of meters and wave heights of several meters.

Accepted Answer

The answer of If a tsunami in deep water has...

Question 14

Most of Greenland's electricity comes from hydroelectric facilities. These produce a total of about 300 million $\mathrm { kWh }$ annually. If wave power on Greenland's southeast coast was developed, how many kilometers of coast would be required at a wave power generation efficiency of $25 \%$, to replace the hydroelectric generation?

Accepted Answer

The answer of Most of Greenland's electricity comes from hydroelectric...

Question 15

North Carolina has some of the highest average wave energies in the United
States along its roughly 500 km of coastline (Figure 12.1). Would it be reasonable for North Carolina to utilize wave energy for all of its electricity? Assume that a total annual per-capita electricity requirement of 15 MWhe and that 5% of the coastline is used for wave electricity generation at an average efficiency of 20%.

Accepted Answer

The answer of North Carolina has some of the highest...

Question 16

In a storm, waves may have a height of 12 m and a period of 15 s. Compare
the power per meter of wave front to the average values shown in Figure 12.1.

Accepted Answer

The answer of In a storm, waves may have a...

Question 17

A section of ocean shoreline has incident waves with a constant period of $15 \mathrm {~s}$. During one day the waves have a height of $2 \mathrm {~m}$ for 12 hours, $3 \mathrm {~m}$ for 8 hours, and $5 \mathrm {~m}$ for 4 hours.
(a) What is the total energy available per km of coast?
(b) During which period of different wave heights is the greatest energy available?
(c) What is the average wave power per km of coast during the day?

Accepted Answer

The answer of A section of ocean shoreline has incident...

Question 18

An ocean wave has a height of 5 m and a period of 12 s.
(a) Calculate the wavelength of this wave.
(b) Calculate the power per unit wave front for this wave.
(c) Calculate the energy per unit wave front for one period of this wave.

Accepted Answer

The answer of An ocean wave has a height of...

Question 19

(a) If all other factors are the same, which is the single most important
parameter in maximizing the energy per unit length of wave front for one period of the wave: wavelength, wave height or wave velocity.?
(b) What is the effect of doubling each of these parameters on the wave energy per period?

Accepted Answer

The answer of (a) If all other factors are the...

Quiz 12: Wave Energy

An ocean wave has a velocity of 20 m/s. (a) What is its period? (b) What is its wavelength? (c) If the energy per meter of wave front is $1 \mathrm { MJ }$ per period, what is the wave height?

The LIMPET wave generating facility in Scotland has a width of 21 m. Estimate its efficiency.

The average wave power available in Hawaii (Figure 12.1) is 100 kW/m. For waves with a period of 10 s, what is the average wave height?

A tsunami in deep water has a period of $5.7 \times 10 ^ { 2 } \mathrm {~s}$ (a) What is its velocity? (b) What is its wavelength? (c) If the tsunami has a height of 2.5 m, what is its power per km of wave front?

An ocean wave has a height of 2.5 m and a period of 10 s. What is the total power available from 20 m of wave front for this wave?

An ocean wave has a wavelength of 120 m and a period of 16 s. (a) Calculate the wave's velocity. (b) If the amplitude of the wave is 1.5 m, calculate the energy per km of wave front in one period.

An ocean wave has a velocity of 8.5 m/s. What are its period and wavelength?

In a storm, waves may have a height of 12 m and a period of 15 s. Compare the power per meter of wave front to the average values shown in Figure 12.1.

An ocean wave has a height of 5 m and a period of 12 s. (a) Calculate the wavelength of this wave. (b) Calculate the power per unit wave front for this wave. (c) Calculate the energy per unit wave front for one period of this wave.

(a) If all other factors are the same, which is the single most important parameter in maximizing the energy per unit length of wave front for one period of the wave: wavelength, wave height or wave velocity.? (b) What is the effect of doubling each of these parameters on the wave energy per period?