Plane-Wave Summary
A two-dimensional plane wave may be expressed as
f(x, y, t) = Re A ei(kx+l y−νt)=Re A eiθ (1)
•x, y and tare independent variables (space and time).
•kand lare the xand ywavenumbers (units: m−1).
•Ais the wave amplitude.
•θ=kx +ly −νt is the wave phase angle.
•The wave propagates normal to lines of constant phase angle.
At any instant in time [tfixed; (x, y) varies]:
θ = constant
x
y
wave propagation
∆
θ
θ = 0
θ = π
θ = 2π
θ = 3π
Plot of θ as a function of (x,y) for fixed t.
x
y
Plot of Re{exp(i θ)} as a function of (x,y) for fixed t.
L
L
HH
HH
1
-1
-1
1
X
λ
•θ=kx +ly +C;θis a linear function of space.
•θis constant on lines of kx +ly.
•eiθ =ei(θ+ 2πn), where nis an integer, are lines of constant phase (e.g. highs and lows).
•~
K=∇θ=ˆ
ik +ˆ
jl is the wave vector;K=|~
K|is the wavenumber.
•λ=2π
Kis the wavelength: the distance between lines of constant phase.
