Grass waves

Presented by Bulanchuk Pavel

Build a theory of grass waves excitation

Structure

• Features• Models:

А) Independent cones B) CollisionsC) Thick grass

• Details of “Thick grass”• Demonstration

Features

• 2 types of waves: “quick small waves” and “slow big waves”

• Weak wind → only quick waves

Independent cones(quick waves)

• Waves - a consequence of heterogeneity of air flow (turbulence)

• Disturbances propagate at the speed of wind

Collisions

• Cones collide → disturbances propagate

• Problem: quick dying-out

Thick grass

• The higher the blade above the rest → → The stronger it is bend → → The grass starts to sway →

→ effect of self-oscillations

Thick grass: theory

1) 1D model2) Stable wind3) Grass surface is described by

( , ),y y n t t R n Z

Thick grass: theory

2 21n n n n ny y y y y

Wind elasticity fading

(*)

• Linear approximation

Thick grass: theory

Separate for sine waves

Solve individually

Sum solutions

2 21n n n n ny y y y y

Thick grass: theory

• • Dispersion law:

2 21n n n n ny y y y y ( )i kn x ty e

2 2 21( ) 4 4 1

2ik xk e

i

x

Thick grass: theory

• Disturbance will exponentially rise if

2

2 2

sin( )1

cos( )

k x

k x

2 21n n n n ny y y y y

Dispersion lawIm( )

k

22 2 No waves

Conclusion

• Slow and fast waves• Self-oscillations in constant wind• Critical wind velocity

Thank you for attention!

Experiment

Artificial Real2 2100 /m с 2 2300с

12 8c

2 2100 /m с 2 2100с

12 4c

4кр m 10кр m

Thick grass: analysis

• Phase speed:

• Group speed:

1

222 2 4 2 2 214 16 4

2фV k

k

2 2 2 2

4

4 2 2 2 4 2 2 2

4 2

16 ( 4 ) 4 16 ( 4 )гр

kV

k k

tt x ty y y y

Dispersion lawIm( )

k

Dispersion lawRe( )

k