hich fo h E 2j sin k,h The scattered field has the form of (5.7)but must be odd. Thus A+=-A-and the total field for y h E2( 2JA(kx, a)sin kyy -ou lo()2j sin k, Setting E, =0 at z=d and solving for A+ we find that the total field for this case is +j△ , (. y, 0)=2/[ jk ly-hl 2 i sin k、he jkr sin k, d Adding the fields for the two cases we find that +j△ oulo(o) -jkyIy-hl y dk (5.8) which is a superposition of impressed and scattered fields 5.2 Solenoidal-lamellar decomposition We now discuss the decomposition of a general vector field into a lamellar comp aving zero curl and a solenoidal component having zero divergence. This is known as a Helmholtz decomposition. If V is any vector field then we wish to write where Vs and Vi are the solenoidal and lamellar components of v. Formulas expressing these components in terms of V are obtained as follows. We first write Vs in terms of a vector potential”Aas v=V×A. This is possible by virtue of(B 49). Similarly, we write VI in terms of a"scalar potential v=Vφ ②2001which for y > h is E˜ i z(x, y,ω) = −ωµ˜ ˜I0(ω) 2 2π ∞+  j −∞+ j 2 j sin kyh 2ky e− jky y e− jkx x dkx . The scattered field has the form of (5.7) but must be odd. Thus A+ = −A− and the total field for y > h is E˜ z(x, y,ω) = 1 2π ∞+  j −∞+ j  2 j A+(kx ,ω)sin ky y − ωµ˜ ˜I0(ω) 2 2 j sin kyh 2ky e− jky y  e− jkx x dkx . Setting E˜ z = 0 at z = d and solving for A+ we find that the total field for this case is E˜ z(x, y,ω) = −ωµ˜ ˜I0(ω) 2 2π ∞+  j −∞+ j  e− jky |y−h| − e− jky |y+h| 2ky − − 2 j sin kyh 2ky e− jkyd sin kyd sin ky y  e− jkx x dkx . Adding the fields for the two cases we find that E˜ z(x, y,ω) = −ωµ˜ ˜I0(ω) 2π ∞+  j −∞+ j e− jky |y−h| 2ky e− jkx x dkx + +ωµ˜ ˜I0(ω) 2π ∞+  j −∞+ j  cos kyh cos ky y cos kyd + j sin kyh sin ky y sin kyd  e− jkyd 2ky e− jkx x dkx , (5.8) which is a superposition of impressed and scattered fields. 5.2 Solenoidal–lamellar decomposition We nowdiscuss the decomposition of a general vector field into a lamellar component having zero curl and a solenoidal component having zero divergence. This is known as a Helmholtz decomposition. If V is any vector field then we wish to write V = Vs + Vl, (5.9) where Vs and Vl are the solenoidal and lamellar components of V. Formulas expressing these components in terms of V are obtained as follows. We first write Vs in terms of a “vector potential” A as Vs =∇× A. (5.10) This is possible by virtue of (B.49). Similarly, we write Vl in terms of a “scalar potential” φ as Vl = ∇φ. (5.11)