How are waves created?

By friction of wind with water surface.

What is the Bernoulli equation?

It says that an increase in wave speed will mean a decrease in air pressure. p_air + 1/2 rho v_wind^2 = const (+ impact pressure of additional wind).

Draw a deep ocean wave.

Sinusoidal form. Water moves in circles with radius R = amplitude of the wave. Speed on top is v_wave - v_circle. Speed on bottom is v_wave + v_circle.

Does the water molecules of a wave move laterally?

Not averaged over time.

Calculate the wave speed.

The speed of the circle:

v_{circle }= 2πRf_{w }

∆E_{kin }= 1/2 m ( v^{2}_{bottom }+ v^{2}_{top}) = 1/2 m * (4v_{w }v_{circle}) = 2m v_{w} 2πRfw

This is equal to the change in potential energy ∆E_{pot }= 2mgR

g = 2v_{w}πf_{w} => v_{w }= g/2πf_{w}

Using f_{w} = v_{w}/lambda_{w}

=> v_{w} = sqrt(lambda_{w} g / 2π)

What can be said about the dispersion of deep ocean waves?

They have a large dispersion. The longer the wave lengths, the faster they move.

What is the wave speeds dependence on surface tension?

The surface tension leads to an additional restoring force of a wave, trying to minimize the water surface.

This means that the wave velocity gets an additional term:

v_{w} = sqrt(g*lambda_{w}/2π + 2πσ/rho *lambda_{w})

This means that we have two regimes:

When lambda_{w} < 1cm, we get capillary waves and v_{w} scales with sqrt(1/lambda).

When lambda_{w} > 10 cm, we get gravitational waves, and v_{w} scales with sqrt(lambda).

How does the speed of gravitational waves scale? What is the condition for gravitational waves?

Condition: lambda > 10cm. Scales with sqrt(lambda).

How does the speed of capillary waves scale? What is the condition for capillary waves?

Condition: lambda < 1 cm. Scales with sqrt(1/lambda).

How does the equation for wave velocity look for shallow water waves?

v_{w} = sqrt(g*lambda/2π * tanh(2πd/lambda)).

For d << lambda, tanh x -> x and we get

v_{w} = sqrt(gd), that is the speed is independent on the wavelength.

What is the condition for shallow water waves?

d < lambda/4.

What is the reason for the destructive effect of tsunamis?

When the wave comes in to shallow waters, the speed of the water scales with sqrt(d), where d is the water depth. This means that it ever decreases its speed, and compensates with amplitude heigth to maintain energy conservation.

Estimate the energy of a wavefront of breadth b = 1km, with an amplitude A = 2.5 m, and wavelength lambda_{w} = 50m. Assume sinusoidal shape.

Calculate the mass of the top:

M_{top} = rho*V = rho*b*A * ∫ sin(2πx / lambda) dx from 0 to lambda/2 = rho * b * A * lambda/π.

The potential energy of this wavefront is then:

∆E_{pot} = M*g*2∆h_sp, where ∆h_sp is the center of gravity, which is π/32*A.

In this example, ∆E_{pot} = 2*10^{18}J

Derive the expression for the power per meter of a wave.

dP/dx = ∆E_{pot}/b * f_{w} = MgAπf_{w}/16b

= rho*A^{2}*g*lambda*f_{w }/ 16

Using lambda = g/2πf_{w}^{2}

=> dP/dx = 1/32π * rho * g^{2} * A^{2}/f_{w}

Describe the origin of the solar tide in the global ocean model.

This is caused by a local equilibrium of gravitational force and centrifugal force on the earth while moving around the sun. At the bright-side the gravitational pull is stronger than the centrifugal force, and we have a tide. On the dark side the centrifugal force is stronger than the gravitational pull, and we also have a tide here.

What is the average centrifugal acceleration of the earth in its orbit?

a_{cf} = r_{SE}*omega^{2}

With omega = 2π/1 year = 2*10^{-7} s^{-1 }and r_{SE} = 1.5 * 10^{11} m

we get a_{cf} = 6*10^{-3} m s^{-2}^{ }

What is the variation of the centrifugal acceleration from the sun?

∆a_{cf} = 2R_{earth}omega^{2} = 2.5 * 10^{-7 }m s^{-2}

What is the average gravitational acceleration on the earth by the sun?

g = G m_{sun} / r_{SE}^{2}

What is the variation in gravitational acceleration from the sun through the earth?

∆g = |dg/dr| * 2R_{earth} = 2Gm_{sun}/r_{SE}^{3} * 2R_{earth} ≈ 10^{-6}

Which of the changes of centrifugal acceleration and gravitational acceleration contributes most to the solar tides?

The changes in gravitational acceleration is the highest, and contribues the most to the solar tides.

What is the expected tidal period?

12h.

Describe the origin of the lunar tides.

Similar to the solar tides. Here the Earth rotates around center of gravity, which lies inside the earth, with a period of 27.3 days.

From the models we have learned about lunar and solar tides, how big tidal waves can we expect based on a global ocean model?

50 cm.

How can tidal heights be amplified?

By coasts and by so-called lambda/4 resonances.

What is lambda/4-resonances when talking about waves?

A lambda/4-resonance is an effect we get if we have bays that have a length which are a fourth of the wavelength. Then we have a phase shift of π of the ingoing and outgoing waves. Due to the phase shift of π from reflection, we get a total phase shift of 2π. We get a constructive interference between incoming and reflected tidal waves, and we can get tidal amplitudes up to 10 m.

Why has the Earth's rotation slowed over the years?

Due to mechanical energy from the Earth-moon system being lost to friction (water towards coastlines etc). Similar effects of the sun.

Name two systems of harnessing wave power.

Tapered channel: Bring waves into a tapered channel, and extract energy when water is returned into ocean.

Oscillating water column: waves causes air to flow up through a turbine. When water receeds, the turbine continues to go the same way due to clever design (Wells turbine).

Name one way of harnessing tidal power.

By having a turbine in a bay. When the tide comes in, the turbine is run, and when the tide goes out the turbine is run again.

Name one big advantage of wave power compared to for example solar power.

The available energy corresponds to our demand for energy.