Question

Let's imagine waves hitting the entire \(2,000 \mathrm{~km}\) Pacific coastline of the U.S. that are different from those evaluated in the text. This time, the waves have \(50 \mathrm{~m}\) wavelength, arriving every 10 seconds, and \(2 \mathrm{~m}\) crest-to-trough amplitude. How much power does the coast receive under these more active conditions?

Short answer

Under these more active conditions, the coast receives approximately \(5.09825\times 10^9 \mathrm{~W}\) of power.

Step-by-step solution

Convert length of coastline to meters

To proceed with our calculations, we need to convert the length of the coastline to meters. $$L = 2000 \mathrm{~km} * 1000 \mathrm{~m/km} = 2,000,000 \mathrm{~m}$$

Calculate wave speed

We can calculate the wave speed (v) by dividing the wavelength (λ) by the period (T): $$v = \frac{\lambda}{T} = \frac{50 \mathrm{~m}}{10 \mathrm{~s}} = 5 \mathrm{~m/s}$$

Calculate wave frequency

Next, we can calculate the frequency (f) of the waves by taking the inverse of the period (T): $$f = \frac{1}{T} = \frac{1}{10 \mathrm{~s}} = 0.1 \mathrm{~Hz}$$

Calculate wave energy density

We will use the wave energy density formula to calculate the energy density (E) of the waves: $$E = \frac{1}{8}*\rho*g*A^2$$ Where ρ is the density of seawater, g is the acceleration due to gravity, and A is the amplitude of the waves. Assuming the density of seawater is 1025 kg/m³ and the acceleration due to gravity is 9.81 m/s², we can find the wave energy density: $$E = \frac{1}{8}\times 1025 \mathrm{~kg/m^3}\times 9.81 \mathrm{~m/s^2}\times (2 \mathrm{~m})^2 = 5098.25 \mathrm{~J/m^2}$$

Calculate the power received by the coast

Now, we can calculate the power received by the coast (P) using the wave energy density (E), the wave speed (v), the length of the coastline (L), and the frequency (f): $$P = E \times v \times L \times f = 5098.25 \mathrm{~J/m^2} \times 5 \mathrm{~m/s} \times 2,000,000 \mathrm{~m} \times 0.1 \mathrm{~Hz}$$ $$P = 5.09825\times 10^9 \mathrm{~W}$$ Therefore, under these more active conditions, the coast receives approximately \(5.09825\times 10^9 \mathrm{~W}\) of power.

Key concepts

Wave Calculations

Wave calculations are essential for understanding how much energy is carried by waves as they travel through water. To begin, it's crucial to convert all measurements into consistent units. For instance, converting the length of the coastline from kilometers to meters helps in carrying out uniform calculations, which is what we did here by making the length 2,000,000 meters. The speed of a wave relates to its wavelength and the period of time it takes to pass a point. The formula is: \[ v = \frac{\lambda}{T} \] where \( v \) is the speed, \( \lambda \) is the wavelength, and \( T \) is the period. For our specific case, the speed computed was 5 meters per second. Frequency, another critical calculation, is simply the inverse of the period, giving us an idea of how often waves hit the shore. Here, a 0.1 Hz frequency was calculated for waves arriving every 10 seconds. Finally, to get the power that the coast receives, we multiply the wave energy density by the wave speed, coastline length, and frequency. The ultimate result in this scenario was a power output of approximately \(5.09825 \times 10^9\) watts.

Physics of Waves

The physics of waves reveal fundamental principles that describe how they propagate and carry energy. Waves possess characteristics such as wavelength, frequency, and amplitude, all of which define their energy potential.

• Wavelength (\( \lambda \)): The distance from one wave crest to the next.
• Frequency (\( f \)): How often waves reach a point per second.
• Amplitude (\( A \)): The vertical distance from the crest to the trough. Higher amplitude means more energy.

The energy contained in a wave is critically dependent on the amplitude squared, following the formula for wave energy density:\[ E = \frac{1}{8} \times \rho \times g \times A^2 \] Density of seawater and gravity also play roles, and it’s easy to see how varying these factors changes the wave power significantly. Understanding these relationships helps in predicting the amount of energy we can harness from waves.

Renewable Energy

Wave energy, as a form of renewable energy, harnesses the natural movement of water to create power. It's incredibly consistent, considering that waves crash on shores around the clock, compared to other forms like solar or wind energy which can be intermittent.

• Sustainability: Unlike fossil fuels, wave energy is clean and emits no greenhouse gases.
• Abundance: With vast coastlines and persistent wave activity, there's a vast potential to tap into energy.
• Efficiency: Waves carry a lot of energy, making it a density-rich source of power.

Moreover, as technology advances, the methods of capturing wave energy become more effective and less invasive to marine life. Investing in research and infrastructure for wave power can provide a significant boost to renewable energy efforts worldwide, contributing to the goal of reducing global carbon emissions.