Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16



      

  

  

x q

 



, ,

     

   

  



 





    

K x



n

 

 





i

in

d dx 

e

n 

x



 

  

x q





0



  

 

, ,

Finally, we derive the expression for the function’s transformation n k  

through the transformations of its jump and the

jump of its normal derivative





  





  







n

n

n  





 



P

P

sin

cos

cos





k

n

k

n k   

(10)

 k  

 2

2

t

1/ 2

D ERIVING THE FINAL FORMULA

T

he inverse Legendre’s transformation is applied to (10)



k n  

    

  cos n 





P

(11)



n k 

n

kn k

 







k n 

k

1/ 2

!





where





kn

k n 

!

Then the inverse Kantorovich-Lebedev transformation is applied to the obtained expression

0  

,    

  

K x

x q

      i    





sh

d

 



n

n

x

Bearing in mind that the expressions for the transformations of the wave potential jumps and its normal derivative have the following form



      

    

  



,

 

     

n

   





q

  

0  

K





n  n

  



i



d



 



  

  



n   

,

 

q

in the Fourier’s transformation domain, the wave potential has the following form

  

  K x K

  

0 0    



,    

  

sh



x q

k n  





n





i

i









P

sin

cos

 

n

kn k

 k  



 

 

2



x

2



1/ 2

   

   

  

  

  

  















n

n

n 









 









P

P

d d

cos

,

cos

,

k

k

n

q

q

The integral in the last formula is known [ 2.16.52(11), 20]

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