# Speed of a Wave

 Materials: Computer and School Network Time Allotment: 4 Class Days

### Purpose:

The purpose of this lab is to investigate some variables which effect the speed of a wave and to use graphical analysis methods to derive and interpret wave speed equations based on collected data.

### 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. Once the Physics Explorer-Waves application; chose Open... from the File menu.
4. A directory dialogue box should appear. Open the Waves folder, and then the Exploring Waves 1 Makeup file by double-clicking on its icon.
5. Click on the button at the bottom of the monitor which is titled "Stiffness and Mass."

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. The properties of the medium (and of the wave) can be set by the user using data input boxes.

### Part A - Effect of Spring Constant and Mass on Speed:

In the first part of this activity you will investigate the effect of two media properties (stiffness and mass) upon wave speed.

1. The stiffness or elasticity of a medium through which a wave is moving is often measured in terms of the spring constant. The higher the spring constant, the more stiff (and the less elastic) that the medium is. If the medium is stiffer (if the spring constant of the springs connecting the individual masses is increased), do you think that the pulse will travel faster or slower along the string? __________ Explain your answer.

2. Keeping the mass m = 1 kg, you will make the necessary measurements to calculate the speed for various values of the spring constant (k). Use spring constant values between 1 and 16. The speed can be determined by measuring the distance traveled per time. There is a timer in the window on the left. This timer can be used to measure the time for the pulse to move down the medium and back. The distance down the medium is 50 meters; so down and back would be a total distance of 100 meters.

 Sping Constant, k (N/m) Distance (m) Time (s) Speed (m/s) ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________

3. Now consider the effect of the particles' mass on the wave's speed. How do you think an increase in the masses of the particles of the medium will influence the wave's speed of travel? Explain your answer.

4. To check your answer, keep k = 10 N/m and measure the speed for a particles of varying mass between 1 and 16 kg. Record your measurements and calculations in the table.

### Mass, m (kg) Distance (m) Time (s) Speed (m/s) ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________ ________

For each set of data, utilize the Graphical Analysis program (found on all school servers) to determine the quantitative relationship between the variables under study. Construct a plot of speed vs. spring constant (speed is plotted along the vertical axis) and speed vs. mass ( speed is plotted along the vertical axis). Be sure that the Regression Line, Point Protectors, and Statistics options are selected under the Graph menu and that each axis is labeled with a quantity symbol (v, k, and m) and a unit. For each graph, raise the quantity plotted along the horizontal axis to some power in order to obtain a line with the best-fit (refer to the Graphical Analysis manual or relevant Survival Packet pages). Once a best-fit line has been obtained and a COR of 0.996 or less is obtained, print the graphs (with Statistics showing) and submit with your final lab report.

In the table below, write the equations representing the quantitative relationship between the two variables plotted along each graph.

### Speed vs. Mass

________________________

________________________

### Part B - Effect of Frequency and Wavelength upon Wave Speed:

Many waves are not just single disturbances, but are continous phenomenon, periodic in both time and space. In the Part B of this activity you will investigate the effect of two wave properties (frequency and wavelength) upon wave speed.

1. Click on the button at the bottom of the monitor which is titled "To Part 2." A different window will appear at the bottom of the monitor. Click on the "Traveling Waves" button in this window. Then check that the values in the "Traveling Waves" window are as shown:

Run the simulation and watch the wave train travel down the string.

2. Use the following steps to calculate the speed of propagation of the traveling wave.
1. Note the time at which the first peak of the wave "hits" the first detector (i.e., the particle begins its initial upward displacement from rest); the first detector is positioned on the 10th particle.
2. Continue the simulation and find the time for the wave to "hit" the seond detector (i.e., the particle begins its initial upward displacement from rest); the second detector is positioned on the 40th particle.
3. From your results calculate the time for the wave to pass from the first detector to the second detector.
4. Select the RULER tool (see diagram at right) and measure the horizontal distance between the two detectors.
5. Calculate the speed of the wave by dividing the distance into the time.

How do your results compare to the results in Part A for the same values of mass (1 kg) and spring constant (10 N/m)?

3. There are two important parameters characterizing a periodic waves:
• wavelength - the distance bewtween two consecutive crests of a wave (or the distance between any two corresponding points on adjacent waves).
• frequency (f) - the number of complete oscillations per second.

1. Set the frequency to the first value as shown in the table below. Run the simulation.
2. Measure the wavelength with the RULER tool.

3. Calculate the velocity. Record all results in the table below.

 Frequency (Hz) Wavelength (m) Time (s) Distance, D2-D1 (m) Speed (m/s) _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________ _________

4. Based on the table, how are the frequency, wavelength and speed related? That is what effect does an increase in frequency have upon the wavelength and the wave speed? (This relation holds for a variety of phenomenon, such as waves of sound, light, water, and strings.)

### Part C - Investigation of Wave Speed for a Standing Wave:

A standing wave is a wave in which there are points along the medium which appear to be standing still and still other points along the medium which undergo a maximum displacement. Standing waves are the result of two waves within the same medium and traveling different directions such that they interfere and produce a wave pattern known as a standing wave. In Part C of this lab, you will investigate the effect of a few variables upon the speed of a standing wave. You will do this by simulating the motion of a guitar string. A string with both ends fixed is brought into the shape of a full sine curve and then left alone to start vibrating.

1. Click on the button "Standing Waves" in the bottom window.

Check that the "Standing Waves" window shows these values.

Run the simulation and watch the string begin moving.

• Describe the differences and the similarities between the "standing wave" in Part C and the traveling waves observed in Parts A and B.

2. The graph in the "Standing Waves" window is set to draw the displacement vs. time for the three detectors (particles #13, #22, #37). Use the graph, and if necessary, move the detectors to answer the following questions.
• Which particles along the medium appear not to move at all? (Particles that do not move are termed "nodes.")

• Which particles along the medium appear to move with a maximum amount of displacement? (Particles that have maximum amplitude of motion are termed "antinodes.")

3. Run the simulation and find the time it takes for the "wave" in this activity to return to its intial position. (This is the period of the oscillation.)

Change the number of starting wavelengths found between the ends of the medium and repeat the simulation. Mesure the wavelengths in each case and complete the table below. Use the reciprocal of the period to calculate the frequency. Use the frequency and the wavelength to calculate the speed of the wave.

 Number of Wavelengths Wavelength (m) Period (s) Frequency (Hz) Speed (m/s) __________ __________ ________ __________ ________ __________ __________ ________ __________ ________ __________ __________ ________ __________ ________ __________ __________ ________ __________ ________

What is the relationship between these values? Explain.

### Conclusion:

Using a well-written paragraph, discuss all the variables which were studied in this lab and identify their effect (if any) upon the wave speed. Do a bang-up job!