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