Purpose: To observe and study the behavior of standing transverse waves and to determine the speed of standing and traveling waves. Equipment: Cenco String Vibrator Hooked Mass Set Pulley Table Clamp String Meter Stick Balance Theory: Traveling harmonic waves can be either transverse, longitudinal, or a combination of both. Transverse waves travel in a direction perpendicular to the displacement of the media, i.e. if the velocity is in the x direction, then the displacement is in the y direction. Waves on a string are transverse because the wave travels down the string, while the string particles are actually displaced up and down. Sound waves are longitudinal. As the compression wave travels through a medium such as air, the air molecules are displaced back and forth, in the same direction as the wave travels. In spite of this difference in directions and terminology, the same mathematical description relates the displacement, speed, frequency, and wavelength of both longitudinal and transverse waves: (for waves traveling to the righ, and d ( x, = d sin( kx ω t + φ) Eq. 1a d ( x, = d sin( kx + ω t + φ) Eq. 1b (for waves traveling to the lef, where v = ω k Here, d represents the displacement of the medium from its equilibrium position, and the wave is traveling in the x direction 1 of 6
For transverse waves, d is often replaced with y, the vertical displacement of the particle (on the string). For longitudinal waves, d represents the distance a particle at position x and time t is displaced from its equilibrium position. In practice is far easier to measure the pressure change produced by the compression of a traveling wave. Since this pressure change is related to the change in the displacement (see the tex, the pressure at a given point (x, can be described by: P ( x, = P cos( kx ω t + φ) Eq. 2 Traveling waves can and will add together, according to the Principle of Superposition, which states that the displacement of a particle is the sum of independent displacements. In other words, d(x, = d 1 (x, + d 2 (x, + d 3 (x, + In particular, waves of the same amplitude, but traveling in different directions add to give the net displacement as a function of time, as with: d ( x, = d1( x, + d 2 ( x, = d sin( kx ω t + φ) + d sin( kx + ωt + φ) This gives us the following wave equation: d( x, = 2d sin( kx)cos( ω Eq. 3 To apply this equation to a particular situation, such as traveling waves moving back and forth on a stretched string, we must consider the boundary conditions. These are physical constraints about the displacement of the string at a given point x, and time t. For example, the displacement of a string fastened at x = 0, and x = L is zero at both endpoints for all times. Thus, d(0,0) = d(0,l) = 0 (t = zero is selected for simplicity). To satisfy this relationship, sin(kx) = 0 at x = 0, and x =L. This first case, x = 0 is true for all values of k. For the second case: sin( kl ) = 0, where kl=nπ, and k=(nπ)/l However, since k also equals 2π/λ, then the following is true: 2π λ L = nπ which gives us: λ L = n Eq. 4 2 The physical meaning of this relation is that only an integer number of half wavelengths can fit on a string fixed at both ends. Since the wavelength and frequency are related by: Eq. (4) can be written as: v Eq. 5 ω = = λ f k v 2 f n L n = Eq. 6 where f n is the frequency that produces n half wavelengths on the string of length L. 2 of 6
For a string under tension T, Newton's laws can be used to derive the wave velocity. The result is: T v = Eq. 7 µ where T is the tension in the string, and µ is the mass per unit length. Experiment: 1. Attach a pulley to the end of the lab table with a table clamp. Place the vibrator at the other end of the table. Tie a length of string to the center hole in the tongue of the vibrator, and attach a hooked mass to the other end of the string. Run the string over the pulley so that the pulley turns freely and the mass hangs below it. Measure and record the horizontal length of the string (from the vibrator to the pulley). Set up your equipment so that this length is between 150 and 200 cm. Add masses to the bottom hook so that the total hanging mass is about 190-200 g. Cenco Wave Generator Pulley Wavelength Amplitude Mass Illustration by Jay Ayers 2. Turn on the vibrator, and adjust the hanging masses until the string vibrates in an evident standing wave pattern. (The string may not necessarily vibrate in the vertical plane; you should observe it from above as well as from the side when watching for a pattern to appear.) Record the total mass hanging on the string. Sketch the shape of the standing wave and use the meter stick to measure the location of any nodes. Note whether the vibrator is at a node or an anti-node or somewhere in between. Also estimate roughly the amplitude of vibration of the string and of the vibrator tongue. 3 of 6
3. Repeat the procedure, varying the mass used so as to make the string vibrate in different standing wave patterns. Measure at least three wave patterns if you can without moving the vibrator and the pulley. If you do need to move the pulley, be sure to record the new length L, and use it where appropriate in calculations (If the tongue of the vibrator clatters noisily against its housing, immediately turn it off or reduce the mass in the hanger.) Analysis: 1. Use Eq. 7 to determine the theoretical values of the wave speed, v th. Remember that µ, the mass per unit length, is equal to the mass of the entire string divided by the entire length of the string, not just the length from the vibrator to pulley. 2. Now we must resolve the vibration frequency; 30, 60, 90 and 120 Hz vibration modes are possible (the frequency must be an integer multiple of 30). Use Eqs. 5 and 7 to determine which vibration frequency is most likely (i.e. solve Eq. 7 for the velocity, measure the wavelength, and plug into Eq. 5 to find the frequency). 3. Using this "most likely" frequency determine the wave speed, v exp, from Eq. 6. These are your experimental values. 4. Compare the results of Eqs. 6 and 7. If they are not sufficiently close to being equal, it may be because the end of the vibrator is not exactly a node. This can be remedied 2 f nl by determining a better value of L. Start with Eq. 6, v =, where v is the n theoretical wave speed as calculated in Step 1, f is the most likely frequency as calculated in Step 2, count n, the number of anti-nodes, and solve for L. This new value compensates for the end of the vibrator moving slightly. 5. Once you determine a better value of the string length, L, recalcuate the wave speed as given by Eq. 6. 6. Using estimated and calculated uncertainties, compare the results of Eqs. 6 and 7. Do they agree within uncertainty? If not, why not? Use a one-dimensional graph to display your results and uncertainties. Also make sure to use one or more tables to display your data. 7. Repeat Steps 1-6 for each separate hanging mass and string length combination. 8. Question: is there any additional experimental error? Is it systematic (all in the same direction), or are the errors equally likely to be positive or negative? What do you suppose the causes of error may be? How would you propose to redesign the experiment so as to reduce or eliminate them? 4 of 6
Results: Write at least one paragraph describing the following: what you expected to learn about the lab (i.e. what was the reason for conducting the experiment?) your results, and what you learned from them Think of at least one other experiment might you perform to verify these results Think of at least one new question or problem that could be answered with the physics you have learned in this laboratory, or be extrapolated from the ideas in this laboratory. 5 of 6
Clean-Up: Before you can leave the classroom, you must clean up your equipment, and have your instructor sign below. How you divide clean-up duties between lab members is up to you. Clean-up involves: Completely dismantling the experimental setup Removing tape from anything you put tape on Drying-off any wet equipment Putting away equipment in proper boxes (if applicable) Returning equipment to proper cabinets, or to the cart at the front of the room Throwing away pieces of string, paper, and other detritus (i.e. your water bottles) Shutting down the computer Anything else that needs to be done to return the room to its pristine, pre lab form. I certify that the equipment used by has been cleaned up. (student s name),. (instructor s name) (date) 6 of 6