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					  Mechanical Wave

          Lecture 3:Standing Waves on a
                    String          1
• Question: What happens to the
  propagating pulse or sinusoidal
  wave as it arrives at the end of the
  string?                                                               Incident wave
• For a fixed end of a string:
   At the end, the wave exerts a force on the
   support-the support exert a force back on
   the string an sets up a reflected wave.
   A reflected wave has its amplitude inverted
   relative to that of the incident wave (see
   figure (a))
• For a free end of a string:
   At the end, the string slides along the
   support and come to rest momentarily, the                             reflected wave
   string is now stretched increasing the
   tension and resulting with the reflected
   But the amplitude of the reflected wave is
   not inverted relative to that of the incident.

 The condition at the end of the string, fixed or free are called the boundary
 As the wave arrive at the boundary, all or part of the wave is reflected back to
 propagate in the opposite direction of the incident wave.                                                 2
• The formation of the reflected pulse is
  equivalent to the overlap of two pulses
  travelling in opposite directions.
• Consider two pulses with the same
  shape, one inverted with respect to the
  other and travelling in opposite direction.
• As the pulses overlap and pass each
  other, the total displacement of the string
  is the algebraic sum of the displacement
  at the point in the individual pulse.
• Because the two pulses have the same
  shape and amplitude, the total
  displacement at point O is zero.
• The two pulses, on the right side
  corresponds to the incident and the
  reflected pulse in a one fixed boundary
  situation.                3
•   If the two pulses were not inverted relative to
    each other, then the total displacement at
    point O will be twice that of the individual
•   Slope at point O is zero corresponding to the
    absence of any transverse force at this point.
•   Combining the displacement of the separate
    pulses at each point to obtain the actual
    displacement is an example of the principle
    of superposition.
• It states that when two waves overlap,
  the actual displacement at any point on
  the string at any time is obtained by
  adding the displacement the point
  would have if only the first wave was
  present and the displacement it
  would have if only the second wave
  was present.                      4
•   Let the wave function of the incident and reflected wave be y1(x, t)
    and y2(x, t) respectively. The actual resulting wave at the point of
    overlap is the sum of the individual wave overlapping.

•   Each wave function, yi(x, t), of wave i must satisfy the wave
    equation which requires linearity of the wave function.
    Linear function

  Nonlinear function
• Summation of linear function is also linear.
• As a results, if any N linear functions, satisfies the wave equation
   separately, then their sum also satisfies it and it is therefore a
   physical possible motion.
• Because this principle depends on the linearity of the wave equation
   and the corresponding linear-combination property of its solution, it
   is called principle of linear superposition.                                           5
Standing waves
• Question: What happen when
   a sinusoidal wave is reflected
   at the boundary?
• The resulting motion when the
   two waves combine: no longer
   looks like two waves travelling
   in opposite direction.
• The string appears to be
   subdivided into a number of
 • The wave pattern remains in the same position along the string,
    and its amplitude fluctuates.
 • Because the wave pattern does not appear to be moving either
    direction along the string and it is called a standing wave.
    Recap :Travelling wave: In a wave that travels along the string, the amplitude is
    constant and the wave pattern moves with speed ν.
•   The points of the standing wave that never moves at all are called the
    Nodes (N).
•   The points midway between the nodes are referred to as antinodes (A).                                                    6
• Using the principle of superposition, we
  can now explain how the incident and
  reflected wave combined to form a
  standing wave.
   The red curve shows a wave travelling to the left.
   The blue curve indicates the wave travelling to
   the right.
   Both waves have the same (λ,ν and A)
   The black curve shows the resulting wave
   formed by adding the two waves travelling in
   opposite directions.
• At t = 4T/16, the two waves are
  completely out of phase with each
  other, and thus the total wave at that
  instant is zero.
• At t = 8T/16, the two wave patterns are
  exactly in phase with each other, and
  the resulting wave (black) has twice the
  amplitude.                       7
• At the nodes, the resulting displacement is always zero. At this point,
  the displacements of the two waves in red and blue are always equal
  and opposite and cancel each other resulting with a interface called
  destructive interference
• At the antinodes, the displacements of the two waves in red and blue
  are always identical giving a large resultant displacement, and this kind
  of interference is known as the constructive interference.
• The wave function of the individual wave are

Travelling to the left:

Travelling to the right:

  • The phase of the reflected wave (travelling to the right) is out of phase
    by 180o relative to that of the incident wave.

 • Using the principles of linear superposition, the wave function
   describing the resulting wave is given as:                                        8
Standing wave on a string:

  •   Shows that at each instant, the shape of the   •   The wave shape stays in the same
      string is a sine curve                             position, oscillating up and down

 • At the nodes, sin(kx) = 0 for the displacement to be always zero,

• A standing wave, unlike a travelling wave, does not transfer energy
  from one and to the other. The two wave that forms it would individually
  carry equal amounts of power in opposite directions. There is a local
  flow of energy from each node to the adjacent antinode and back but
  the average rate of energy transfer is zero at every point.                                                   9
• Question: What happen if the string has a definite length L, rigidly held
  at both ends? .g. string of a guitars and violins
• The standing wave that results must have a node at both ends of the
  string. Since the nodes are always (λ/2) apart, the length (L) of the
  string must be:

For the standing wave to exist. The standing-wave wavelength is given as

• Note: if l=2L/n, individual waves can still exist but their summations
  cannot be a standing waves
 • The standing wave frequencies is therefore:                                          10
• Where f1 is the smallest frequency corresponding to the largest
  wavelength (n = 1),
• f1 is called the fundamental mode.
• The higher frequency are called harmonics or in musician language
  are called overtones.
    n                              Harmonics    Overtones
    2                              2nd          1st
    3                              3rd          2nd

    n                              nth          (n -1)’s overtones

 • Each of the wavelength given by, λn, for a string fixed at both end,
   corresponds to a possible normal modes pattern and frequency.
 • Normal modes of an oscillating system is a motion in which all
   particles of the system move sinosoidally with the same frequency.                                     11