a wave source at x = 0 that oscillates with simple …...in-class activity 1 - transverse waves on a...
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© 2017 Pearson Education, Inc. Slide 16-1
Sinusoidal Waves
A wave source at x = 0 that oscillates with simple
harmonic motion (SHM) generates a sinusoidal
wave.
© 2017 Pearson Education, Inc. Slide 16-2
Above is a history graph for a sinusoidal wave, showing
the displacement of the medium at one point in space.
Each particle in the medium undergoes simple
harmonic motion with frequency f, where f = 1/T.
The amplitude A of the wave is the maximum value of
the displacement.
Sinusoidal Waves
© 2017 Pearson Education, Inc. Slide 16-3
Above is a snapshot graph for a sinusoidal wave,
showing the wave stretched out in space, moving to the
right with speed v.
The distance spanned by one cycle of the motion is
called the wavelength λ of the wave.
Sinusoidal Waves
© 2017 Pearson Education, Inc. Slide 16-4
A wave on a string is traveling
to the right. At this instant, the
motion of the piece of string
marked with a dot is
QuickCheck
A. Up.
B. Down.
C. Right.
D. Left.
© 2017 Pearson Education, Inc. Slide 16-5
or, in terms of frequency:
The distance spanned by one cycle of the motion is
called the wavelength λ of the wave. Wavelength is
measured in units of meters.
During a time interval of exactly one period T, each
crest of a sinusoidal wave travels forward a
distance of exactly one wavelength λ.
Because speed is distance divided by time, the
wave speed must be
Sinusoidal Waves
© 2017 Pearson Education, Inc. Slide 16-6
The Mathematics of Sinusoidal Waves
Define the wave number and angular frequency:
Wave function:
This wave travels at a speed v = ω/k.
𝜔 =2𝜋𝑣
λ=2𝜋
𝑇
𝑦(𝑥, 𝑡) = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡 + ϕ𝑜)
Particle velocity and acceleration in a sinusoidal wave
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𝜕2𝑦(𝑥, 𝑡)
𝜕𝑥2=
1
𝑣2𝜕2𝑦(𝑥, 𝑡)
𝜕𝑡2
Wave equation:
All wave behavior obeys the wave equation. Likewise,
any physical behavior that satisfies the wave equation
can be modeled as a wave.
© 2017 Pearson Education, Inc. Slide 16-8
Which of the following
equations satisfy the wave
equation?
QuickCheck
A.
B.
C.
D. Both A and B.
𝜕2𝑦(𝑥, 𝑡)
𝜕𝑥2=
1
𝑣2𝜕2𝑦(𝑥, 𝑡)
𝜕𝑡2
Acos(kx+ωt); .
Acos(kx+ωt); .
𝑦(𝑥, 𝑡) = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡)
𝑦(𝑥, 𝑡) = 𝐴𝑐𝑜𝑠(𝑘𝑥 − 𝜔𝑡)
𝑦 𝑥, 𝑡 = 𝐴𝑐𝑜𝑠 𝑘𝑥 + 𝐴𝑐𝑜𝑠(𝜔𝑡)
© 2016 Pearson Education, Inc.
Example 1 - A water wave traveling in a straight line on a lake is described by
the equation
y(x,t) = 2.75cos(0.410x+6.20t) cm
(a)How much time does it take for one complete wave pattern to go past a
fisherman in a boat at anchor?
(b) What horizontal distance does the wave crest travel in that time?
(c) What is the wave number?
(d) What is the number of waves per second that pass the fisherman?
(e) How fast does a wave crest travel past the fisherman?
(f) What is the maximum speed of his cork floater as the wave causes it to bob
up and down?
© 2016 Pearson Education, Inc.
In-class Activity 1 - Transverse waves on a string have wave speed 8.00
m/s, amplitude 0.0700 m, and wavelength 0.320 m. The waves travel in
the -x direction, and at t = 0 the x = 0 end of the string has its maximum
upward displacement.
(a) Find the frequency, period, and wave number of these waves.
(b) Write the wave function describing this wave.
© 2017 Pearson Education, Inc.
If wave 1 displaces a particle in the medium by y1
and wave 2 simultaneously displaces it by y2, the net
displacement of the particle is y1 + y2.
The Principle of Superposition
Slide 17-11
It can be easily shown that the superposition of waves
still satisfies the wave equation.
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The figure shows the
superposition of two waves
on a string as they pass
through each other.
The principle of
superposition comes into
play wherever the waves
overlap.
The solid line is the sum at
each point of the two
displacements at that
point.
The Principle of Superposition
Slide 17-12
© 2017 Pearson Education, Inc.
QuickCheck 17.1
Two wave pulses on a
string approach each
other at speeds of
1 m/s. How does the
string look at t = 3 s?
Slide 17-13
© 2017 Pearson Education, Inc.
QuickCheck 17.2
Two wave pulses on a
string approach each
other at speeds of
1 m/s. How does the
string look at t = 3 s?
Slide 17-14
FIGURE 16.20
Constructive interference of two identical waves produces a wave with twice the
amplitude, but the same wavelength.
FIGURE 16.21
Destructive interference of two identical waves, one with a phase shift of 180°(𝜋 rad) ,
produces zero amplitude, or complete cancellation.
Superposition of two waves
with identical amplitudes,
wavelengths, and frequency,
but that differ in a phase shift.
The resultant wave has a
modified amplitude and
phase shift but in other ways
it is similar to the original
waves.
𝑦𝑛𝑒𝑡(𝑥, 𝑡) = 2𝐴𝑐𝑜𝑠(ϕ𝑜
2)𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡 +
ϕ𝑜
2)
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.
© 2016 Pearson Education, Inc.
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.
Example 2 – 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.
© 2016 Pearson Education, Inc.
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.
© 2016 Pearson Education, Inc.
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.
© 2016 Pearson Education, Inc.
Example 3 - 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.
(c) Find the max speed of the string.
In-class Activity #2 – 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?