The speed of a wave on a string
Transcript of The speed of a wave on a string
The speed of a wave on a string
• One of the key properties of any wave is the wave speed.
• Consider a string in which the tension is F and the linear
mass density (mass per unit length) is μ.
• We expect the speed of transverse waves on the string v
should increase when the tension F increases, but it should
decrease when the mass per unit length μ increases.
• It is shown in your text that the wave speed is:
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The four strings of a musical instrument are all made of the same
material and are under the same tension, but have different
thicknesses. Waves travel
A. fastest on the thickest string.
B. fastest on the thinnest string.
C. at the same speed on all strings.
QuickCheck
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Example 1
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Example 2 – One end of a 2.00-kg rope is tied to a support at the top of a
mine shaft 80.0 m deep. The rope is stretched taut by a 20.0-kg box of
rocks attached at the bottom. A geologist at the bottom of the shaft
signals to a colleague at the top by jerking the rope sideways. What is
the speed of a transverse wave on the rope? Also, if a point on the rope
is in transverse SHM with f = 2.00 Hz, how many cycles of the wave are
there in the rope’s length?
Reflection of a wave pulse at a fixed end of a string
• What happens when a wave
pulse or a sinusoidal wave
arrives at the end of the string?
• If the end is fastened to a rigid
support, it is a fixed end that
cannot move.
• The arriving wave exerts a
force on the support (drawing
4).
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Reflection of a wave pulse at a fixed end of a string
• The reaction to the force of
drawing 4, exerted by the
support on the string, “kicks
back” on the string and sets up a
reflected pulse or wave traveling
in the reverse direction.
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Reflection of a wave pulse at a free end of a string
• A free end is one that is
perfectly free to move in the
direction perpendicular to the
length of the string.
• When a wave arrives at this
free end, the ring slides along
the rod, reaching a maximum
displacement, coming
momentarily to rest (drawing
4).
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Reflection of a wave pulse at a free end of a string
• In drawing 4, the string is now
stretched, giving increased
tension, so the free end of the
string is pulled back down, and
again a reflected pulse is
produced.
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Standing waves on a string
• Waves traveling in opposite directions on a taut string
interfere with each other.
• The result is a standing wave pattern that does not move on
the string.
• Destructive interference occurs where the wave
displacements cancel, and constructive interference occurs
where the displacements add.
• At the nodes no motion occurs, and at the antinodes the
amplitude of the motion is greatest.
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Standing Waves
Time snapshots of two sine waves.
The red wave is moving in the −x-
direction and the blue wave is
moving in the +x-direction. The
resulting wave is shown in black.
Consider the resultant wave at the
points x = 0 m, 3 m, 6 m, 9 m, 12
m, 15 m and notice that the resultant
wave always equals zero at these
points, no matter what the time is.
These points are known as fixed
points (nodes).
In between each two nodes is an
antinode, a place where the medium
oscillates with an amplitude equal to
the sum of the amplitudes of the
individual waves.
FIGURE 16.27
𝑦 𝑥, 𝑡 = 2𝐴𝑠𝑖𝑛 𝑘𝑥 cos(𝜔𝑡)
The sine function dictates the position of the standing waves
while the cosine function expresses how the shape
oscillates with time.
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What is the wavelength
of this standing wave?
QuickCheck
A. 0.25 m
B. 0.5 m
C. 1.0 m
D. 2.0 m
Slide 17-12
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Example 3
Example 4 – A guitar string is plucked and creates a standing sinusoidal
wave with amplitude 0.750 mm and frequency 440 Hz. The wave velocity
is 143 m/s.
(a) Find the equation of the standing wave.
(b) Locate the nodes
(c) Find the maximum speed and acceleration of the string.
Standing waves on a string
• This is a time exposure of a
standing wave on a string.
• This pattern is called the
second harmonic.
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Standing waves on a string
• As the frequency of the
oscillation of the right-hand
end increases, the pattern of
the standing wave changes.
• More nodes and antinodes
are present in a higher
frequency standing wave.
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Normal modes
• For a taut string fixed at both
ends, the possible wavelengths
are and the possible
frequencies are fn = n v/2L =
nf1, where n = 1, 2, 3, …
• f1 is the fundamental
frequency, f2 is the second
harmonic (first overtone), f3 is
the third harmonic (second
overtone), etc.
• The figure illustrates the first
four harmonics.
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What is the mode number
of this standing wave?
QuickCheck
A. 4
B. 5
C. 6
Slide 17-18
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QuickCheck
A standing wave on a string vibrates as shown.
Suppose the string tension is reduced to 1/4 its
original value while the frequency and length are
kept unchanged. Which standing wave pattern is
produced?
Slide 17-19
Standing waves and string instruments
• When a string on a musical instrument is plucked, bowed or
struck, a standing wave with the fundamental frequency is
produced:
• This is also the frequency of the sound wave created in the
surrounding air by the vibrating string.
• Increasing the tension F increases the frequency (and the
pitch).
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Example 5 - Adjacent antinodes of a standing wave on a string are 15.0 cm
apart. A particle at an antinode oscillates in simple harmonic motion with
amplitude 0.850 cm and period 0.0750 s. The string lies along the +x-axis and
is fixed at x = 0.
(a) How far apart are the adjacent nodes? How far from antinodes?
(b) Find the wavelength, amplitude, and speed of the standing wave.
(c) Find the wavelength, amplitude, and speed of the traveling waves.
(d) Find the max speed of the string.
In-class Activity #1 – A standing wave on a wire has an amplitude of 2.40
mm, an angular frequency of 934 rad/s, and wave number 0.750π rad/m.
The left end of the wire is at x = 0. At what distances from the left end are
(a) the nodes of the standing wave?
(b) the antinodes of the standing wave?
(c) What is the node to antinode distance?
Organ pipes
• Organ pipes of different sizes produce tones
with different frequencies (bottom figure).
• The figure at the right shows displacement
nodes in two cross-sections of an organ pipe
at two instants that are one-half period
apart. The blue shading shows pressure
variation.
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Harmonics in an open pipe
• The fundamental frequency of an open pipe is shown.
• The shading indicates the pressure variations.
• The red curves are graphs of the displacement along the pipe
axis at two instants separated in time by one half-period.
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Harmonics in an open pipe
• Higher harmonics in an open pipe have frequency:
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Harmonics in an open pipe
• Higher harmonics in an open pipe have frequency:
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Harmonics in a stopped pipe
• The fundamental frequency of a stopped pipe is shown.
• The shading indicates the pressure variations.
• The red curves are graphs of the displacement along the pipe
axis at two instants separated in time by one half-period.
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Harmonics in a stopped pipe
• Higher harmonics in a stopped pipe have frequency:
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Harmonics in a stopped pipe
• Higher harmonics in a stopped pipe have frequency:
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QuickCheck
An open-open tube of air has
length L. Which is the
displacement graph of the m = 3
standing wave in this tube?
Slide 17-30
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QuickCheck
An open-closed tube of air of length
L has the closed end on the right.
Which is the displacement graph of
the m = 3 standing wave in this tube?
Slide 17-31
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Example 6 – One a day when the speed of sounds is 345
m/s, the fundamental frequency of a particular stopped-organ
pipe is 220 Hz. How long is this pipe? The second overtone
of this pipe has the same wavelength as the third harmonic of
an open pipe. How long is the open pipe?