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 nZ Thick grass: theory • Linear approximation yn 2 yn yn1 Wind elasticity 2 yn yn (*) fading yn 2 yn yn1 2 yn yn Thick grass: theory Separate for sine waves Solve individually Sum solutions Thick grass: theory i ( knx t ) • ye • Dispersion law: yn 2 yn yn1 2 yn yn 1 2 2 2 ik x (k ) 4 4 1 e 2i x yn 2 yn yn1 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 300с2 2 100с2 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