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```					  Mechanical Wave

Lecture 3:Standing Waves on a
String

http://physics.ukzn.ac.za/~sonot          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
wave.
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
conditions
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.
http://physics.ukzn.ac.za/~sonot                                                 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.

http://physics.ukzn.ac.za/~sonot                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
pulse
•   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
would have if only the first wave was
present and the displacement it
would have if only the second wave
was present.

http://physics.ukzn.ac.za/~sonot                      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.

http://physics.ukzn.ac.za/~sonot                                           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
segments.
• 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).
http://physics.ukzn.ac.za/~sonot                                                    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.

http://physics.ukzn.ac.za/~sonot                       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:

http://physics.ukzn.ac.za/~sonot                                        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,
hence

• 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.

http://physics.ukzn.ac.za/~sonot                                                   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:

http://physics.ukzn.ac.za/~sonot                                          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.

http://physics.ukzn.ac.za/~sonot                                     11

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