Lo2

and Mountain Waves2nd 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--explained






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






Another misconception is that the amplitude is constant for all positions in a given waveThis is true in travelling waves, as the wave propagates through the entire

medium, but NOT TRUE for standing wavesIn 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 :

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/Satellite_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














Lo2

Education

Transcript of Lo2