The tsunami propagation through time can also be determined precisely by using the bathymetry data with 100 km step size and EXCEL. Directions for the calculations with the original bathymetry data can be found below.
A. The data files
For each of the three tsunami paths there is a data file with bathymetry measurements: profileA.dat, profileB.dat and profileC.dat. The first few lines of profileA.dat are:
95.85 3.32 0 -756.407
95.2708 2.63133 100 -1781.25
The first column gives the longitude in degrees east, the second column the latitude in degrees north, the third column the distance from the epicentre in kilometres and the fourth column the water depth D as a negative number in meters.
B: Reading data into EXCEL (for EXCEL 2003) You can read in the data per column as follows:
i Open a new EXCEL file. Put your cursor on cell A4; this way you leave some lines open above the data.
ii Choose Data >> Import external Data >> Import Data. Browse for the right data file and choose All Files for the option File Types. Click Open.
iii Every time you click Next, new options will appear. The Data Format screen is especially important. Select (with Shift-Click) all columns, choose Advanced and switch the decimal separator to . (point) and the thousands separator to ‘ (quotation mark).
iv Click Finish and check to see that the starting cell is A4. [This is, of course, not obligatory, but the example below assumes your data starts at A4.]
C: The bathymetric profile
You can image the bathymetric profile right away: Click on the depth D column and choose Insert >> Chart >> Line. You do not need the graph for any calculations, but you might want to use graphs for step E.
D: The calculations Methods in short:
Determine the velocity in each 100-km segment; determine the time needed for the tsunami to cross each segment; sum the segment travel times. You can give these calculations in the first data line and then copy them up to the last line of data. EXCEL will then automatically do the other calculations for you. (Be careful to start calculations with an = sign.)
Methods in detail:
– Use cell F4 for the tsunami velocity (
v
gh
). Type: = SQRT(9.81 · - D4) [the reference to cell D4 can be given by clicking on D4].– Use cell G4 for the travel time per segment. Each segment is 100 km long and the velocity in F4 is in meter per second. Compose a proper equation for the travel time in seconds with a reference to cell F4.
– Sum up travel times in column H. In the cells of column H, you sum up the value of the neighbouring G-cell and the H-cell above. So, for the first data line, you sum up G4 and H3. H3 is empty, but EXCEL will attribute it value 0, or you can assign it value 0 yourself.
- Now you have your first complete line of calculations. Copy these calculations by selecting cells F4-G4-H4, putting your cursor on the right lower corner of the selection and dragging it to the last line of data. Column H will now tell you when the wave passed each distance in column C.
– Column H gives the time in seconds; make an extra column that gives the time in hours. E: Improvements and further investigations
a. Do you think your calculations would be more accurate if you were to use the average of the velocities at the beginning and end of each segment as segment velocity? You can try this method with EXCEL or predict its effects yourself.
c. You can also construct other graphs. For example, a plot of distance versus time. Do this by copying column C to the right of your last column, selecting both columns and making a Chart of the scatter-type.
Figure 3.20: The bathymetric profile along path C.
Figure 3.20: The bathymetric profile along path B.
Final exercise Ch3. Answer the section questions and the main question
a. Answer the six section questions and the main question from the beginning of this chapter. b. If you find you have new questions after reading this chapter, write them down.
Optional exercise 3-1: Earthquakes and seismic waves (From a final exam in Physics, Havo 1999-I, question 6)
During an earthquake, longitudinal and transverse waves (the P- and S-waves of Section 3.1) travel through the Earth.
a. What is the difference between longitudinal and transverse waves?
In a certain type of rock, transverse waves have a velocity of 3.4 km/s and a frequency of 1.2 Hz. b. Calculate the wavelength of the transverse waves in this rock.
Seismographs register earthquake vibrations. Figure 3.21 shows a simple type of seismograph: A heavy block hangs suspended on a spring and can move freely, but only in the vertical plane (because of hinge A). During an earthquake, the spring-block system is not allowed to resonate with the earthquake vibrations. To prevent resonance, the frequency of the spring and block is small (only 0.37 Hz) compared to the frequency of the earthquake’s vibrations. The mass of the block is 4.2 kg.
c. Calculate the spring constant.
The velocity of longitudinal waves differs from that of transverse waves. Due to this difference, the waves do not arrive at a seismic station at the same time. Figure 3.22 shows a recording of the seismograph of an earthquake in Greece, measured by the KNMI in De Bilt, The Netherlands. The L denotes the arrival of the longitudinal waves, the T that of the transverse waves. You can see that the longitudinal waves arrived first. It is assumed that both types of waves followed the same path. The earthquake took place 2300 km away from the seismograph. Take the average velocity of transverse waves as 3.4 km/s.
d. Determine the average velocity of the longitudinal waves in two significant digits.
Figure 3.22: Registration of an earthquake in Greece. Figure 3.21: A seismographs. Spring
Optional exercise 3-2: Resultant rotations as resultant vectors
In this chapter we have investigated the resultant motion of two plates, the motion of one plate with respect to the other. Focus on the equators especially on the equators belonging to those motions, to prove the fourth point stated just above Exercise 3-5: when you use vectors in the direction of the rotation axis with a length proportional to the angular velocity to represent rotations, you can use vector addition.
Study Figure 3.23. Rotation poles A and B are indicated together with their angular velocities ωA and ωB. (In this case, ωA : ωB
= 2 : 1, but the following deductions are valid for every velocity ratio.) Equators cA
and cB belong to poles A and B,
respectively, and cross in point P. The actual velocity of P as a point on the A- shell is vA and that of P as part of the B-
shell is vB.
a. Explain why |vA| : |vB| = ωA : ωB. Hint:
Use v = ω·R, with R the radius of the Earth (see BINAS).
b. Explain why the angle between the axis of rotation of A and that of B is equal to the angle between vA and vB.
Using the above definitions, we can draw two vectors, starting in the centre M, along the rotation axes of A and B with their length ratio equal to the ratio ωA : ωB. Call the vectors mA and mB; mA is
already drawn in. c. Also draw in mB.
Vector couples vA-vB and mA-mB are similar (they have equal angles and proportional lengths).
The resultant velocity vC, the difference between the two velocity vectors vA and vB, also works on
point P. We know the resultant motion has a rotation pole C as well. C should lie on the great circle through A and B, because in C the motion around A should equal the motion around B.
d. We are going to try to draw C and the arc CP. CP is perpendicular to vC. Why?
e. Draw the equator of C and then construct arc CP and rotation pole C.
The angular velocity ωC is determined by the magnitude of vC. The following relationship holds,
which is also valid if you replace B by A: |vC| : |vB| = ωC : ωB.
f. Explain why the above relationship holds. Starting at M you can draw resultant vector mC.
g. Why does mC lie along the rotation axis of C?
Using mC, you can determine C and angular velocity ωC directly, without using P.
h. Draw in vectors mA and mB along the axes of A and B with the same magnitude ratio as ωA and
ωB.
i. Determine resultant vector mC.
The direction of vector mC gives the position of C and its magnitude gives ωC.
To summarize: The arrows in the direction of the axes of the rotation poles and with magnitudes proportional to the angular velocities act like vectors in case of resultant motions. You can determine the resultant vector through vector addition.
Chapter 4.
Volcanoes
The main questions for this optional chapter are: