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and Mountain Waves
2nd LO submission PHYS 101 LG2 (Green)
Acrophobia
Mountain (Lee) Waves
 Mountain Waves are atmospheric standing waves that exist on the lee
side of mountains (side sheltered from wind by mountains)
 They form standing waves due to periodic changes of temperature
and atmospheric pressure that lead to vertical displacement
 Characteristic to standing waves, mountain waves involve the
interference of two waves of identical frequency moving in opposite
directions, resulting in a constant amplitude at each position
 They pose great danger to aircrafts flying at low altitudes near
mountainous regions as they can cause sudden and dramatic vertical
displacement/ turbulence
Mountain Waves
 In order to help your friend to battle a fear of heights, you decide to
take him/her for a ride on a small airplane your rich friend has kindly
decided to lend you
 However, upon hearing the dangers of mountain waves, you take
extra pre-cautions and try to study standing waves beforehand
 Assumptions:
We assume that the presence of the airplane will not affect the
standing wave
We assume the waves that are travelling in opposite directions forming
the standing wave have the same frequency, wavelength, amplitude
and no phase constants
Standing Waves—Misconceptions
 Which of the below is true about the amplitude of standing waves?
a) The amplitude is constant and the same for a given wave
b) The amplitude changes over time
c) Due to interference, the amplitude at all points of the standing
wave is always higher than either of the component waves
d) The amplitude is position-dependent
Standing Waves—Misconceptions
 Which of the below is true about the amplitude of standing waves?
a) The amplitude is constant and the same for a given wave
b) The amplitude changes over time
c) Due to interference, the amplitude at all points of the standing
wave is always higher than either of the component waves
d) The amplitude is position-dependent
Standing Waves—Misconceptions--explained
 Which of the below is true about the amplitude of standing waves?
a) The amplitude is constant and the same for a given wave
b) The amplitude changes over time
c) Due to interference, the amplitude at all points of the standing wave is
always higher than either of the component waves
d) The amplitude is position-dependent
 A common mistake with standing waves is that amplitude is mistaken for displacement
 The amplitude is the maximum possible displacement, not the displacement at a
given time and position
 Also, interference is not always constructive but can also be destructive, therefore the
amplitude is not always higher than both of the component waves
Amplitude
at a given
position Displacement at given position
and time
Standing Waves—Misconceptions--explained
 Which of the below is true about the amplitude of standing waves?
a) The amplitude is constant and the same for a given wave
b) The amplitude changes over time
c) Due to interference, the amplitude at all points of the standing wave is
always higher than either of the component waves
d) The amplitude is position-dependent
 Another misconception is that the amplitude is constant for all positions in a given
wave
This is true in travelling waves, as the wave propagates through the entire medium,
but NOT TRUE for standing waves
In standing waves, amplitude is position-dependent and constant for each position,
but are not all the same across different positions!
Remember, amplitude of a standing wave: A(x)=2Asin(kx), where A is the
amplitude of the component waves
Amplitude 2
Amplitude 1
Let’s Fly!
 Now that you’re ready to fly, you
quickly ascend to above the mountains
 You decide to fly parallel to the
mountain waves in order to not be
affected by its turbulence.
 Your friend is starting to feel uneasy and
you try to show him/her cool tricks in the
air in order to calm him/her down
 You remember that mountain waves
actually create clouds at every second
anti-node and you look for clouds to fly
through in order to impress you friend
-Clouds formed by mountain waves:
From far away, all the clouds seem to
connect even though they form only
at every second anti-node (since they
spread)
Distance between Anti-Nodes
 1)Flying on the same axis as the standing wave (parallel, without
coming into contact), you pass right by a cloud. How much further do
you need to fly from this cloud in order to reach another cloud, given
that you will reach a node after flying 5km?
 Knowing that A(x)=2A sin(kx)
and that the maximum
value from a sine function is
1 and the lowest
(magnitude-wise) is 0, we
can solve for the
wavelength.
 Remember: Antinode
occurs at the position with
highest amplitude and
node at the position with an
amplitude of 0
 Wavelength = 20km
 Knowing that the distance
between each consecutive
anti-node is λ/2, and that
λ = 20km, the plane must
travel 20km since clouds
are formed only at every
second node and each
node is 10km apart
Displacement of Standing Wave Elements
 Suddenly, your friend starts to feel very sick and demands that you land as soon
as possible. In order to land, the fastest and shortest would be to pierce right
through the mountain waves—however, the turbulence is dangerous!.
 Right as your friend faints, you realize that you could pierce through the
mountain waves at a time when there is minimal to no turbulence anywhere
within a segment between nodes of the standing wave. The displacement
function of the standing wave is given by :
𝐷 𝑥, 𝑡 = 2𝐴𝑠𝑖𝑛
2𝜋
𝜆
𝑥 𝑐𝑜𝑠
2𝜋
𝑇
𝑡
 Given that T (period) is 10 seconds and at t=0, each element is at its maximum
displacement, at what time should you pierce through the mountain waves to
avoid as much turbulence as possible?
 Hint: to have no turbulence anywhere along a segment between 2 nodes of the
mountain wave is to have no displacement throughout that segment
Displacement of Standing Wave Elements
 To do this question, we must realize that the segment between two nodes of a
standing wave displaces each proportionally to their amplitude—so after time t,
each element of the segment will have displaced the same fraction of their
individual amplitudes
 Remember that standing
waves vary in displacement
over time as well, even though
they are always shown at their
individual amplitudes
Displacement of Standing Wave Elements
 So we find the time when each element of the segment of the standing wave is
at the equilibrium position, where D(x,t)=0
 At t=2.5s, you will be able to pierce the mountain wave with minimal turbulence
(when a segment is all at the equilibrium position)
Picture Sources
 http://en.wikipedia.org/wiki/Lee_wave
 https://simpleharmonicmotion-tgn05.wikispaces.com/Standing+Waves
 http://tornado.sfsu.edu/geosciences/classes/m415_715/monteverdi/Satellit
e_Lectures/sat.html
 http://www.atsb.gov.au/publications/2009/mountain-wave-and-
associated-turbulence.aspx
 http://rses.anu.edu.au/education/potential-projects/laboratory-and-
numerical-modeling-lee-wave-breaking
 http://xeeming.deviantart.com/art/The-Amityville-Project-Phobos-
Acrophobia-399766756

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Lo2

  • 1. and Mountain Waves 2nd LO submission PHYS 101 LG2 (Green) Acrophobia
  • 2. Mountain (Lee) Waves  Mountain Waves are atmospheric standing waves that exist on the lee side of mountains (side sheltered from wind by mountains)  They form standing waves due to periodic changes of temperature and atmospheric pressure that lead to vertical displacement  Characteristic to standing waves, mountain waves involve the interference of two waves of identical frequency moving in opposite directions, resulting in a constant amplitude at each position  They pose great danger to aircrafts flying at low altitudes near mountainous regions as they can cause sudden and dramatic vertical displacement/ turbulence
  • 3. Mountain Waves  In order to help your friend to battle a fear of heights, you decide to take him/her for a ride on a small airplane your rich friend has kindly decided to lend you  However, upon hearing the dangers of mountain waves, you take extra pre-cautions and try to study standing waves beforehand  Assumptions: We assume that the presence of the airplane will not affect the standing wave We assume the waves that are travelling in opposite directions forming the standing wave have the same frequency, wavelength, amplitude and no phase constants
  • 4. Standing Waves—Misconceptions  Which of the below is true about the amplitude of standing waves? a) The amplitude is constant and the same for a given wave b) The amplitude changes over time c) Due to interference, the amplitude at all points of the standing wave is always higher than either of the component waves d) The amplitude is position-dependent
  • 5. Standing Waves—Misconceptions  Which of the below is true about the amplitude of standing waves? a) The amplitude is constant and the same for a given wave b) The amplitude changes over time c) Due to interference, the amplitude at all points of the standing wave is always higher than either of the component waves d) The amplitude is position-dependent
  • 6. Standing Waves—Misconceptions--explained  Which of the below is true about the amplitude of standing waves? a) The amplitude is constant and the same for a given wave b) The amplitude changes over time c) Due to interference, the amplitude at all points of the standing wave is always higher than either of the component waves d) The amplitude is position-dependent  A common mistake with standing waves is that amplitude is mistaken for displacement  The amplitude is the maximum possible displacement, not the displacement at a given time and position  Also, interference is not always constructive but can also be destructive, therefore the amplitude is not always higher than both of the component waves Amplitude at a given position Displacement at given position and time
  • 7. Standing Waves—Misconceptions--explained  Which of the below is true about the amplitude of standing waves? a) The amplitude is constant and the same for a given wave b) The amplitude changes over time c) Due to interference, the amplitude at all points of the standing wave is always higher than either of the component waves d) The amplitude is position-dependent  Another misconception is that the amplitude is constant for all positions in a given wave This is true in travelling waves, as the wave propagates through the entire medium, but NOT TRUE for standing waves In standing waves, amplitude is position-dependent and constant for each position, but are not all the same across different positions! Remember, amplitude of a standing wave: A(x)=2Asin(kx), where A is the amplitude of the component waves Amplitude 2 Amplitude 1
  • 8. Let’s Fly!  Now that you’re ready to fly, you quickly ascend to above the mountains  You decide to fly parallel to the mountain waves in order to not be affected by its turbulence.  Your friend is starting to feel uneasy and you try to show him/her cool tricks in the air in order to calm him/her down  You remember that mountain waves actually create clouds at every second anti-node and you look for clouds to fly through in order to impress you friend -Clouds formed by mountain waves: From far away, all the clouds seem to connect even though they form only at every second anti-node (since they spread)
  • 9. Distance between Anti-Nodes  1)Flying on the same axis as the standing wave (parallel, without coming into contact), you pass right by a cloud. How much further do you need to fly from this cloud in order to reach another cloud, given that you will reach a node after flying 5km?
  • 10.  Knowing that A(x)=2A sin(kx) and that the maximum value from a sine function is 1 and the lowest (magnitude-wise) is 0, we can solve for the wavelength.  Remember: Antinode occurs at the position with highest amplitude and node at the position with an amplitude of 0  Wavelength = 20km
  • 11.  Knowing that the distance between each consecutive anti-node is λ/2, and that λ = 20km, the plane must travel 20km since clouds are formed only at every second node and each node is 10km apart
  • 12. Displacement of Standing Wave Elements  Suddenly, your friend starts to feel very sick and demands that you land as soon as possible. In order to land, the fastest and shortest would be to pierce right through the mountain waves—however, the turbulence is dangerous!.  Right as your friend faints, you realize that you could pierce through the mountain waves at a time when there is minimal to no turbulence anywhere within a segment between nodes of the standing wave. The displacement function of the standing wave is given by : 𝐷 𝑥, 𝑡 = 2𝐴𝑠𝑖𝑛 2𝜋 𝜆 𝑥 𝑐𝑜𝑠 2𝜋 𝑇 𝑡  Given that T (period) is 10 seconds and at t=0, each element is at its maximum displacement, at what time should you pierce through the mountain waves to avoid as much turbulence as possible?  Hint: to have no turbulence anywhere along a segment between 2 nodes of the mountain wave is to have no displacement throughout that segment
  • 13. Displacement of Standing Wave Elements  To do this question, we must realize that the segment between two nodes of a standing wave displaces each proportionally to their amplitude—so after time t, each element of the segment will have displaced the same fraction of their individual amplitudes  Remember that standing waves vary in displacement over time as well, even though they are always shown at their individual amplitudes
  • 14. Displacement of Standing Wave Elements  So we find the time when each element of the segment of the standing wave is at the equilibrium position, where D(x,t)=0  At t=2.5s, you will be able to pierce the mountain wave with minimal turbulence (when a segment is all at the equilibrium position)
  • 15. Picture Sources  http://en.wikipedia.org/wiki/Lee_wave  https://simpleharmonicmotion-tgn05.wikispaces.com/Standing+Waves  http://tornado.sfsu.edu/geosciences/classes/m415_715/monteverdi/Satellit e_Lectures/sat.html  http://www.atsb.gov.au/publications/2009/mountain-wave-and- associated-turbulence.aspx  http://rses.anu.edu.au/education/potential-projects/laboratory-and- numerical-modeling-lee-wave-breaking  http://xeeming.deviantart.com/art/The-Amityville-Project-Phobos- Acrophobia-399766756