# Boundary Behavior

 Materials: Computer and School Network Time Allotment: 3 Class Days

### Purpose:

The purpose of this lab is to investigate the behavior of a wave at the boundary between two different media and to make some generalizations about the relationships between the energy and amplitude of the incident wave (the wave approaching the boundary) and the energy and amplitude of the reflected and transmitted waves..

### Getting Started:

1. Log on to the student server in the usual manner.
2. Open the Math/Science folder; then open the Science Apps folder; open the Physics folder.
3. Open the Physics Explorer-Waves application; chose Open... from the File menu.
4. A directory dialogue box should appear. Open the Physics 163 folder, the Waves folder, and then the Core 2 - Reflection and Transmission file by double-clicking on its icon.

The Physics Explorer - Waves software models the motion of a wave (or a pulse) along a medium. The medium is depicted as a string of 50 masses (representing the particles of the medium) connected by springs. As a disturbance or pulse is introduced into the medium, it moves from one mass to the next. The masses, being connected to each other by springs, interact with each other in order to transmit the energy of the disturbance along the medium. Two media can be linked together such that a pulse sent from one end will pass from one medium into another medium near the middle of the simulation window.

### Part A - Energy Distribution:

If you look at a window you will see your face reflected in the glass. A friend looking at you from the other side of the window will see you through the glass. Both of you can see your face because some of the light rays reflected from your face are reflected from the glass window and back to your eyes and some are transmitted through the glass window to the eyes of your friend. How much is transmitted and how much is reflected? How do these quantities depend on the properties of the medium carrying the waves? These questions, and others are the subject of this investigation.

1. If the values of spring constant (k) and mass (m) are the same for both parts of the string, then what will happen to the energy in medium 1 as a wave progresses across the boundary to medium 2? Sketch your prediction of the graph of energy in medium 1 vs. time on the graph below.

Show how you expect the graph of total system energy (in media 1 and 2) graph to look as the wave passes the boundary.

2. Open the Energy Distribution window by clicking on the "Energy Data" button found in the window on the left. Run the simulation and observe the energy in medium 1 vs. time graph and the total system energy bar graph. Were your predictions correct?

1. Before the wave reaches the boundary: What percent of the total energy is found in medium 1?

2. Before the wave reaches the boundary: What portion of the total system energy is in medium 2?

3. When the wave has completely crossed the boundary, what is the distribution of energy between the two media? That is, what percent is in medium 1 and what percent is in medium 2?

3. Now repeat the simulation with several higher values of mass (m) for medium 2.
1. What happens to the distribution of energy after the wave meets the boundary? That is, by what percentages is the total system system energy distributed between media 1 and 2?

2. Look closely at the wave motion. Can you see where the energy goes that is not being transmitted across the boundary?

4. Experiment with different values and combination of values for spring constant (k) and mass (m) for medium 2. Observe the results and answer the following questions.
1. What is the minimum transmission of energy which you can attain?

2. With what combination of mass (m) and spring constant (k) values did you observe this?

5. The energy not transmitted is reflected.

When the relationship between the particle masses and the spring constants of the two media is such that all of the energy is not transmitted across the boundary, a wave is returned (reflected) from the interface or boundary. This reflected wave carries that portion of the energy that is not transferred into medium 2. We will now investigate the amplitude and the energy distribution among the incident, transmitted and reflected pulses.

### Part B - Amplitude and Energy Distribution:

When a wave reaches the end of a medium, a portion of its energy is transmitted into the next medium and a portion of its energy is reflected off the boundary and remains in the same medium. Part B of this lab allows you to study the variables which effect the amount of energy transmitted and reflected at the boundary.

1. Set the values of spring constant (k) and mass (m) to the following values:

Click on the "Amplitude Data" button. The window that opens contains displays of the total energy in each part of the string, and the amplitude of motion of a particle in the middle of each section of the string. Note that the two particles being traced - #12 and #37 - are indicated by the detector mark ("-") above them.

2. To examine the incident wave, run the simulation. Stop the motion when the crest of the wave is almost at particle #12. Use the "Single Step" button to advance the wave until #12 is just at the crest of the wave. Record the maximum displacement or amplitude (of particle #12) and the total energy in medium 1.
Incident wave:

Amplitude, Ai = __________________

Energy, Ei = __________________

3. Allow the wave to continue until it reaches the boundary between the media and watch for the reflected wave. When the reflected wave passess particle #12, "Single Step" the simulation until the crest of the reflected wave has reached particle #12. Record the maximum displacement or amplitdue (of particle #12) and the total energy in medium 1 as you did for the incident wave.
Reflected wave:

Amplitude, Ai = __________________

Energy, Ei = __________________

4. Rerun the simulation and use the same method to record the maximum displacement or amplitude of the transmitted wave (using the data from particle #37) and the energy in medium 2.
Transmitted wave:

Amplitude, Ai = __________________

Energy, Ei = __________________

5. Using the same method as in steps 3 and 4, repeat the simulation several times for the following values of k2 and m2 (spring constant and mass for medium 2). Complete the table below. (Important: Keep the mass of medium 1 set to 2 kg and the spring constant of medium 1 set to 2 N/m.)

Spring
Constant
of
medium 2

Mass of
medium 2

Incident
Wave

 Ei Ai

Reflected
Wave

 Er Ar

Transmitted
Wave

 Et At

2

8

 ____ ____

 ____ ____

 ____ ____

4

8

 ____ ____

 ____ ____

 ____ ____

8

4

 ____ ____

 ____ ____

 ____ ____

2

0.5

 ____ ____

 ____ ____

 ____ ____

_________

_________

 ____ ____

 ____ ____

 ____ ____

Some of the results may surprise you; you can check results that seem "strange" by running the simulation again.

### Part C - Energy Analysis:

The amount of energy transmitted and reflected at the boundary is dependent upon the properties of the two media on each side of the boundary. The total incident energy is divided into both media, and the fraction which is proportioned to the two media can be represented by the coefficient of R and a coefficient of transmission T.

1. Let's look at the energies transported by the waves (incident wave, reflected wave, and transmitted wave) in Part B of this lab.
• Based on your tabulated values of Ei, Er and Et, what appears to be the algebraic relationship between the total energies of the three waves? Write an equaion relating these three energies.

Check that your relationship holds for another set of values of spring constant (k2) and mass (m2) for medium 2. Record your data in the last row of the table in Part B, section 5.

2. We can define a coefficient of reflection R and a coefficient of transmission T as follows:

R and T are the fraction of incident wave energy that is reflected and transmitted, respectively. Use your results from the previous section (Part B) to calculated values of R and T for your previous simulations. (Remember: The mass of medium 1 was set to 2 kg and the spring constant of medium 1 was set to 2 N/m.)

k2
(N/m)

#### m2 (kg)

R

T

2

8

_________

_________

4

8

_________

_________

8

4

_________

_________

2

0.5

_________

_________

_________

_________

_________

_________

• What relationship (direct, inverse, etc.) exists between R and T? _________________ Explain.

• The values of R and T depend upon the similarity of the two media. Two media are most similar if their ratio of k/m is the same. From analysis of the data in the table above, what can be said about the magnitude of R and T (large, small) when the media are similar? Support your answer by making reference to specific trials.

### Part D - Amplitude Analysis:

The energy carried by the wave was only apparent by obseving the energy charts in the Physics Explorer software. A more visible parameter to observe is the amplitude of each of the three waves (incident wave, reflected wave and transmitted wave). Data from Part B of this lab can be used to make some generalizations about the amplitudes of the three waves.

1. Make the following predictions about the relationships between the amplitude of the three waves:
• What relationship would you expect among the amplitudes of the three waves?

• Should this parallel the relationship among energies which you discovered in Part C? Explain.

2. Analyze the values you obtained for Ai, Ar, and At in the table of Part B. Can you find a mathematical relationship between them? Write that relationship below.

This somewhat surprising result (which you wrote above) is due to the fact that at the instant when the incident, reflected, and transmitted waves are all at the boundary between the media, the amplitude of the two waves (incident and reflected) on the medium 1 side of the boundary must equal the amplitude of the waves (transmitted) on the medium 2 side of the boundary. Otherwise, the two sides would not meedt and there would have to be a break in the string!

3. Try additional values of k and m for both media to see if the relationship holds. Record your findings below.

### Part E - Properties of the Media:

In the final part of this lab you will explore how the properties of the two adjoining media effect the amount of transmission and reflection.

1. For the following values of k and m for the two media, measure energies of the three waves (incident wave, reflected wave and transmitted wave) as you did in Part B of this lab; record your data in the table below; and calculate the coefficients of transmission (T) and reflection (R) for the following combinations of masses (m) and spring constants (k) of the media.

Trial

Medium 1
 k m

Medium 2
 k m

Coefficients
 T R

1

 2 2

 1 1

 ____ ____

2

 4 1

 1 1

 ____ ____

3

 4 4

 1 1

 ____ ____

4

 10 10

 1 1

 ____ ____

5*

 1 1

 4 4

 ____ ____

*Reduce the T-factor to 0.3 for this measurement.

• Which of the five trial results in your table appear to yield the same division of energy among the two media (although the k and m values are different)?

• Can you find another set of values that give the same results as trial #4? Record your findings below.

2. By varying values of k and m for both media, try to find a combination for which there is no reflected wave, i.e., T =1 and R = 0. The simple solution, of course, is when the media are indistingusihable, so that k1 = k2 and m1 = m2. Other cases exist too. See if you can find such cases. Record your findings below.

### Conclusion:

Write a short paragraph in which you summarize the complete behavior of a wave at the boundary between two media and the variables which effect this behavior. Do a bang-up job!