LECTURE 3
TRAVELING WAVES
Instructor: Kazumi Tolich
Lecture 3
2
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Reading chapter 15-1 to 15-2
¤ Transverse and longitudinal waves
¤ Moving displacement of
¤ Speed of
waves
¤ Harmonic waves
a wave pulse
Quiz: 1
3
Transverse & longitudinal waves
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Mechanical wave is the disturbance, that travels through a medium, carrying energy
with it.
Transverse waves are waves in which the motion of the medium is perpendicular to the
direction of propagation of the disturbance.
¤ Waves on a string
Longitudinal waves are waves in which the motion of the medium is along (parallel to)
the direction of propagation of the disturbance.
¤ Sound waves
Demo 1 & 2
5
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Wave on Rope
¤ An
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example of transverse waves.
Longitudinal Waves (Large Spring or Slinky)
¤ An
example of longitudinal waves.
Speed of waves
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Speed of waves relative to the medium in which they are traveling depends
on the medium.
Speed of waves on a string is given by
¤ FT
is the tension in the string.
¤ µ is the mass density (mass per unit length) of the string.
Demo 3
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Tension Dependence of Wave Speed
Quiz: 2 & 3
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Harmonic waves
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If disturbances occur due to periodic motion, they produce a periodic wave.
If a harmonic wave is traveling through a medium, each point of the medium
oscillates in simple harmonic motion.
Wavelength, 𝜆, of a wave is the distance that wave travels in one cycle.
Period, 𝑇, is the duration of one cycle.
The speed of the wave propagation is given by
Moving displacement
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Suppose we wish to move a parabolic function 𝑓 𝑥 = 𝑥 & so that it is centered at greater and
greater values of the independent variable 𝑥. Then we change 𝑥 to 𝑥 − 2 , 𝑥 − 4 , etc.
Similarly, if we wish to displace some arbitrary function 𝑓 𝑥 by a distance 𝑑 in the positive 𝑥direction, we replace 𝑓 𝑥 by 𝑓 𝑥 − 𝑑 .
If we wish to make the displacement increase with time, i.e., move the curve with speed 𝑣, we
make 𝑑 = 𝑣𝑡, so that the function becomes 𝑓 𝑥 − 𝑣𝑡 .
Similarly, to displace the function in the negative 𝑥 direction, we use 𝑓 𝑥 + 𝑣𝑡 .
Harmonic transverse wave function
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For a transverse harmonic wave traveling in the positive x-direction, the wave
function is
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k is called wave number and is given by
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ω is the angular frequency.
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A is the amplitude, the maximum displacement in the y-direction.
Quiz 3-4
Example 1
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The wave function for a harmonic wave
on a string is
𝑦 𝑥, 𝑡 = 1.00 mm sin 62.8 m:; 𝑥 + 314 s :; 𝑡 .
a)
b)
c)
In what direction does this wave travel,
and what is the wave’s speed?
Find the wavelength, frequency, and
period of this wave.
What is the maximum speed of any point
on the string?
Energy transfer via a wave on a string
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As a wave moves along a string, energy is transferred from one segment to the next.
Average power are given by
;
𝑃>? = & 𝜇𝑣𝜔& 𝐴&
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The average energy flowing at a point P during Δt is
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This energy is distributed over a length Δx = vΔt, so the average energy in length Δx is
Example 2
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A harmonic wave on a string that has
a mass per unit length of
µ = 0.050 kg/m and a tension of
FT = 80 N has an amplitude of
A = 5.0 cm. Each point on the string
moves with simple harmonic motion at
a frequency of f = 10 Hz. What is the
power carried by the wave
propagating along the string?
Demo 4
16
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Transverse Waves (Bell Labs Wave Machine)
¤ Torsional waves
sent on a series of rods.
¤ The ends of the rods move up and down.
¤ The shorter the rods, the faster the wave travels. (Each rod connected to the
wire acts like a torsion pendulum. The shorter rod has a smaller I, so the
period of oscillation is shorter.)
Harmonic sound waves
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A harmonic sound wave traveling in the positive xdirection can be described by
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s0 is the amplitude, the maximum displacement in the x-direction.
The pressure wave is given by
𝑝 𝑥, 𝑡 = −𝑝D cos 𝑘𝑥 − 𝜔𝑡 where 𝑝D = 𝜌𝜔𝑣𝑠D.
¤
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p stands for the change in pressure from the equilibrium
pressure
p0 is the pressure amplitude, the maximum value of p
Energy of sound waves
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The average energy of a harmonic sound wave in a volume element
ΔV is given by
Energy per unit volume is given by
Demo 5
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Siren in Vacuum
¤ Demonstration
of sound wave traveling in the air.