Grass waves
Presented by Bulanchuk Pavel
Build a theory of grass waves excitation
Structure
• Features
• Models:
А) Independent cones
B) Collisions
C) 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 →
• Problem:
disturbances propagate
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 model
2) Stable wind
3) Grass surface is described by
y  y (n, t ),
t R nZ
Thick grass: theory
• Linear approximation
yn  
2
 yn  yn1   
Wind
elasticity
2
yn   yn (*)
fading
yn   2  yn  yn1    2 yn   yn
Thick grass: theory
Separate for sine waves
Solve individually
Sum solutions
Thick grass: theory
i ( knx t )
• ye
• Dispersion law:
yn  

2
 yn  yn1   
2
yn   yn
1
 (k ) 
   2  4 2  4 2 1  eik x 
2i

x

yn   2  yn  yn1    2 yn   yn
Thick grass: theory
• Disturbance will exponentially rise if
 sin(k x)
2
   cos(k x)
2
2
1
 Im( )
Dispersion law
2  2  
2
No waves
k
Conclusion
• Slow and fast waves
• Self-oscillations in constant wind
• Critical wind velocity
Thank you for attention!
Experiment
Artificial
Real
 2  100 m / с2
 2  100 m / с2
 2  300 с 2
 2  100с 2
2  8c
2  4c
1
кр  4 m
1
кр  10 m
ytt   yx   y   yt
Thick grass: analysis
• Phase speed:
1 
Vф 

2k 

2
 4

2 2

 16 k    4 

4
2
2
2
1
2
• Group speed:
Vгр 
4 2k 4
16k 2 4  (4  2   2 ) 2 4  2   2  16k 2 4  (4  2   2 ) 2
 Im( )
Dispersion law
k
 Re( )
Dispersion law
k