We want to find the expression for a plane that is perpendicular to
k
⃗
k
⃗
,
where
k
⃗
k
⃗
is a vector in the direction of propagation of the wave.The plane is the
set of points that has the same projection onto the vector
k
⃗
k
⃗
That is any point
r
⃗
r
⃗
that satisfies
k
⃗
⋅
r
⃗
=
c
o
n
s
t
a
n
t
k
⃗
⋅
r
⃗
=
c
o
n
s
t
a
n
t
is a point on the planeNow consider the function
ψ
(
r
⃗
)
=
A
e
i
k
⃗
⋅
r
⃗
ψ
(
r
⃗
)
=
A
e
i
k
⃗
⋅
r
⃗
we see that the magnitude of
ψ
(
r
⃗
)
ψ
(
r
⃗
)
is the same over every plane that is defined by
k
⃗
⋅
r
⃗
=
c
o
n
s
t
a
n
t
k
⃗
⋅
r
⃗
=
c
o
n
s
t
a
n
t
we want to construct harmonic waves, ie. they should repeat every
wavelength along the direction of propagation so they should satisfy
ψ
(
r
⃗
)
=
ψ
(
r
⃗
+
λ
k
⃗
k
)
ψ
(
r
⃗
)
=
ψ
(
r
⃗
+
λ
k
⃗
k
)
where
λ
λ
is the wavelengththen we must have
A
e
i
k
⃗
⋅
r
⃗
=
A
e
i
k
⃗
⋅
(
r
⃗
+
λ
k
⃗
k
)
=
A
e
i
k
⃗
⋅
r
⃗
e
i
k
⃗
⋅
k
⃗
λ
/
k
=
A
e
i
k
⃗
⋅
r
⃗
e
i
k
λ
A
e
i
k
⃗
⋅
r
⃗
=
A
e
i
k
⃗
⋅
(
r
⃗
+
λ
k
⃗
k
)
=
A
e
i
k
⃗
⋅
r
⃗
e
i
k
⃗
⋅
k
⃗
λ
/
k
=
A
e
i
k
⃗
⋅
r
⃗
e
i
k
λ
This is true if
e
i
λ
k
=
1
=
e
i
2
π
e
i
λ
k
=
1
=
e
i
2
π
or
λ
k
=
2
π
λ
k
=
2
π
k
=
2
π
λ
k
=
2
π
λ
This should have a familiar look to it! Finally we want these waves to
propagate in time so you should be able to guess the answer from our work on
mechanical waves
ψ
(
r
)
=
A
e
i
(
k
⃗
⋅
r
⃗
∓
ω
t
)
ψ
(
r
)
=
A
e
i
(
k
⃗
⋅
r
⃗
∓
ω
t
)
