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Learning Goal: To practice Problem-Solving Strategy 16.1 Standing waves. An air-filled pipe is found to have successive harmonics at 390 Hz , 650 Hz , and 910 Hz . It is unknown whether harmonics below 390 Hz and above 910 Hz exist in the pipe. What is the length of the pipe? PROBLEM-SOLVING STRATEGY 16.1 Standing waves We can use the same general strategy for any standing wave. PREPARE For sound waves, determine what sort of pipe or tube you have: open-open, closed-closed, or open-closed. For string or light waves, the ends will be fixed points. For other types of standing waves, such as electromagnetic or water waves, the mode diagram will be similar to one of the cases outlined in Synthesis 16.1 (Page) . You can then work by analogy with waves on strings or sound waves in tubes. Determine known values: length of the tube or string, frequency, wavelength, and positions of nodes or antinodes. Sketch a visual overview, including a picture of the relevant mode. SOLVE Once you have determined the mode diagram, you can use the appropriate set of equations in Synthesis 16.1 (Page) to find the wavelength and frequency. These equations work for any type of wave. =v. 2L v = an and fin inven =m (32), where m = 1,2,3,4,.... For sound waves in an open-open or closed-closed tube, the allowed frequencies and wavelengths are given by = BD and f;, =m (sr) = mf, where m = 1, 2,3,4,.... For sound waves in a open-closed tube, the allowed frequencies and wavelengths are given by aS AL and fn =m(z) = mf, where m = 1,3,5,7,.... ASSESS Does your final answer seem reasonable? Is there another way to check on your results? For example, the frequency times the wavelength for any mode should equal the wave speed you can check to see that it does. PREPARE Determine what type of pipe you have and identify the known values for each pipe. v PartA What type of pipe do you have? View Available Hint(s) open-open closed-closed open-closed either open-open or closed-closed ut)