94 Prof. C. G. Knott on Reflexion and Refraction of 



detailed numerical calculation of the ratios of the A and B 

 quantities in any particular case, the energy equation supplies 

 an important criterion of the accuracy of the work. 



The equations (1) give the relations among the quantities 

 c c' y y for any assumed values of the densities and elastic 

 constants. Hence for any chosen value of c, that is for any 

 chosen angle of incidence, the corresponding- values of c', y, 

 y' are readily calculated ; and the numerical values of the 

 coefficients in equations (2) can be filled in. We are thus 

 left with four simple numerical linear equations from which 

 any four of the live quantities can be determined in terms of 

 the fifth*. B and B' follow at once; and finally, by calcu- 

 lating the terms in the energy equation (3) and dividing 

 throughout by cpB 2 , we obtain numbers showing the partition 

 of energy among the reflected and refracted waves. 



When any one of the quantities d, y, y', becomes zero or 

 imaginary, there is no wave of that type. In such cases the 

 A and B quantities may work out in the form 



and we must then take the expression p 2 + g' 2 as the number 

 on which the energy depends. 



II. Distortional Wave at the Interface of an Elastic Solid 

 and a Fluid. 



There is no distortional wave in the second medium. The 

 term in B' has therefore no existence ; and we have only 

 three surface conditions. Obviously the second condition in 

 the general problem is the one that must be dropped ; while 

 in the fourth condition the right-hand side becomes zero. 



We may work out the solution ab initio, or we may get the 

 necessary equations by putting B' = 0, n' = 0, and c' 2 =x> in 

 the first;, third, and fourth of equations (1). 



Also, c being infinite, 



n f) d + 1 p v ' 



* I have found it both quicker and more accurate to till in the 

 numerical values in the equations as they stand, and then solve the 

 equations for every individual case, than to write out the several algebraic 

 expressions for each ratio and then substitute. Except when n and n' 

 are equal, or when either vanishes, the expressions are unwieldy. With 

 a table of squares and square roots and with Crelle's RecltentAtfeln at 

 hand, the four equations with numerical coefficients can be worked out 

 ■with great ease. 



