Shoreline Systems

Wave Velocities

 

The velocity at which a wave (the energy and not the particles in the medium) moves is related to the water depth and the wavelength. Experiments have shown that

v2 = (gL/2p) tanh (2pd/L) where

 

v = velocity of the wave crest (m/s)
g = acceleration of gravity (9.8 m/s2)
L = wavelength (m)
d = depth of water (m)
p = 3.1415
tanh = hyperbolic trigonometric function

 

1. Estimate the velocity of a wave along a straight stretch of shoreline where the wavelength is 30 m and the depth of the water is 5 m. To do this you will need to calculate in Excel using the above formula. The value 2pd/L is expressed in radians. Using the TANH function to find the tangents of numbers will simplify the process and save you time. To use the tanh function in Excel go to Insert>Function and find TANH and click on it to insert it in an empty cell. Place the cursor in the parentheses and identify the cell with the value you want to convert. Press return. You can insert the square root function (SQRT) the same way. (Here is an instructional video in QuickTime or m4v format).

 

2. Using a constant wavelength of 30 m, construct a graph showing the calculated wave velocity (y-axis) versus depth (x-axis).  Do the calculations over the range 0.1 m to 50 m depth. Enter at least 20-30 values so your curve will be well-constrained. What is the effect of depth on the wave velocity? Include your graph on your submitted answer sheet. 

 

3. Estimate the depth of the sea floor along a straight stretch of shoreline where the wave velocity is 5 m/s and the wavelength is 30 m. The easiest way to solve this problem is to read off the depth from the graph you constructed in the previous step. If you used a spreadsheet to make the graph you can easily plug in a variety of depths until you find a velocity of 5 m/s.  Alternatively, you could rearrange the equation and solve for depth (d), but this involves taking the inverse of the hyperbolic tangent.

 

4. How does this quantitative relationship help explain wave refraction? See the figure below.

5. Tsunamis are dramatic waves that occasionally inundate shorelines. Vist this website (http://earthweb.ess.washington.edu/tsunami/) for a more detailed look at tsunamis. Play the computer-generated model of the tsunami from the 2004 Indonesian earthquake (https://www.youtube.com/watch?v=4yFNOuo_YxI), and look at the photographs of the damage at http://cwis.usc.edu/dept/tsunamis/2005/tsunamis/041226_indianOcean/sumatra/sumatra.html. Other visualizations of the tsunami can be seen at http://www.noaanews.noaa.gov/video/tsunami-worldpropagation2004.mov.

5a. What are the principal differences between regular ocean waves and a tsunami?
5b. About how high is a tsunami as it traverses the open ocean?
5c. About how high might one be when it strikes a shoreline?

5d. Why are they so much higher along the shoreline?
5e. What controls how fast a tsunami moves?