Waves and patterns Science & Computers PHY307/PHY607

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Waves and patterns Science & Computers PHY307/PHY607 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.)

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## 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 visual.graph import *

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

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