OCR Text |
Show is in approximate accord with the existing observations of avalanche movement. We are dealing with a secondary effect in the consideration of this internal friction or viscosity, which can be neglected for wet snow or powder snow avalanches. Taking into account the buoyancy of air of density y, = 1.25 kg/ m , for a portion of the sliding snow layer of which the horizontal projection is equal to the unit of surface, the equation of motion is: Force = Mass x Acceleration ( Y - YL ) h ' s i n t - ( Y - YL ) h ' C° S ^ - Y/ g v2 = Y h'/ g dv/ dt or dv/ dt = ( g / Y h ' ) [ h* ( y ^^ j ( sin^ _. ^ cos^) . ( y / £ ) v 2 ] The solution of this differential equation, with k = Ph/ g and tanh = hyperbolic tangent, is ( 3) v = v tanh [( v A) ( t) ] max max in which the maximum velocity is: ( 4) v = [£ h' Cl - Yr / Y) ( s i n * - HVC O S ^ ) ] 1 / 2 max L< In most cases Y L Y and f- can be neglected; one can then write: "•> v2 m a x z , h. Si„ + If the snow layer at the beginning of the distance traversed ( s = 0) already has an initial velocity v0 , then the proper time, tQ , is determined from Equation ( 3) and for the study of the subsequent process the time t + t is inserted in the place of t. The distance traversed, s, is obtained from the integration of Equation ( 3): ( 5) s = k in cosh [( vmax A ) ( t) ] B^? bS!! S = cosine For the initial interval t* to reach 80% of the maximum velocity, vmax it is true that tanh( t* vmax/ k) = 0.8 and by the substitution of the value t* determined in this way into Equation ( 5): ( 6) s* " Z 0.5 k That is, for £ = 500 m/ sec2 , one obtains s* = 25 h*. 23 |