lo Lamb, Waves in a Medium with Periodic Structure, 



The intensity (/) of the transmitted waves, that of 

 the incident waves being taken as unity, is given by the 

 square of the modulus of this expression, and is therefore 



j___ sin^ Q sin^ ka . . 



"~ sin^ ^iv^ka cos'' (;^ + i)f^ + (i - cos Q cos kaf sin^ (« + i)^ ^^ ' 



Again, from (50), (51), (52) we find 



- 2/sin - (0 + ka) sin -(0 - ka) sin {n + i)9 



A = ? ? Jgg-inTza 



sin Q sin ka 



_ /(cos - cos koi) sin (« + i)0 e^'^ . . 



sin sin ka cos (^^ + i)^ + /(i - cos 6* cos ka) sin(;^ + 1)^ \^^i' 



The intensity (/') of the reflected waves is therefore 

 given by 



(cos0-cos^«)2sin2(;2 + 1)0 

 ~ sin^ Q sin^ ^« cos^ (« + 1)6/ + (i - cos 6^ cos kaf sin^ (« + i)0 ^^ '' 



It is easily verified that 



^+^'=1 • • . (57), 

 as should obviously be the case, since there is no dissipa- 

 tion of energy in our medium. 



When the number (/2+1) of particles in the loaded 

 portion of the string is considerable, a very slight variation 

 in the value of ka (and consequently of 0) will cause 

 great fluctuations in the values of cos (;2 4- i)0 and 

 sin(;^-|-i)0, and thence in the values of / and /'. The 

 mean value (/) of the former expression, for wave-lengths 

 in the neighbourhood of 27r//^, is easily calculated* from 

 the formula 



5 / d^ i_ 



2 



Thus 



/= 



Try a2cos2 + /32sin20-a/3 



o 



sin sin ka 



I - cos Q cos ka 

 Cf. KirchhofF, Optik, p. 165. 



(59), 



