In ocean sediments it is common to find a series of layers of almost constant 

 properties. This model will be a good representation of these cases. In other places thick 

 layers (approximately one wave length) are found with continuous change of properties. 

 This continuous change in this case can be represented by a large number of constant 

 property layers. Later it will be shown there is good agreement between the gradient model 

 and the multilayered model. 



LINEAR GRADIENT MULTILAYER MODEL 



We developed another method for modehng the change of sediment properties with 

 depth due to increasing compaction and temperature. In this approach changes in sound 

 speed and density are accounted for by using single or multiple Hquid layers where Airy 

 functions can be used to represent the sound energy. This method was first used by Morris 

 (1970) and the use and development of the model has continued (Morris, 1972, 1975). This 

 model is used to explain low values of bottom loss at small grazing angles and low frequency. 

 In this case we will use a somewhat different function x so we can account for a density 

 change in the layer. The general wave equation for the case where there is variation in both 

 sound speed and density (Brekhovskikh, 1960) is 



where 



P=\/pX 



Waveeq. V- X + K^X = 



k2 = (co/v)2 + ^ (d^ p/dzh - 3/4[y (dp/dz)j' 



If K^ can be represented as a linear function of depth then the potential function x can be 

 written as the sum of the Airy functions Ai and Bi. The argument of the Airy functions is 

 defined in terms of the horizontal wave number k, the profile parameters Kq and |3 and the 

 depth z. To add the effect of absorption in the Hquid an imaginary term ia;/8.686 is added 



toK^. 











If 



k2 



= K, 



a^d+iSz) 





then 



X = 



= A • 



' Aj(^) + B 



°Bj(^) 



where 



1 = 



k2. 



-"°' (, 







^ 

 ^o 



,r'^ ^' 



To add a(dB/unit length) attenuation: K„ ^ K„ + i 



8.686 



A multilayered model composed of Hnear K^ and constant K (constant sound speed) 

 layers is shown in Figure 18. We can start at the bottom and work up through the layers 

 using the interface conditions of which the pressure and the vertical component of particle 

 velocity are continuous functions. In this case P, equal to the pressure, isy/px and Q, equal 



50 



