ELASTIC WAVE THEORY 

 Mechanical Properties 



From the theory- of elastic waves in an isotropic medium, the longitudinal 

 plate-wave velocity is given by the relation 



Fp={£/[pi(l-<T==)])i/== 

 where 



E = Young's modulus 



Pi = density of medium (ice) 



o- = Poisson's ratio 



The shear wave velocity. Van = (j^/pi)^^^) in which /a= E/[2(1 + o')] where fi = 

 shear modulus. 



If the medium is isotropic and the above two wave velocities can be detected, 

 then the elastic properties of the medium can be described by any two elastic con- 

 stants of the above equations. It should be noted that Vp is a function of two elastic 

 constants whereas Vgh is a function of only one elastic constant. In addition, both 

 of these wave velocities are a function of density. 



The longitudinal wave velocity for rods, in which the diameter of the rod is 

 small in comparison to its length, is given by the relation® 



Vr=(E/piy/^ 



From the plate velocity and the rod velocity equations, the following relation can 

 be written: 



Vp = Vr/(l - <t2)1/2 



which gives the plate-wave velocity in terms of the rod velocity. 



The theory of flexural waves^ in a floating ice sheet over deep water gives the 

 relation for the phase velocity C as 



^, _ {l/iWy^Vp" + {[(p„/pi)^A]/(4^^)} 



1 + {pjpi) [(1 - C^)/F»2] -1/2 (2^y)-i 



where 



y = Hf/C 

 H = thickness of ice sheet 



/ = frequency of flexural wave 



C = phase velocity 

 p„ = density of underlying water 



Pi =: density of ice 



A = wavelength in ice 

 Fw = velocity of longitudinal wave in water 

 Vp = velocity of longitudinal plate wave in ice 



g = acceleration of gravity 



Flexural waves are usually generated by an explosive shot being placed in the 

 ice or in the air above the ice. The frequency of these waves changes with the 

 varying velocities of the shots. However, if the explosive shots are set ofif in the 

 air or on the surface of the ice, the frequency remains constant. Theory'' shows 

 these constant-frequency waves build up to a maximum amplitude, and then fall 

 oflF rapidly in amplitude. The point of maximum amplitude represents the passage 



