1

Mathematical justification that the equation “Ԧ𝐤 ⋅ Ԧ𝐫 = constant”

describes a plane perpendicular to Ԧ𝐤

By setting

(1)

eqn (1) can be expressed as

(based on Hecht, Optics)

Derivations related to 3D plane waves:

2

Further discussion of “Ԧ𝐤 ⋅ Ԧ𝐫 → Ԧ𝐤 ⋅ Ԧ𝐫 − 𝜔𝑡” for a propagating plane wave.

(a) (b)

Suppose the 𝑡 = 0 profile of a plane wave is described by

𝜓 = 𝐴sin(𝑘𝑥)

Then, the 𝑥 and 𝑡 dependence of the traveling plane wave can be written as

(1)

𝜓 = 𝐴sin(𝑘𝑥 − 𝜔𝑡) (2)

For the waveform described by eqn (2), the wave disturbance at an arbitrary point in space, defined by the vector Ԧ𝐫 in Figure (a), is the same as for the point 𝑥 along the 𝑥-axis, where 𝑥 = 𝑟cos𝜃. Eqn (1) may then be written as

𝜓 = 𝐴sin(𝑘𝑟cos𝜃) (3)

3

Eqn (2) can be generalized if the propagation number 𝑘 = 2𝜋/𝜆 is now considered to be a vector quantity, pointing in the direction of propagation.

“Ԧ𝐤 ⋅ Ԧ𝐫 → Ԧ𝐤 ⋅ Ԧ𝐫 − 𝜔𝑡" (con’t)

Then 𝑘𝑟cos𝜃 = Ԧ𝐤 ⋅ Ԧ𝐫 and the harmonic wave of eqn (2) can be written as

𝜓 = 𝐴sin(Ԧ𝐤 ⋅ Ԧ𝐫 − 𝜔𝑡) (4)

Eqn (4) can represent plane waves propagating in any arbitrary

direction given by Ԧ𝐤, as shown in Figure (b).

In the general case,

Ԧ𝐤 ⋅ Ԧ𝐫 = 𝑘𝑥𝑥 + 𝑘𝑦𝑦 + 𝑘𝑧𝑧

where (𝑘𝑥, 𝑘𝑦, 𝑘𝑧) are the components of the propagation direction

and (𝑥, 𝑦, 𝑧) are the components of the point in space where 𝜓 is evaluated.

(5)

4

Phase Velocity of a Plane Wave (based on Hecht, Optics)

From the figure, the scalar component of Ԧ𝐫 in the direction

of Ԧ𝐤 is 𝑟𝑘.

Hence, the terms in the exponents in the last two terms are equal:

=