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Topic 11: Wave phenomena 11.1 Standing (stationary) waves

11.1.1 Describe the nature of standing (stationary) waves. Be sure to consider energy transfer, amplitude and phase. 11.1.2 Explain the formation of one-dimensional standing waves. Know what nodes and antinodes are, and how they form.

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Topic 11: Wave phenomena 11.1 Standing (stationary) waves

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  1. 11.1.1 Describe the nature of standing (stationary) waves. Be sure to consider energy transfer, amplitude and phase. 11.1.2 Explain the formation of one-dimensional standing waves. Know what nodes and antinodes are, and how they form. 11.1.3 Discuss the modes of vibration of strings and air in open and in closed pipes. The lowest-frequency mode is known either as the fundamental or as the first harmonic. The term overtone will not be used. 11.1.4 Compare standing waves and traveling waves. 11.1.5 Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves FYI Topic 11.1 is an extension of Topic 4.5. Review it!

  2. Standing Waves & Resonance A standing wave is created from two traveling waves, having the same frequency and the same amplitude and traveling in opposite directions in the same medium. Using superposition, the net displacement of the medium is the sum of the two waves. When 180° out-of-phase with each other, they cancel (destructive interference). When in-phase with each other, they add together (constructive interference).

  3. Formation of Standing Waves • Standing waves form as a result of constructive and destructive superposition of two waves moving in opposite directions with equal amplitude and frequency. • If at an instant when the waves overlap any points are in-phase, constructive superposition occurs. • - diagram • If each wave moves ¼ of a cycle in opposite directions, the waves become ½ a cycle out of phase and destructive interference occurs. • Diagram • Please watch • http://www.youtube.com/watch?v=3BN5-JSsu_4&list=PL2248ACCDFEAEAEB8&index=10

  4. Describe the nature of standing (stationary) waves. The principle of superposition yields a surprising resultant for two identical waves traveling in opposite directions. Snapshots of the blue and the green waves and their red resultant follow: Because the resultant red wave appears to not be traveling it is called a standing wave. Topic 11: Wave phenomena11.1 Standing (stationary) waves

  5. N N N N N N N Explain the formation of standing waves. Know what nodes and antinodes are, and how they form. The standing wave has two important properties. It does not travel to the left or the right as the blue and the green wave do. Its “lobes” grow and shrink and reverse, but do not go to the left or the right. Any points where the standing wave has no displacement is called a node (N). The lobes that grow and shrink and reverse are called antinodes (A). Topic 11: Wave phenomena11.1 Standing (stationary) waves A A A A A A

  6. Discuss the modes of vibration of strings and air in open and in closed pipes. You may be wondering how a situation could ever develop in which two identical waves come from opposite directions. Well, wonder no more. When you pluck a stringed instrument, waves travel to the ends of the string and reflect at each end, and return to interfere under precisely the conditions needed for a standing wave. Note that there are two nodes and one antinode. Why must there be a node at each end of the string? Topic 11: Wave phenomena11.1 Standing (stationary) waves L N N A Because it is fixed at each end.

  7. L N N A Discuss the modes of vibration of strings and air in open and in closed pipes. Observe that precisely half a wavelength fits along the length of the string. Thus we see that  = 2L. Since v = f we see that f = v/(2L) for a string. This is the lowest frequency you can possibly get from this string configuration, so we call it the fundamental frequencyf1. The fundamental frequency of any system is called the first harmonic. Topic 11: Wave phenomena11.1 Standing (stationary) waves 1st harmonic fundamental frequency f1 = v/(2L)

  8. L Discuss the modes of vibration of strings and air in open and in closed pipes. The next higher frequency has another node and another antinode. We now see that  = L. Since v = f we see that f = v/L. This is the second lowest frequency you can possibly get and since we called the fundamental frequency f1, we’ll name this one f2. This frequency is also called the second harmonic. Topic 11: Wave phenomena11.1 Standing (stationary) waves N N N 2nd harmonic A A f2 = v/L

  9. Nodes and Antinodes Points oscillating with the biggest amplitude in a stationary wave are called antinodes. Points undergoing zero oscillation are called nodes.

  10. λ = 2L Frequency = f λ = L Frequency = 2f λ = 2/3 L Frequency = 3f

  11. The lowest resonant frequency of the system is called the fundamental frequency (or 1st harmonic frequency). The next are called the 2nd harmonic, 3rd harmonic etc. For a stationary wave in a string: Harmonic frequency = no. of loops x natural frequency

  12. . Because a standing wave consists of two traveling waves carrying energy in opposite directions, the net energy flow through the wave is zero. Topic 11: Wave phenomena11.1 Standing (stationary) waves

  13. f1 = v/2L f2 = v/L f3 = Solve problems involving standing waves. PRACTICE: Complete the table below with both sketch and formula. Remember that there are always nodes on each end of a string. Add a new well-spaced node each time. Decide the relationship between  and L. We see that  = (2/3)L. Since v = f we see that f = v/(2/3)L = 3v/2L. Topic 11: Wave phenomena11.1 Standing (stationary) waves 3v/2L

  14. Discuss the modes of vibration of strings and air in open and in closed pipes. We can also set up standing waves in pipes. In the case of pipes, longitudinal waves are created (instead of translational waves), and these waves are reflected from the ends of the pipe. Consider a closed pipe of length L which gets its wave energy from a mouthpiece on the left side. Why must the mouthpiece end be an antinode? Why must the closed end be a node? Topic 11: Wave phenomena11.1 Standing (stationary) waves (1/4)1 = L (3/4)2 = L (5/4)2 = L f3 = 5v/4L f2 = 3v/4L f1 = v/4L Source. Air can’t move.

  15. Discuss the modes of vibration of strings and air in open and in closed pipes. In an open-ended pipe you have an antinode at the open end because the medium can vibrate there (and, of course, at the mouthpiece). Topic 11: Wave phenomena11.1 Standing (stationary) waves (1/2)1 = L 2 = L (3/2)2 = L f3 = 3v/2L f2 = 2v/2L f1 = v/2L FYI The IBO requires you to be able to make sketches of string and pipe harmonics (both open and closed) and find wavelengths and frequencies.

  16. Solve problems involving standing waves. PRACTICE: A tube is filled with water and a vibrating tuning fork is held above the open end. As the water runs out of the tap at the bottom sound is loudest when the water level is a distance x from the top. The next loudest sound comes when the water level is at a distance y from the top. Which expression for  is correct, if v is the speed of sound in air? A.  = x B.  = 2x C.  = y-x D.  = 2(y-x) v = f and since v and f are constant, so is . The first possible standing wave is sketched. The sketch shows that  = 4x, not a choice. Topic 11: Wave phenomena11.1 Standing (stationary) waves

  17. y-x Solve problems involving standing waves. PRACTICE: A tube is filled with water and a vibrating tuning fork is held above the open end. As the water runs out of the tap at the bottom sound is loudest when the water level is a distance x from the top. The next loudest sound comes when the water level is at a distance y from the top. Which expression for  is correct, if v is the speed of sound in air? A.  = x B.  = 2x C.  = y-x D.  = 2(y-x) The second possible standing wave is sketched. Notice that y – x is half a wavelength. Thus the answer is  = 2(y - x). Topic 11: Wave phenomena11.1 Standing (stationary) waves

  18. Solve problems involving standing waves. PRACTICE: This drum head set to vibrating at different resonant frequencies has black sand on it, which reveals 2D standing waves. Does the sand reveal nodes, or does it reveal antinodes? Why does the edge have to be a node? Topic 11: Wave phenomena11.1 Standing (stationary) waves Nodes, because there is no displacement to throw the sand off. The drumhead cannot vibrate at the edge.

  19. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Alternate lobes have a 180º phase difference. See Slides 3 and 10.

  20. L Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Make a sketch. Then use v = f. antinode antinode v = f f = v/  / 2 = L f = v/(2L)  = 2L

  21. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Reflection provides for two coherent waves traveling in opposite directions. Superposition is just the adding of the two waves to produce the single stationary wave.

  22. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves A snapshot of Slide 3 shows the points between successive nodes. For every point between the two nodes f is the same. But the amplitudes are all different. Therefore the energies are also different.

  23. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Energy transfer via a vibrating medium without interruption. The medium itself does not travel with the wave disturbance. Speed at which the wave disturbance propagates. Speed at which the wave front travels. Speed at which the energy is transferred.

  24. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Frequency is number of vibrations per unit time. FYI: IB frowns on you using particular units as in “Frequency is number of vibrations per second.” FYI: There will be lost points, people! Distance between successive crests (or troughs). Distance traveled by the wave in one oscillation of the source.

  25. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Each of the waves traveling in opposite directions carry energy at same rate in both directions. Thus there is NO energy transfer. The amplitude is always changing and reversing.

  26. L L Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves v = f f = v/ Q / 2 = L P / 4 = L  = 2L  = 4L fQ = v/(2L) fP = v/(4L) v = 4LfP fQ = 4LfP /(2L) fQ = 2fP

  27. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves The tuning fork is the driving oscillator (and is at the top). The top is thus an antinode. The bottom “wall” of water allows NO oscillation. The bottom is thus a node.

  28. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Sound is longitudinal in nature. Small displacement at P, big at Q.

  29. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves If the lobe at T is going down, so is the node at U.

  30. Solve problems involving standing waves. Topic 11: Wave phenomena11.1 Standing (stationary) waves Pattern 1 is a 1/2 wavelength. Pattern 2 is a 3/2 wavelength. Thus f2 = 3f1 so that f1 / f2 = 1/3.

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