Introduction to Waves If you do not have access to equipment, the following experiments can be observed here: http://tinyurl.com/lupz3dh 1.1 There is a tray with water in it. This can model throwing a rock into a pond or pushing a beach ball down in a pool. Throw a rock in, or push down on a ball, and observe what happens to the water. What do you observe about the water after the rock/ball interacts with it? 1.2 Grab a slinky and a partner. Place the slinky horizontally on the floor stretch it holding at both ends. Let one of you generate an abrupt pulse by shaking your hand side to side and then stopping. Observe the motion of the pulse. a. What do you observe? b. What is similar to the experiment with the rock in the pond? What is different? 1.3 Devise a mechanism that explains how the pulse propagates along the slinky. a. What do you think affects the speed at which the pulse propagates? b. What is the role of the person shaking the end of the slinky? c. What is the role of the coils? 1.4 Push or strum the slinky instead of shaking it. To do this stretch the slinky between two students and pull several coils of the slinky toward your hand. Let go. A pulse will travel down the length. How is this wave different from the one in part 1.3 and how is it similar? Longitudinal and transverse waves In a longitudinal wave the vibrational motion of the particles or layers of the medium is parallel to the direction of propagation of the disturbance. In a transverse wave the vibrational motion of the particles or layers of the medium is perpendicular to the direction of propagation of the disturbance.
1.5 The Wave Wiggler ( https://serc.carleton.edu/dmvideos/videos/wave_properties.html ) Open the video ( http://s3-us-west-2.amazonaws.com/dmvideos.org/players/waves_grid/waves_grid_3d.html ). Position the video so you can read these instructions and view the video on your screen. Press play and watch. A high-speed (slow motion) camera attached to the ceiling shows the wave moving through the spring. Here s how the wave is produced: The aluminum rod at the left end of the spring wiggles. The rod is wiggled by a large sub-woofer from a car stereo. The sub-woofer is activated by an amplifier, which receives its signal from signal-generating software on a laptop computer. The student sitting on the floor controls the amplitude and frequency of the wave. We affectionately call this apparatus the wave wiggler. There are tools you can use to study the wave motion: 1. Your keyboard spacebar will play and pause the video. 2. If the video is paused, the arrow keys on your keyboard will move forward or backward one frame at a time. 3. Click the ruler button to bring a floating ruler that you can drag and position anywhere on the video. 4. The stopwatch button brings up a stopwatch. You can position the stopwatch where ever you like on the video. The stopwatch shows frames and time. If the video is paused, you can press set to 0 to reset the stopwatch. 5. Using the drop-down boxes at the lower left of the video, you can adjust three parameters: the wave frequency, wave amplitude, and spring tension. Activities I. Waves investigations 1. Choose the point marked with an orange dot on spring on the video "frequency 1 amplitude E Tension 1 and observe it carefully. Does it move vertically? horizontally? Can you determine the period of its motion? Draw y(t) graph for this point. What causes that point on the spring to move the way it does? 2. Play with the video, changing available physical quantities. Describe in words what you observe when you change each variable. 3. Draw vertical displacement vs. horizontal position graph at one particular instant of wave motion (a snapshot of the wave at a specific instant in time) Label the vertical axis y and horizontal axis - x. Explain why points with different x-values have different y-displacements. How would this graph look different a short time later? 4. The time it takes one point on the spring to complete one cycle of its motion is called the period. The distance between two closest points along the spring that are moving the same way is called a wavelength. Investigate what physical quantities affect the period and which affect the wavelength. 5. Based on your investigations so far, how would you describe wave motion to a person who had never seen it?
Period T in seconds is the time interval for one complete vibration of a point in the medium anywhere along the wave s path. Frequency f in Hz (s -1 ) is the number of vibrations per second of a point in the medium as the wave passes. Amplitude A is the maximum displacement of a point of the medium from its equilibrium position as the wave passes. Speed v in m/s is the distance a disturbance travels during a time interval, divided by that time interval. Wavelength λ equals the distance between two nearest points on a wave that at any clock reading have exactly the same displacement and shape (slope). It is also the distance between two consecutive wavefronts. v λ = v T = f II. Observational Experiments: Wave speed 1. Devise a method to determine the horizontal speed of the wave. Describe your method and write your results. 2. Brainstorm possible variables that would affect the speed of the pulse on the cord and then conduct experiments to qualitatively observe whether and how they affect that speed. Describe your process and results. 3. Determine qualitatively which wave properties affect (or do not affect) the speed of the wave. Describe your process and results. Make sure you investigate the frequency of the wave, the amplitude and any other parameters that you can change. 4. Did you find that one of the physical quantities in the video affects the speed of the wave? What evidence do you have to support your conclusions? Why, do you think it does? Did you find any variables that do not affect the speed of the wave? Why, do you think, they do not? 5. For the physical quantity that affects the speed of the wave, what mathematical function do you think would best represent the speed as the function of this quantity? For example, will it be a i.e., linear, quadratic, inverse, etc. relation? Each of the above possibilities can become hypothesized functions. How can you test each hypothesis? Make a prediction based on each hypothesis and check whether it matches the outcome of the new experiment(s) that you conduct. These testing experiments can be conducted using the simulation at http://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html
III Wave properties 1. Wavelength is defined as the distance between two nearest points in the wave that have exactly the same displacement from equilibrium and are moving in the same direction. Mark the wavelength on the graph y ( x ) that you made in experiment 3. Investigate qualitatively what wave properties affect the wavelength. 2. Let s focus on what affects the wavelength in a spring of a specific tension. Decide what data you need to collect to determine how the other properties, amplitude and frequency, affect the wavelength. To do this you need to decide what data to collect and how to represent them. 3. How do these two variables affect the wavelength? 4. Devise a way to test your answer using a spring with a different tension. IV Wave Speed 1. What factors affect wave speed? We have operationally defined wave speed as the distance a disturbance travels in a medium during a time interval divided by that time interval. This does not explain why a wave has a certain speed. The following data were collected by two people holding the ends of a slinky. They measure the distance a pulse travels and the time it takes to travel that distance. Use the data below to explore how amplitude, frequency, and pulling force affect wave speed. Effect of amplitude Change the amplitude of the pulses Amplitude (m) Distance (m) Time Interval (s) Speed (m/s) 0.1 4.0 1.0 4.0 0.2 4.0 1.0 4.0 0.3 4.0 1.0 4.0 Effect of frequency Change the frequency of the pulses (we send several pulses one after another) Frequency (Hz) Distance (m) Time Interval (s) Speed (m/s) 2.0 4.0 1.0 4.0 4.0 4.0 1.0 4.0 6.0 4.0 1.0 4.0 Effect of force Change the force pulling on the end of the Slinky (the length of the Slinky also changes) Force (N) Distance (m) Time Interval (s) Speed (m/s) 2 4.0 1.0 4.0 4 8.0 1.0 8.0 6 12.0 1.0 12.0 2. What patterns are present in the data?
Pulling harder on the end of the slinky increases wave speed. As we stretch the Slinky coils farther apart, this reduces its linear density (mass per unit length). The force pulling on the slinky and its linear density seem to affect the speed of a pulse. If we do similar experiments with stiff springs that stretch less than a Slinky, we find that the wave speed is proportional to the square root of the force pulling on the end of the spring (or the tension, T, in the spring). This applies to other objects as well, such as strings. Other experiments indicate that the speed is also affected by the mass per unit length of vibrating particles. The speed of a wave is inversely proportional to the square root of the mass per unit length ( m / L ) of the medium, or linear density (μ). We can combine these two factors into one equation: v = T = T m L μ