To approach this issue, we deal with superposition of two waves that have
different phases and different amplitudes. To generalize the problem, let ψ 1 and
ψ 2 be two cosine functions described as
ψ 1 ¼ a 1 cos b 1 and ψ 2 ¼ a 2 cos b 2 :
ð8:180Þ
Their addition is expressed as
ψ ¼ ψ 1 þ ψ 2 ¼ a 1 cos b 1 þ a 2 cos b 2 :
ð8:181Þ
Here we wish to unify (8.181) as a single cosine (or sine) function. To this end, we
modify a description of ψ 2 such that
ψ 2 ¼ a 2 cos b 1 þ b 2 À b 1
ð
Þ
½
¼ a 2 cos b 1 cos b 2 À b 1
ð
ÞÀsin b 1 sin b 2 À b 1
ð
Þ
½
¼ a 2 cos b 1 cos b 1 À b 2
ð
Þþsin b 1 sin b 1 À b 2
ð
Þ
½
:
ð8:182Þ
Then, the addition is described by
ψ ¼ ψ 1 þ ψ 2
¼ a 1 þ a 2 cos b 1 À b 2
ð
Þ
½
cos b 1 þ a 2 sin b 1 À b 2
ð
Þsin b 1 :
ð8:183Þ
Putting R such that
R ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a 1 þ a 2 cos b 1 À b 2
ð
Þ 2 þ a 2
2 sin 2 b 1 À b 2
ð
Þ
½
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
a 1
2 þ a 2
2 þ 2a 1 a 2 cos b 1 À b 2
ð
Þ
p
,
ð8:184Þ
we get
ψ ¼ R cos b 1 À θ
ð
Þ,
ð8:185Þ
where θ is expressed by
tan θ ¼
a 2 sin b 1 À b 2
ð
Þ
a 1 þ a 2 cos b 1 À b 2
ð
Þ
:
ð8:186Þ
Figure 8.14 represents a geometrical diagram in relation to the superposition of two
waves having different amplitudes (a 1 and a 2 ) and different phases (b 1 and b 2 ) [5].
To apply (8.185) to the superposition of two electromagnetic waves that are
propagating forward and backward to collide head-on with each other, we change
the variables such that
332
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media
different phases and different amplitudes. To generalize the problem, let ψ 1 and
ψ 2 be two cosine functions described as
ψ 1 ¼ a 1 cos b 1 and ψ 2 ¼ a 2 cos b 2 :
ð8:180Þ
Their addition is expressed as
ψ ¼ ψ 1 þ ψ 2 ¼ a 1 cos b 1 þ a 2 cos b 2 :
ð8:181Þ
Here we wish to unify (8.181) as a single cosine (or sine) function. To this end, we
modify a description of ψ 2 such that
ψ 2 ¼ a 2 cos b 1 þ b 2 À b 1
ð
Þ
½
¼ a 2 cos b 1 cos b 2 À b 1
ð
ÞÀsin b 1 sin b 2 À b 1
ð
Þ
½
¼ a 2 cos b 1 cos b 1 À b 2
ð
Þþsin b 1 sin b 1 À b 2
ð
Þ
½
:
ð8:182Þ
Then, the addition is described by
ψ ¼ ψ 1 þ ψ 2
¼ a 1 þ a 2 cos b 1 À b 2
ð
Þ
½
cos b 1 þ a 2 sin b 1 À b 2
ð
Þsin b 1 :
ð8:183Þ
Putting R such that
R ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a 1 þ a 2 cos b 1 À b 2
ð
Þ 2 þ a 2
2 sin 2 b 1 À b 2
ð
Þ
½
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
a 1
2 þ a 2
2 þ 2a 1 a 2 cos b 1 À b 2
ð
Þ
p
,
ð8:184Þ
we get
ψ ¼ R cos b 1 À θ
ð
Þ,
ð8:185Þ
where θ is expressed by
tan θ ¼
a 2 sin b 1 À b 2
ð
Þ
a 1 þ a 2 cos b 1 À b 2
ð
Þ
:
ð8:186Þ
Figure 8.14 represents a geometrical diagram in relation to the superposition of two
waves having different amplitudes (a 1 and a 2 ) and different phases (b 1 and b 2 ) [5].
To apply (8.185) to the superposition of two electromagnetic waves that are
propagating forward and backward to collide head-on with each other, we change
the variables such that
332
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media