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Physics 1C: More on
Mechanical Waves
!
!
Wednesday, 8 April 2015
Reminders
• The first quiz will be this Friday at 1pm!
•
Don’t forget your homework and reading assignment too.
•
You will be given all of the equations you need and more on the
test… Etot=K+U, F=−kx, x=Acos(ωt+φ0), v=dx/dt, a=d2x/dt2,
ω =√(k/m), etc. No need to memorize them!
•
Focus on understanding the physics concepts and on what the
equations mean and describe. Think through each question
carefully and step-by-step. Draw a picture or graph. Remember
what cosine and sine functions look like.
•
note: on Monday we’ll have a guest instructor (being interviewed
for a lecturer position at UCSD): sound waves & Doppler effect
Concepts included on Friday’s Quiz
•
positions, velocities, and accelerations of simple harmonic
motion
•
frequency and period
•
force and energy: potential, kinetic, conservation; probably
not momentum
•
simple pendulum, but not physical pendulum
•
maybe damped oscillations, but not forced oscillations
•
transverse and longitudinal mechanical waves: wavelength,
speed, etc., but probably not energy, power, pressure (that is,
the test only goes up to section 13.2 or 13.3)
Mechanical Waves: What is a Wave?
• a disturbance that carries energy and momentum from
one location to another without a transfer of matter
• a wave is created by a source and typically needs a
medium to travel through
• examples: ripples in water, sound
• waves of light and radiation are not mechanical waves
What is a wave?
What is a wave?
!
!
Transverse wave: motion of medium is
perpendicular to direction of motion
Longitudinal wave: motion of medium is
parallel to direction of motion
Physics 112
Clicker Question
19A-3
Properties
of Sinusoidal
Waves
Properties
Waves
What of
is Sinusoidal
the wavelength
of the
following
graph (all numbers arewave
given
in centimeters)?
velocity:
4 m/s to the right
!y(mm)
x(m)
•
•
•
Wavelength: the distance between two corresponding
points
a wave
A) on
0.1!cm.
D) 2.0!cm.
Frequency: the number of cycles
a wave undergoes in
B) 0.2!cm.
a given
amount of time.
E) 3.0!cm.
Wave
C) speed
1.0!cm.
Physics 1C
Clicker
Question
19A-3
Sinusoidal Waves
Sinusoidal
Waves
What is the
wavelength of the following
graph (all numbers are
given
in centimeters)?
wave
velocity:
4 m/s to the right
!y(x, t= 0 s) [mm]
t = 4/8 s
x [m]
How do we determine the period of the wave?
T = 1/2 s
A) 0.1!cm.
D) 2.0!cm.
B) we
0.2!cm.
How do
determine the frequency
of the wave?
E) 3.0!cm.
C) 1.0!cm.
f = 1/T = 2 Hz
angular frequency: ⇥ = 2 f = 12.57 rad/s
Physics 1C
Physics 1C
Sinusoidal
Waves
Clicker
Question
19A-3
Sinusoidal
Waves
What is the
wavelength of the following
graph (all numbers are
given
in centimeters)?
wave
velocity:
4 m/s to the right
!y(x, t= 0 s) [mm]
t = 4/8 s
x [m]
Period is the amount of time necessary for the wave to
travel by one wavelength:
A) 0.1!cm.
D) 2.0!cm.
speed = wavelength / period
B) 0.2!cm.
E) 3.0!cm.
C) 1.0!cm.
v= f
Physics 1C
!
Traveling Wave:
More Definitions
•
wavelength λ (inversely related
to k) and amplitude A
!
•
period T and frequency (number
of crests that pass a given point
in a unit time interval: f=1/T)
generalized wave function (for a
wave moving to the right):
Traveling Wave: water ripples
think of water ripples y(x) at
a given snapshot in time t.
or think of y(t) at a given x.
example of a wave function
(for wave moving to the right):
y is a function of both x and t
Example: Traveling Sinusoidal Wave
A sinusoidal wave traveling in the positive x direction
has an amplitude of 15.0 cm, a wavelength of 40.0 cm,
and a frequency of 8.00 Hz.
The vertical position of an element of the medium at t=0
and x=0 is also 15.0 cm.
a) Find the wave number k, period T,
angular frequency ω, and speed v of the
wave.
You’d use these familiar equations:
k=2π/λ, T=1/f, ω=2πf, v=λf
Example: Traveling Sinusoidal Wave
A sinusoidal wave traveling in the positive x direction
has an amplitude of 15.0 cm, a wavelength of 40.0 cm,
and a frequency of 8.00 Hz.
The vertical position of an element of the medium at t=0
and x=0 is also 15.0 cm.
b) Determine the phase constant and write a
general expression for the wave function.
15.0=(15.0) sin φ, so φ=π/2 rad
Clicker
Question
19A-3
Another Example
Clicker-6A
What is the wavelength of the following
graph (all numbers are
given
in centimeters)?
wave
velocity:
4 m/s to the right
!y(x, t= 0 s) [mm]
x [m]
If we halve the frequency, but do not alter the medium,
what happens to this graph?
A) 0.1!cm.
D) 2.0!cm.
A. nothing
B) B.0.2!cm.
stretched in the horizontal direction
E) 3.0!cm.
C.
compressed
in
the
horizontal
direction
C) 1.0!cm.
D. stretched in the vertical direction
E. compressed in the vertical direction
Physics 1C
Clicker Question 19A-3
Another Example Clicker-6B
*credit forWhat
correct is
answer
the wavelength of the following
graph (all numbers are
given
in centimeters)?
wave
velocity:
4 m/s to the right
!y(x, t= 0 s) [mm]
x [m]
If we alter the medium so that the speed is halved, but
keep the frequency the same, what happens to this graph?
A) 0.1!cm.
D) 2.0!cm.
A. nothing
B) B.0.2!cm.
stretched in the horizontal direction
E) 3.0!cm.
C.
compressed
in
the
horizontal
direction
C) 1.0!cm.
D. stretched in the vertical direction
E. compressed in the vertical direction
Physics 1C
Speed of Transverse Wave
net force acting on piece (or “element”)
of string under tension T:
applying the sinusoidal wave function
to the linear wave equation gives us the
speed of the wave on a string
(μ is the string’s mass per
unit length)
Rate of Energy Transfer
Waves transport energy (and momentum) when they
propagate through a medium
We model each piece of string as a simple harmonic oscillator;
each element has the same total energy
Kinetic energy: dK = (1/2) dm v2
using Acos(kx-ωt), then we get:
K = (1/4) μ ω2 A2 λ
Potential energy:
Total Energy:
Rate of Energy Transfer
Total Energy:
The power, P, or rate of energy transfer TMW,
associated with a mechanical wave is then:
Sound
Waves
Sound
• Longitudinal Wave traveling through the air
• speed of sound at room temperature ~ 340 m/s
• audio frequency range: 20-20,000 Hz
Physics 1C
Sound Waves
Clicker-6C
The red dot represents an air molecule that oscillates in
simple harmonic motion with a period, T. What
physical factors determine the period of its oscillations?
A. the frequency of the vibrator on the left
B. the speed of sound through the air
C. the distance between the vibrator and the red dot
D. both B and C
E. A, B and C
Physics 1C
Reflection & Transmission, Wavefronts
• Wavefronts: a way to draw a “snapshot graph,” but in multiple
dimensions
• lines represent the peaks of the waves
• as the wave propagates, the wavefronts move
For Friday:
1. make sure you turn in your homework and reading quiz
on www.webassign.net
2. Good luck on the test!
3. we’ll continue with chapter 13 on Monday—be ready for
sound waves and Doppler effect then