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Friday, July 12, 2024

OK Surfers, How Much Would It Cost to Power Your Own Wave?

If you are into surfing, you have a problem: You can only surf where there are waves. Even if you’re lucky enough to live near an ocean, surfable waves aren't consistent. One day the break is perfect, and the next day the water is flat as glass. Well, there's a simple solution to this problem: Make your own.

That's exactly what happens at places like Surf Ranch, which has created a barreling wave that travels 2,300 feet. Making an artificial wave machine starts with a body of water. In this case, it's basically a rectangular pond. On one of the long sides is a type of car on rails with a hydrofoil, essentially a wing that runs through the water. As the hydrofoil moves, it creates surfable waves. Also, by adjusting the hydrofoil, the waves can be custom-made (to a degree) to produce waves between 6 and 8 feet high with speeds between 10 and 20 miles per hour.

But let’s say you don't live close to any surf parks, and you decide to build your own. Let’s not sweat the technical details. We’ll just assume you already have a small lake and can build a track next to it that pushes your giant electric-motor-powered hydrofoil. Instead, let’s get to the real question: How much is a single wave going to cost you?

When you pay your electricity bill every month, you are paying for energy in the form of electrical current. Electricity prices vary by location, but we can estimate the amount of energy needed to create one fake wave and then use average prices to find out how much that would cost.

Imagine a very simple wave. If you looked at it from the side, it might take a shape like this:

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Don't worry about the actual dimensions, but we are going to need a few values for our calculation. I have illustrated a simple triangular wave with a height h and width w. You can't see it in the picture, but the wave also has a length, which is probably the same width as the lake. Let's call that L. Finally, there's the wave speed, which will be represented by v. (In the picture, the wave is traveling to the right.)

This simple wave will have two types of energy once it's in motion: kinetic energy and gravitational potential energy. Kinetic energy is associated with the motion of an object (the wave, in this case), and it depends on both the object's mass and its velocity. We can calculate the kinetic energy with the following equation:

Yes, we don't yet know the mass of the wave—but just hold on.

The other type of energy the wave has is gravitational potential energy. This is associated with the gravitational interaction between the water and the Earth. As an object moves away from the surface of the Earth, it has an increase in gravitational potential energy. Since this wave sticks out above the surface of the water, it's going to have some potential. We almost always use U to represent potential energy, and we can calculate it like this:

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What about that g variable? That's the gravitational field. It's a measure of the strength of the gravitational interaction. On the surface of the Earth, it has a value of 9.8 newtons per kilogram. If you want to build your surf park on another planet, there will be a different value for the gravitational field. For example, on Mars, g = 3.75 N/kg because of the weaker gravitational interaction.

Be careful with the height (h) in this equation. Different parts of the wave are at different heights above the surface. Since it's a triangular wave, most of the water is close to the surface and just a tiny little bit is at the top. Instead of using the height of the triangular wave, we can instead use the height of the center of mass of the wave. Fortunately, since this is a triangle we know that the center of mass would be 1/3 the height of the wave. Nice.

Both the kinetic and the gravitational potential energy depend on the mass of the wave. Assuming the wave is made of water (I mean, there are other options to consider), then we know the density is 1,000 kilograms per cubic meter.

Now I just need to find the wave volume (V) to determine the mass. Since this simple wave is just a triangular prism, I can find the volume, no problem. Together with the density (ρ), I can get the mass, like this:

Putting all of this together, I get the following expression for the total energy of one wave:

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That expression doesn't look nice, but at least now the energy calculation is in terms of things that we actually know or can estimate. All we need to do is convert our estimates from imperial units to metric and we are all set. Using a wave traveling at 20 miles per hour, with these estimations I get a wave energy of 16 million joules.

Is this a lot of energy? Here are some quick numbers for comparison. Suppose you pick up a textbook from the floor and put it on a table. This takes roughly 10 joules. Your smartphone’s battery stores about 10,000 joules. A full tank of gasoline, or 12 gallons, is about 1.5 billion joules.

OK, now that we know the energy it takes to make a wave, we have some options for how to create this thing. Suppose you use an electric motor to pull the hydrofoil. If the motor is 85 percent efficient, then you will actually need to put 19 million joules into it in order to get 16 million joules into the wave.

The average price of electricity in the US is 23 cents per kilowatt-hour. Power is a measure of how fast you use energy, and we can calculate that as P = E/ΔT, where T is time. If energy is in joules and time is in seconds, then the power would be in watts. So 1 kilowatt-hour is the energy you would get running 1,000 watts for 1 hour (3,600 seconds), or 3.6 million joules. That's how much energy you get for just 23 cents. If you want 19 million joules, it would cost you $1.23.

What about a gasoline-powered hydrofoil? In the US, you normally buy gasoline by the gallon; in other parts of the world, it’s sold by the liter. Gasoline stores about 34 million joules per liter (or 128 million joules per gallon).

However, a gasoline engine has a much lower efficiency than an electric motor. At best, it would be 40 percent efficient. That means we would need to use 40.9 million joules, or 1.2 liters (0.32 gallons) of gasoline. Assuming you pay $3 per gallon (which is a bit lower than the US average in July 2023), that would cost close to $1, or about the same price as an electric-generated wave.

Now, suppose you are really off the grid and wanted to make waves with human power. Obviously you can't pull a hydrofoil by yourself. But maybe you could pedal a bike to raise some large mass using pulleys, and once the mass had enough energy, you’d let it fall and pull the hydrofoil. Let’s say this whole system is 50 percent efficient, so that you would need to produce enough energy to store 32 million joules.

Let’s suppose that you can output 100 watts of power. How long would it take you to get that much energy stored for your wave? Let’s calculate this:

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That’s about 89 hours to store that amount of energy. Even if you work in 10-hour shifts, it's going to take more than nine days to get enough energy for a single wave. Technically, this wave is priceless, but it’s still going to cost you. At least if you are making your waves this way, you will have plenty of time to think about your poor decisions as you prepare for the next surfing session. I would probably go with electric-powered waves instead.

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