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Waves and patterns Science & Computers PHY307/PHY607PowerPoint Presentation

Waves and patterns Science & Computers PHY307/PHY607

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### Waves and patternsScience & ComputersPHY307/PHY607

PHY307, Fall 2002

HWK MC #2

- Random walks: the summation of random changes.
- Bacteria moves “forward”, but buffeted randomly by surrounding molecules.
- The DIRECTION accumulates random changes (a random walk on the sphere of directions.)

- Stock prices: Nobel prize winning model (Black-Scholes) for estimating future values of stocks and options.
- The PRICE accumulates random changes (multiplied by a random number.)

PHY307, Fall 2002

from random import *

stockchart = gcurve()

price = 100.0

for week in range(50):

growth = 0.9 + 0.25 * random()

price *= growth

stockchart.plot(pos=(week,price))

print `Final price is`, price

Lab MC #2

- Average displacement grows as the square root of the time (since average of the square of the position grew proportional to time.) Characteristic feature of a simple random walk.
- The density of a diffusion created aggregate decreases as the aggregate grows: fractal structure with a dimension less than 3.

PHY307, Fall 2002

Few coordinate systems

- Projectiles.
- Coordinates might be x, y, and z positions and x, y, and z velocities.

- Planets.
- Similar coordinates.

- Get periodic orbits and chaos.

PHY307, Fall 2002

Extended (many-coordinate) systems

- Can’t be described by a few coordinates
- (Often can for some purposes. For example, Earth is extended, but treated as a point in the solar system.)

- “Many degrees of freedom”
- Can exhibit quite complex phenomena, including periodic waves, pattern formation, chaotic waves, and turbulence.

PHY307, Fall 2002

movie

Why waves?

- Exchange between neighboring positions of
- energy
- material
- whatever else

- Also, convert from one form to another
- Example: Height of water to motion of water.

PHY307, Fall 2002

Oscillation

- Exchange of energy in a few coordinate system as an example (not a wave yet.)
- In a spring, the energy of distortion, which depends on position, is exchanged with the energy of motion, which depends on velocity.
- Python demo.
- Change in position is velocity
- Change in velocity (acceleration) is force/mass, with force = - k * (position – rest_position)

PHY307, Fall 2002

Concepts and Procedures for Simulating Waves

- Approximate continuous space by a grid.
- Need a list of the values of the quantities (positions, velocities, chemical concentrations.) This list gives the values of the quantities at the grid points.
- Need rules for how quantities affect their values at neighboring grid points.

PHY307, Fall 2002

Example

- Demo of slinkie.
- How do we describe the motion and simulate it?
- FOR CODE DEVELOPED DURING LECTURE, SEE Codes page
- http://physics.syr.edu/courses/PHY307/Codes/WaveInClass.py

PHY307, Fall 2002

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