standing wave

19-Jul-10 PHYS102

KFUPM PHYSICS Department of Physics revised 19/07/2010 Dhahran 31261 68

Cal Lab (EXCEL 2007 version)

Simulation of Standing Waves on a String A step by step, quick tutorial on how to make sine waves in EXCEL, make them travel and make them stand by summing two traveling waves! Background Assume that we have a string two meters long (L=2 m) fixed at both ends (as in your experiment Resonance in a String) with the frequency of the vibrator given by = 2f =10 rad/sec. Enough mass is hung to make it resonate in four segments (n = 4). And choose the amplitude ym = 0.2 m. With these choices of values, we have = 1 m (from L= n/2 for resonance), and k = 2. The wave function for the waves traveling to the right will be given by y1 = ym sin(kx - t) = 0.2 sin(2x - 10t) and the wave function for the waves traveling to the left will be given by: y2 = ym sin(kx + t) = 0.2 sin(2x + 10t) In this exercise using EXCEL, we will create data points for the two waves:

y1 = 0.2 sin(2x - 10t) and y2 = 0.2 sin(2x + 10t)

and the superposition of the two waves as y1 + y2 = 0.2 sin(2x - 10t) + 0.2 sin(2x + 10t)

for each values of x starting from one fixed end of the string (x=0) to the other fixed end (x = 2m) in increment of x = 0.01m. Initially, the value of time t will be set to 0. Then y1, y2, and y1+y2 are plotted against x in the same graph. We then obtain the animation effect by dynamically changing the value of t in increment of t=0.01s continuously. As new set of data points are created for each value of t, EXCEL will dynamically re-plot the points each time in the chart creating the animation effects of the two travelling waves and the resultant standing wave. NOTE that y1+y2 is calculated at EACH point numerically using the simple sum y1+y2 = 0.2 sin(2x - 10t)+ 0.2 sin(2x + 10t). We do NOT resort to the analytical result of the formula for standing wave. Thus we are going to see what happens when we send two identical waves in the opposite direction in a string satisfying the resonance condition (L = 2). Standing Wave Animation using EXCEL

1. Open Microsoft Office Excel 2007. Type in the numbers and words exactly as shown below in the cells from A1 to C6:

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2. Select the cell B1 and then click the Name box area. Type k in the Name Box (NOT in the cell B1) and press Enter. This step gives the absolute name k to this cell. From now on we can use the name k to refer to this cell.

3. Continue giving names in the same manner for other cells B2, B3, B4, B5, and B6 as follows:

Cell Name B2 ym B3 Delta_x B4 Delta_t B5 Omega B6 t

4. Enter the headings and the formula in cells from D1 to G3 as shown below:

5. When you have completed step 3, you will see the end result like this:

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6. Select the cells starting from D3 all the way to G202. With these selected, on the

Home tab, in the Editing group, click Fill and choose Down.

You will then see that those cells are automatically calculated for each x values starting from 0.01 m to 2.00 m in steps of 0.01 m.

7. Now select the cells from D1 (with the headings) all the way to G202. After selecting

the cells scroll up to the top of the sheet so you can see the 1st row.

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8. On the Insert tab, in the Charts group, click Scatter and choose Scatter with only markers chart type (the first choice).

9. When you are done with step above, you will see the x-y scatter chart with y1, y2 and y in the vertical axis plotted against x in the horizontal axis. If the chart is placed above any data columns, move it away such that it is in an empty area.

You are not seeing the data points for y1 (in blue here) as it is masked by the data points of y2 (red here). If you want to see the blue wave (y1) change the time t in cell B6 to 0.05 and see what happen.

Fixing the vertical axis Scale

10. Right click (click with the right mouse) anywhere on the numbers of vertical axes of the graph and choose Format Axes. In the Format Axes box, in the Axes Option, just make sure the Fixed Radio buttons are checked in Minimum, Maximum, Major unit, and Minor Unit. Do not change the numbers; leave them at the default values chosen by your computer.

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Fixing the horizontal axis Scale

11. Right click any of the x-axis numbers, and choose Format Axes. In the Format Axes box, in the Axes Option, make sure the Fixed Radio buttons are checked in Minimum, Maximum, Major unit, and Minor Unit, AND change Maximum value to 2.0 m.

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Can you tell why these steps (9 & 10) to disable the automatic setting of the scales, are important?

You are now ready to Animate the Waves

12. Click the Office Button , and click Excel Options. Choose Formulas on the left items, and make changes in the Calculation Options as shown below:

Make sure Enable Iterative Calculation box is checked and the Maximum Change value is set to 1.

13. Go to cell B6 and type =B6+Delta_t. Note the cell B6 refers to itself (this is called

circular reference)

14. Now press continuously (keep it pressed) the f9 key (this makes EXCEL calculate the

worksheet formulas again and again for the new values generated for time t each time iteration) and you will see the waves animated, y1 moving to the right, y2 moving to the left, and y is standing (not traveling to the right or to the left).

Congratulations! You are done. To restart iterations from t=0, type 0 in the cell B6. Then repeat steps 13 and 14. OPTIONAL: For more intellectually challenged students: 1. See what happens when the amplitudes of y1 and y2 differ 2. See what happens when you add a phase constant to wave y2. Assign different

values ( = /2, 3/2 etc..) 3. See what happens when you make y2 also to travel in the +ve x-direction.

Cal Lab (EXCEL 2007 version)Simulation of Standing Waves on a StringA step by step, quick tutorial on how to make sine waves in EXCEL, make them travel and make them stand by summing two traveling waves!

standing wave

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