a wave source at x = 0 that oscillates with simple
TRANSCRIPT
© 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?
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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)