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Lesson 5: Standing Waves

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  1. Lesson 5: Standing Waves • 5.1 Traveling Waves vs. Standing Waves • 5.2 Formation of Standing Waves • 5.3 Nodes and Anti-nodes • 5.4 Harmonics and Patterns • 5.5 Mathematics of Standing Waves

  2. Traveling Waves vs. Standing Waves • A mechanical wave • is a disturbance that is created by a vibrating object and subsequently travels through a medium from one location to another, • transporting energy as it moves. • The mechanism by which a mechanical wave propagates itself through a medium involves particle interaction; • one particle applies a push or pull on its adjacent neighbor, causing a displacement of that neighbor from the equilibrium or rest position.

  3. Traveling Waves vs. Standing Waves • This wave pattern continues to move in uninterrupted fashion until it encounters • another wave along the medium or • until it encounters a boundary with another medium. • This type of wave pattern that is seen traveling through a medium is sometimes referred to as a traveling wave.

  4. Traveling Waves vs. Standing Waves

  5. Traveling Waves vs. Standing Waves • Traveling waves are observed when a wave is not confined to a given space along the medium. • The most commonly observed traveling wave is an ocean wave.

  6. Traveling Waves vs. Standing Waves • If a wave is introduced into an elastic cord with its ends held 3 meters apart, it becomes confined in a small region. • Such a wave has only 3 meters along which to travel. • The wave will quickly reach the end of the cord, reflect and travel back in the opposite direction.

  7. Traveling Waves vs. Standing Waves • Any reflected portion of the wave will then interfere with the portion of the wave incident towards the fixed end. • This interference produces a new shape in the medium that seldom resembles the shape of the original wave.

  8. Traveling Waves vs. Standing Waves • Subsequently, a traveling wave (a repeating pattern that is observed to move through a medium in uninterrupted fashion) is not observed in the cord. • Indeed there are traveling waves in the cord; it is just that they are not easily detectable because of their interference with each other.

  9. Traveling Waves vs. Standing Waves • In such instances, rather than observing the pure shape of a wave pattern, a rather irregular and non-repeating pattern is produced in the cord that tends to change appearance over time. • This irregular looking shape is the result of the interference of an incident wave pattern with a reflected wave pattern in a rather non-sequenced and untimely manner. • Both the incident and reflected wave patterns continue their motion through the medium, meeting up with one another at different locations in different ways.

  10. Traveling Waves vs. Standing Waves • For example, the middle of the cord might experience a crest meeting a half crest; then moments later, a crest meeting a quarter trough; then moments later, a three-quarters crest meeting a one-fifth trough, etc. • This interference leads to a very irregular and non-repeating motion of the medium. • The appearance of an actual wave pattern is difficult to detect amidst the irregular motions of the individual particles.

  11. Traveling Waves vs. Standing Waves • It is however possible to have a wave confined to a given space in a medium and still produce a regular wave pattern that is readily discernible amidst the motion of the medium. • For instance, if an elastic rope is held end-to-end and vibrated at just the right frequency, a wave pattern would be produced that assumes the shape of a sine wave and is seen to change over time.

  12. Traveling Waves vs. Standing Waves • The wave pattern is only produced when one end of the rope is vibrated at just the right frequency. • When the proper frequency is used, the interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium that appear to be standing still. • Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern.

  13. Traveling Waves vs. Standing Waves • There are other points along the medium whose displacement changes over time, but in a regular manner. • These points vibrate back and forth from a positive displacement to a negative displacement; the vibrations occur at regular time intervals such that the motion of the medium is regular and repeating. • A pattern is readily observable.

  14. Traveling Waves vs. Standing Waves

  15. Traveling Waves vs. Standing Waves • Note that point A on the medium moves from a maximum positive to a maximum negative displacement over time. • The diagram only shows one-half cycle of the motion of the standing wave pattern. • The motion would continue and persist, with point A returning to the same maximum positive displacement and then continuing its back-and-forth vibration between the up to the down position.

  16. Traveling Waves vs. Standing Waves • Note that point B on the medium is a point that never moves. • Point B is a point of no displacement. • Such points are known as nodes.

  17. http://www.youtube.com/watch?v=_S7-PDF6Vzc • http://www.youtube.com/watch?v=no7ZPPqtZEg

  18. Lesson 5: Standing Waves • 5.1 Traveling Waves vs. Standing Waves • 5.2 Formation of Standing Waves • 5.3 Nodes and Anti-nodes • 5.4 Harmonics and Patterns • 5.5 Mathematics of Standing Waves

  19. Formation of Standing Waves • A standing wave pattern is a vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source. • This interference occurs in such a manner that specific points along the medium appear to be standing still. • Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern.

  20. Formation of Standing Waves • Such patterns are only created within the medium at specific frequencies of vibration. • These frequencies are known as harmonic frequencies, or merely harmonics. • At any frequency other than a harmonic frequency, the interference of reflected and incident waves leads to a resulting disturbance of the medium that is irregular and non-repeating.

  21. Formation of Standing Waves • But how are standing wave formations formed? • And why are they only formed when the medium is vibrated at specific frequencies? • And what makes these so-called harmonic frequencies so special and magical? • Let’s consider a snakey stretched across the room, approximately 4-meters from end to end. (A "snakey" is a slinky-like device that consists of a large concentration of small-diameter metal coils.)

  22. Formation of Standing Waves • If an upward displaced pulse is introduced at the left end of the snakey, it will travel rightward across the snakey until it reaches the fixed end on the right side of the snakey. • Upon reaching the fixed end, the single pulse will reflect and undergo inversion. • That is, the upward displaced pulse will become a downward displaced pulse.

  23. Formation of Standing Waves • Now suppose that a second upward displaced pulse is introduced into the snakey at the precise moment that the first crest undergoes its fixed end reflection. • If this is done with perfect timing, a rightward moving, upward displaced pulse will meet up with a leftward moving, downward displaced pulse in the exact middle of the snakey. • As the two pulses pass through each other, they will undergo destructive interference.

  24. Formation of Standing Waves • Thus, a point of no displacement in the exact middle of the snakey will be produced. • http://www.physicsclassroom.com/Class/waves/di.gif

  25. Formation of Standing Waves • An upward displaced pulse introduced at one end will destructively interfere in the exact middle of the snakey with a second upward displaced pulse introduced from the same end if the introduction of the second pulse is performed with perfect timing. • The same rationale could be applied to two downward displaced pulses introduced from the same end. • If the second pulse is introduced at precisely the moment that the first pulse is reflecting from the fixed end, then destructive interference will occur in the exact middle of the snakey.

  26. Formation of Standing Waves • A wave is certainly different than a pulse. • What if there are two waves traveling in the medium? • Understanding why two waves introduced into a medium with perfect timing might produce a point of displacement in the middle of the medium is a mere extension of the discussion.

  27. Formation of Standing Waves • While a pulse is a single disturbance that moves through a medium, a wave is a repeating pattern of crests and troughs. • Since the introduction of a crest is followed by the introduction of a trough, every crest and trough will destructively interfere in such a way that the middle of the medium is a point of no displacement.

  28. Formation of Standing Waves • Of course, this all demands that the timing is perfect. • Perfect timing can be achieved if every wave crest was introduced into the snakey at the precise time that the previous wave crest began its reflection at the fixed end. • In this situation, there will be one complete wavelength within the snakey moving to the right at every instant in time; this incident wave will meet up with one complete wavelength moving to the left at every instant in time.

  29. Formation of Standing Waves • Under these conditions, destructive interference always occurs in the middle of the snakey. • Either a full crest meets a full trough or a half-crest meets a half-trough or a quarter-crest meets a quarter-trough at this point.

  30. Formation of Standing Waves • http://www.physicsclassroom.com/Class/waves/swf.gif • The waves are interfering in such a manner that there are points of no displacement produced at the same positions along the medium. • These points along the medium are known as nodes and are labeled with an N.

  31. Formation of Standing Waves • http://www.physicsclassroom.com/Class/waves/swf.gif • There are also points along the medium that vibrate back and forth between points of large positive displacement and points of large negative displacement. • These points are known as antinodes and are labeled with an AN.

  32. Formation of Standing Waves • There are other ways to achieve this perfect timing. • The main idea behind the timing is to introduce a crest at the instant that another crest is either at the halfway point across the medium or at the end of the medium. • Regardless of the number of crests and troughs that are in between, if a crest is introduced at the instant another crest is undergoing its fixed end reflection, a node (point of no displacement) will be formed in the middle of the medium.

  33. Formation of Standing Waves • The number of other nodes that will be present along the medium is dependent upon the number of crests that were present in between the two timed crests. • If a crest is introduced at the instant another crest is at the halfway point across the medium, then an antinode (point of maximum displacement) will be formed in the middle of the medium by means of constructive interference. • In such an instance, there might also be nodes and antinodes located elsewhere along the medium.

  34. Formation of Standing Waves • Summary • A standing wave pattern is an interference phenomenon. • It is formed as the result of the perfectly timed interference of two waves passing through the same medium.

  35. Formation of Standing Waves • A standing wave pattern is not actually a wave; rather it is the pattern resulting from the presence of two waves (sometimes more) of the same frequency with different directions of travel within the same medium. • The physics of musical instruments has a basis in the conceptual and mathematical aspects of standing waves.

  36. Lesson 5: Standing Waves • 5.1 Traveling Waves vs. Standing Waves • 5.2 Formation of Standing Waves • 5.3 Nodes and Anti-nodes • 5.4 Harmonics and Patterns • 5.5 Mathematics of Standing Waves

  37. Nodes and Anti-nodes • One characteristic of every standing wave pattern is that there are points along the medium that appear to be standing still. • These points, sometimes described as points of no displacement, are referred to as nodes.

  38. Nodes and Anti-nodes • There are other points along the medium that undergo vibrations between a large positive and large negative displacement. • These are the points that undergo the maximum displacement during each vibrational cycle of the standing wave. • In a sense, these points are the opposite of nodes, and so they are called antinodes.

  39. Nodes and Anti-nodes • A standing wave pattern always consists of an alternating pattern of nodes and antinodes. • http://www.physicsclassroom.com/Class/waves/h4.gif

  40. Nodes and Anti-nodes

  41. Nodes and Anti-nodes • The positioning of the nodes and antinodes in a standing wave pattern can be explained by focusing on the interference of the two waves. • The nodes are produced at locations where destructive interference occurs. • Antinodes are produced at locations where constructive interference occurs. • Antinodes are always vibrating back and forth between these points of large positive and large negative displacement.

  42. Nodes and Anti-nodes • Nodes and antinodes should not be confused with crests and troughs. • When the motion of a traveling wave is discussed, it is customary to refer to a point of large maximum displacement as a crest and a point of large negative displacement as a trough. • These represent points of the disturbance that travel from one location to another through the medium.

  43. Nodes and Anti-nodes • An antinode on the other hand is a point on the medium that is staying in the same location. • Furthermore, an antinode vibrates back and forth between a large upward and a large downward displacement.

  44. Nodes and Anti-nodes • And finally, nodes and antinodes are not actually part of a wave. • Recall that a standing wave is not actually a wave but rather a pattern that results from the interference of two or more waves. • Since a standing wave is not technically a wave, an antinode is not technically a point on a wave. • The nodes and antinodes are merely unique points on the medium that make up the wave pattern.

  45. http://www.youtube.com/watch?v=12pjjPIE2IQ&feature=player_embedded#!http://www.youtube.com/watch?v=12pjjPIE2IQ&feature=player_embedded#!

  46. Think about it • What if the wave was big enough to stand on? Where would it be hardest to stand? Easiest?

  47. Lesson 5: Standing Waves • 5.1 Traveling Waves vs. Standing Waves • 5.2 Formation of Standing Waves • 5.3 Nodes and Anti-nodes • 5.4 Harmonics and Patterns • 5.5 Mathematics of Standing Waves

  48. Harmonics and Patterns • Standing wave patterns can only be produced within the medium when it is vibrated at certain frequencies. • There are several frequencies with which the snakey can be vibrated to produce the patterns. • Each frequency is associated with a different standing wave pattern. • These frequencies and their associated wave patterns are referred to as harmonics.