WAVE MOTION
Basic Wave Properties
Learning Objectives
A wave is a self-propagating disturbance in a medium.
After you complete the homework associated with this
lecture, you should be able to:
• Explain how a traveling wave is a disturbance in a media
rather than transport of matter;
• Determine the wavelength and amplitude of a wave;
• Determine a wave’s velocity and that of a particle in the
medium from its mathematical description;
• Describe the cause of the Doppler effect, and use
appropriate mathematical relations to predict sound
frequency heard by a moving observer.
It is the propagation of a state of stress or tension in the
medium. Typical waves phenomena:
Type
Medium
water wave
liquid (duh)
sound wave
air
violin/guitar
string/metal
earthquake
earth
radio & light
vacuum space (hmmm....)
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The material of the medium is NOT moved along the wave.
Instead the particles of the medium (e.g. string segments,
water molecules) are momentarily displaced from their
equilibrium situations.
Wave pulse y(x,t)
is moving to the right.
But the medium's
particles are moving
up and down.
Leading edge is pulled
up, trailing “springs”
back down.
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Let y(x,t=0) = D(x-xpeak) be the
shape of the pulse at time zero. For
convenience, we set origin so that
xpeak(t=0) = 0. We see that in time t,
the peak moves to position
xpeak(t) = vxt .
Since the shape of the pulse is preserved as it moves,
the wave function y(x,t) at any time t is related to that at
time zero by the Galilean Transform:
web source: http://webphysics.davidson.edu/physlet_resources/bu_semester1/c20_wave_propagation_sim.html
y(x,t) = D(x-xpeak) = y(x-vxt,0) , i.e., x Y x - vx t
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Basic Sinusoidal Wave at t = 0
This periodic wave has a sinusoidal form:
⎡ ⎛x⎞
⎤
⎡ 2π
⎤
y ( x) = A sin ⎢ 2π ⎜ ⎟ + ϕ ⎥ = A sin ⎢
x + ϕ ⎥ = A sin [ kx + ϕ ]
⎣λ
⎦
⎣ ⎝λ⎠
⎦
where λ = the wavelength (the repeat length). Note that the
Traveling Sinusoidal Wave
At a frozen snapshot at t = 0, wave is y(x) = A sin(kx + φ).
To get sinusoidal waves to move, we simply remember the
replacement (Galilean transformation): x Y x - vx t
direction
vx = +v (pos x)
vx = –v (neg x)
⎡ 2π
y ( x, t ) = A sin ⎡⎣ k ( x − vxt ) + ϕ ⎤⎦ = A sin ⎢
⎣ λ
sine function argument changes by 2π when x changes by λ.
k=
2π
λ
is called the wavenumber of the wave.
⎤
( x − vx t ) + ϕ ⎥
⎦
= A sin ⎡⎣ kx ± ωt + ϕ ⎤⎦ where ω = kv = 2π f = 2π / T
Use – ω for wave traveling in positive x direction
Use + ω for wave traveling in negative x direction.
web source: http://webphysics.davidson.edu/physlet_resources/bu_semester1/c20_wavelength_period.html
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Several Forms for Expressing Traveling Waves
y ( x, t ) = A sin [ kx ± ωt + ϕ ]
⎡ ⎛x
⎤
⎞
= A sin ⎢ 2π ⎜ ± ft ⎟ + ϕ ⎥
⎠
⎣ ⎝λ
⎦
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What is the velocity of a particle located at x at time t
whose displacement is given by y(x,t) = A sin[Θ(x,t)] ?
CAUTION : The particle moves harmonically along the
y-axis, NOT the x-axis of the wave motion!
= A sin [ Θ( x, t ) ]
What is speed and direction of wave’s motion: vx?
Easy – no matter what the mathematical form!
⎛ ∂Θ ⎞ ⎛ ∂Θ ⎞
y(x,t) = A sin[Θ(x,t)] Y vx = vwave , x = − ⎜
⎟ ⎜
⎟
⎝ ∂t ⎠ ⎝ ∂x ⎠
Y v = wave speed = |vx | = λ f = ω / k
Vy = particle velocity in medium (in y -direction)
∂y
∂Θ
= A cos[Θ( x, t )]
(always changing )
∂t
∂t
≠ vx
(not equal to the constant wave velocity )
=
web: http://www.suu.edu/faculty/penny/Phsc2210/Physlets/PhysletsForWeb/Semester1/wave_equation.html
Ans: Asin[2π(x/λ !f@t)] = 1.8*sin(2*pi*(x/2.75-0.4*t)); i.e., A=1.8, λ=2.75, f=0.4
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Example: If y(x,t) = 3 sin(– 2x – 8t + ¼ π)
1. What is the speed and direction of this traveling wave?
vwave , x
⎡ ∂Θ ⎤
∂ (−2 x − 8t + 1 4 π )
⎢⎣ ∂t ⎥⎦
( −8)
∂t
=−
=−
=−
= −4
∂ (−2 x − 8t + 1 4 π )
(−2)
⎡ ∂Θ ⎤
⎢⎣ ∂x ⎥⎦
∂x
Wave is moving 4 m/s in the negative-x direction.
2. What is the maximum speed of a particle in the medium?
∂y
∂Θ
∂[−2 x − 8t + 1 4 π ]
= A cos[Θ( x, t )]
= 3cos[−2 x − 8t + 1 4 π ]
∂t
∂t
∂t
= 3cos[−2 x − 8t + 1 4 π ]( −8) = − 24 cos[−2 x − 8t + 1 4 π ]
Vy =
Speed of Transverse Wave on Stretched Medium
What is the speed of a transverse wave on a stretched
string wire? Let
FT = tension in the string
µ = the mass per unit length of the string
(linear mass density)
Then the speed of the wave is:
v=
ω
k
=
FT
μ
The greater the mass density, the more sluggish the
wave. The higher the tension, the more it whips along,
overcoming sluggish mass.
Maximum speed of the particle is 24 m/s.
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Power Delivered by Wave
The wave motion is a motion of the medium. It takes
effort to keep the particles of the medium moving up
and down. That means POWER must be delivered to the
particles to cause them to change their kinetic energies.
Imagine a wave terminated at an end point that is
coupled to a power
measuring device.
The average power P
(energy per time)
transported by a one-dimensional string of linear mass
density µ is
P = 12 μω 2 A2 v
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Doppler Effect
A source S of waves is emitting at frequency fS. The speed
of the waves in the medium is v. The frequency fO that an
observer O detects will be affected by the motion of the
source and the observer in the medium, increase if
approaching and decrease if receding.
web source:
web source:
web source:
http://webphysics.davidson.edu/Applets/Applets.html Waves/Doppler
http://www.phy.ntnu.edu.tw/java/Doppler/Doppler.html
http://www.walter-fendt.de/ph11e/dopplereff.htm
Sure-fire formula:
⎛ v − vOx ⎞
fO = ⎜
⎟ fS
⎝ v − vSx ⎠
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