Names:______Thursday 1:30 pm
Waves
Waves are termed deep-water when their orbitals do not reach down far enough to touch or 'feel' the bottom of the ocean. For deep-water waves, the celerity (wave speed in cm / second) is:
CDEEP = (g * WL / 2π)1/2
where WL = wavelength (cm) and g = gravitational constant (approx 980 cm / s2). This can also be expressed using the familiar equation for speed: distance divided by time (Period = seconds per cycle):
CDEEP = WL / Period
Once the ocean is shallow enough for a wave’s orbitals touch the seafloor, it becomes a shallow-water wave. The orbital circles get squashed into ellipses, then into a back and forth sloshing motion as the depth decreases. The wave's entire energy that used to extend deep underwater is concentrated in a small volume near the coast, causing the height of the wave to increase. Water molecules moving in orbitals in contact with the bottom are slowed down by friction with the seafloor and the entire wave’s speed changes. Called shallow-water waves, their celerity depends only on the water depth and not the wave’s properties. The celerity of shallow-water waves is:
CSHALLOW = (g * water depth)1/2 g = 980 cm / s2
Question: A 10 meter high, 100 km long Tsunami wave and a 1 meter high, 50 meter long wind wave are both traveling through water that is 10 meters deep. How fast does each travel?
Celerity of shallow-water traveling waves
This experiment calculates shallow-water waves’ celerity by timing their progress for one complete roundtrip across the aquarium (from one side to the other and back).
Water Depth / Time, 1st try (sec) / Time, 2nd try (sec) / Average Time (sec) / Measured Celerity (cm/s) / Theoretical Celerity (cm/s)1.0 cm
1.5 cm
2.0 cm
2.5 cm
3.0 cm
4.0 cm
To calculate the celerity of the waves just measured, divide the distance by time:
The interior length of the aquarium is 59 cm long, so the distance traveled in one roundtrip is 118 cm.
Measured celerity (cm / sec) = distance (cm) / time (sec) = 118 cm / Average Time
For the theoretical celerity, use the shallow-water wave celerity equation:
CSHALLOW (cm / s) = [980 (cm / s2) * water depth (cm)]½
Graphing the Data
Plot water depth on the independent (horizontal) axis and celerity on the dependent (vertical) axis. (0,0) is also a data point! Plot the data points for measured celerity with circles and connect the points with a smooth curve. Plot the data points for theoretical celerity with ‘X’ and connect with a smooth curve.
Question: Your measured celerity was probably a little bit less than the theoretical celerity; especially for 4 cm, why? (hint: these waves have a very short wavelength and are not traveling in the ocean)
Question: Which will travel faster: A) a shallow-water wave in 10 cm deep water on earth, or B) a shallow-water wave in 10 cm deep water in an imaginary ocean on Mars? (hint: Mars’ gravity is less than half as strong as Earth’s: gMARS = 375 cm / s2). Why?
Standing wave period
Wind waves and tsunamis are progressive waves, they travel and transport energy from one location to another. In harbors, bays, lakes, or any closed body of water (such as a bathtub), waves of a different kind are found. Called standing waves, they do not change location. Rather, water sloshes periodically back and forth across the basin. The disturbing force (energy) for these 'seiche' waves can be the wakes of boats, wind blowing water to one side of the basin, and interactions with the tide (also a wave). The standing waves created in different basins will have very different properties; the basin’s size, shape, depth, and the irregular geometry found in the Real World will produce a wave unique to each environment.
Standing waves obey the same laws of physics as progressive waves and will travel at the shallow-wave celerity:
CSHALLOW = (g * water depth)1/2
Their period can be calculated by dividing their distance traveled in one complete cycle by their speed (the distance traveled in one complete cycle is the wavelength).
Period (s) = wavelength (cm) / celerity (cm / s)
Water Depth / Time 1st Try (sec) / Time 2nd Try (sec) / Average Time (sec) / Measured Period (sec) / Theoretical Period (sec)10 cm
15 cm
20 cm
Period is the time between wave crests, so one roundtrip across the aquarium is one period.
By timing ten roundtrips, then dividing by 10, the duration of a single period is found (taking the average of ten is much more accurate than trying to time just one).
Measured period (sec) = Average time (sec) / 10
For the theoretical celerity, since:
Velocity = Distance / Time so Time = Distance / Velocity
Theoretical Period (s) = 118 cm / CSHALLOW
Question: If the aquarium were twice as long, how much larger would the standing wave’s period be?
Question: Consider again the imaginary ocean on Mars (less than half the gravity of Earth: gMARS = 375 cm/s2). Will an otherwise identical standing wave have a longer period on Earth or Mars? (hint: use the definitions of a standing wave’s period and shallow water wave celerity).
Question: Think about the shallow-water celerity and standing wave period experiments. What errors (both the experiment and its human conductors) might have occurred? (be specific, not just ‘human error’)
Water MovementBeneath Standing Wave
Standing waves will be generated in tanks with sand grains at the bottom to help visualize how the water is moving beneath a standing wave.
Question: Where along the bottom of the aquarium do the sand grains move the most?
The least?
Question: Where are the currents along the bottom of the aquarium the strongest? The weakest?
Question: Compare the movement of the water’s surface at 1) the middle, and 2) the ends of the aquarium. Which moves the most? How does this match up with the currents at the bottom of the aquarium?
Question: What shape are the wave orbitals at the bottom of the aquarium? Why aren’t they perfectly circular?
Capillary wave (ripple) tank demonstration
On the side of the room is a capillary wave (ripple) tank set up with a strobe light and mirror to make the wave crests visible. The end of the tank away from the ripple generator is raised up about 3 cm to create a “shoreline”. The ripples are being generated at an angle to the shoreline.
Question: What is happening to the celerity as the waves approach the shoreline? What about the wavelength?
Question: How does the angle change as the waves move from deeper into shallower water? Why does this happen?
Question: Imagine a coastline with a point sticking out to sea in the middle and deep bays on either side. If waves always bend towards shallower water, what part of this coastline will receive the most wave energy? The least?
Internal waves
The waves considered so far have all occurred at the ocean’s surface. It can support waves because it is a density interface (different density material on either side) between air and water. The vast majority of the ocean has a dramatic change in water density between 10 and 500 meters below the surface. Called the pycnocline, it separates the warm, fresher, lighter surface water from the cold, saltier, heavier deep water. Because it marks a change in density, the pycnocline is able to support waves just like the ocean’s surface.
Waves occurring at the pycnocline’s density interface are called internal waves. Their most common source of energy is the tide; as the tide wave moves through the ocean basin its force interacts with the complex ocean bathymetry (underwater topography) creating internal waves. When looking out over the ocean on a sunny afternoon, notice the bands of different texture and color water. Many of these are the surface signs of internal waves. Their motion creates zones of convergence at the sea surface. These areas collect oil, dirt, and organic material that change the water’s surface tension. Capillary waves become harder to sustain and the surface will appear darker and smoother than surrounding areas.
The instructor has prepared a long, rectangular tank to generate an internal wave along the pycnocline. On the bottom is a layer of 50C, salty, dense water colored blue. A top layer of 300C, 0‰, less dense water colored red/yellow was very slowly added by dripping onto a thin film floating on top of the dense bottom layer.
50 cm Travel Time (sec) / Celerity (cm/sec)Student #1
Student #2
Instructor
Average Celerity:______
Through careful laboratory experiments, oceanographers have derived the following equation for the theoretical speed of an internal wave:
CINTERNAL = [g * ((d1 - d2) / d1) * ((h1 * h2) / (h1 + h2))]1/2
where d1 = lower layer density, d2 = upper layer density, h1 = lower layer thickness, and h2 = upper layer thickness.
Write down these values below and calculate the internal wave’s theoretical celerity.
d1 = ______g / ml, d2 = ______g / ml
h1 = ______cm, h2 = ______cm, Theoretical celerity = ______cm/s
Question: Why was the internal wave celerity you measured different from the theoretical celerity from an equation? (hint: what is different between the real ocean and the internal wave tank?)