J.D. Flood and D.W. Follet Modes of Vibration on Circular Membranes and Vibrating Bars Interlude 6 As we continue our exploration of families of instruments we want to have some understanding of how percussion instruments differ from other families. In this exercise we will focus on the modes of vibration on drumheads and struck bars. All of these activities are based on applets developed by Paul Falstad at http://www.falstad.com/mathphysics.html This link is called “Wave Applets” in Blackboard A. Circular Membrane Modes This figure is from Paul Falstad’s “Circular Membrane Applet”. Note that the modes of vibration do not form a harmonic series. In the mode designations mn (e.g. 01, 11, 21 02 etc.), m refers to the number of nodal lines along diameters of the membrane, and n refers to the number of nodal circles. A grid appears below the simulated membrane. Each block in the grid corresponds to a different mode of vibration, with the simplest modes in the upper left and the most complex in the lower right. 1. Open the “Circular Membrane” applet. Click on the “Fundamental” button in the upper right. After observing the vibration while looking for nodal diameters and nodal circles, identify the mode you are looking at in the space below. 2. In the frequency grid (bottom of screen) de­select the fundamental (the square should go from green to black) and select the next square down in the same column. Sketch the 2­D picture for this mode, identify the mode using the table above, and, in words, compare it to the fundamental.
1 J.D. Flood and D.W. Follet 3. Select the third square in the same column. Make sure no other modes are selected. Sketch the 2­D picture, identify the mode, and compare to the other two modes you’ve observed. 4. Now select the second square in the top row, making sure it is the only frequency selected. How is this pattern different from the others you observed? Sketch the pattern below. 5. Finally, select the second square from the top, in the second column. In words, how does this patter compare to the other modes you’ve looked at? Include a sketch. B. Modes of Vibration of a Free Bar Open the “Bar Waves” applet. Make sure “bar, free” is selected under “Setup. This simulation shows the vibrational modes of a uniform bar, with a cross­section like the one shown below. The row of dots in the middle of the simulation screen allows you to select individual modes of vibration with different frequencies. The lowest available frequency, the fundamental, is 127 Hz. 1. Calculate the first 6 members of a harmonic series based on 127 Hz. Write your results here. 2. Look at the first six modes of vibration of the bar. Make a sketch to show each oscillation and record the associated frequency. You may find it helpful to stop the simulation to make your sketch.
2 J.D. Flood and D.W. Follet 3. Compare the frequencies you calculated in part 1 to the frequencies you observed in part 2. How do they differ? 4. Given the frequencies you found in 2, combined with the idea that several modes of vibration are present together in one note, would you perceive the sound from this bar as a having definite pitch? 5. You probably know from experience that the marimba and xylophone have a well­ defined pitch. This is because the bars on a xylophone or a marimba are not flat bars but instead are shaped somewhat like this: The bottom of the bar has been partially hollowed out near the center. This has the effect of reducing the stiffness of the bar in the central region. Looking at the first two modes (the two lowest frequencies) you sketched in part 2, which one would be most affected by hollowing out the bar? How would the frequency change? (It might be helpful to think about how changing the tension in a stretched string changes its frequency). 6. We can see that the hollowing out procedure affects the harmonics of the bar. What relationships between the first and second modes is the designer of the bar trying to create? How would it affect the note you hear?
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