2 The Ether and the Wave Equation
9
Harmonic waves are exceedingly important. They are special solutions of the
wave equation.
s(x, t) = A · sin(k x − ω t)
and
s(x, t) = A · sin(k x + ω t) .
⎫
⎬
⎭
Harmonic waves (4)
For a constant time t, the elastic deflection s of the mass particle describes a sine
function in space with the wavelength λ = 2π/k. The quantity k is the wave number
(respectively wave vector for the case of spatial propagation). At a fixed point x, the
masses oscillate according to a sine function of time with an oscillation period T =
1/ν = 2π/ω. Frequency multiplied by 2π is called the angular velocity ω = 2π ν
of an oscillation. One can register these as pure notes in the field of acoustics (even
though they sound terrible).
This function (4) is only a solution of the wave equation if the above condition
s(x, t) = f (x − c t) is fulfilled,
s(x, t) = f (x − c t) = A · sin
k
x −
ω
k
t
= A sin[k (x − c t)] .
The wavelength λ, frequency ν and velocity of propagation c of a wave are connected
by the relation c = λ ν = ω/k and or ω = c k. However, the propagation velocity c
for waves of arbitrary frequencies does not in any case need to have one and the same
value, as in our example with the elastic rod. The general dependence of a wavelength
on the frequency is called the dispersion relation. We will come back to this in Chap. 4
when discussing the linear chain. If the angular velocity ω is proportional to the wave
number k with the unchangeable constant c for the velocity of propagation, one states
that the medium (or space) is free of dispersion,
c = λ ν =
ω
k
.
Dispersion free medium (5)
The wave’s velocity of propagation c is independent of its frequency. This has been
fulfilled in near approximation for the field of acoustics and is especially valid for
electromagnetic waves in a vacuum. When light travels through a medium, e.g.
light through glass or water, it gets dispersed or broken. This can be observed in a
spectrum of a glass prism, or in drops of water in a rainbow. The front velocity of
an electromagnetic wave in a medium is dependent on its frequency (its colour). In
this case, more complicated equations are at work, but we will not take them into
consideration.
Taking the dispersion relation ω = c k, we can describe the general solution
of the wave equation as a superposition of harmonic waves with a variable wave
number k,
s(x, t) =
k
[A(k) sin(k(x − c t) + B(k) cos(k(x − c t))] . General solution
(6)
9
Harmonic waves are exceedingly important. They are special solutions of the
wave equation.
s(x, t) = A · sin(k x − ω t)
and
s(x, t) = A · sin(k x + ω t) .
⎫
⎬
⎭
Harmonic waves (4)
For a constant time t, the elastic deflection s of the mass particle describes a sine
function in space with the wavelength λ = 2π/k. The quantity k is the wave number
(respectively wave vector for the case of spatial propagation). At a fixed point x, the
masses oscillate according to a sine function of time with an oscillation period T =
1/ν = 2π/ω. Frequency multiplied by 2π is called the angular velocity ω = 2π ν
of an oscillation. One can register these as pure notes in the field of acoustics (even
though they sound terrible).
This function (4) is only a solution of the wave equation if the above condition
s(x, t) = f (x − c t) is fulfilled,
s(x, t) = f (x − c t) = A · sin
k
x −
ω
k
t
= A sin[k (x − c t)] .
The wavelength λ, frequency ν and velocity of propagation c of a wave are connected
by the relation c = λ ν = ω/k and or ω = c k. However, the propagation velocity c
for waves of arbitrary frequencies does not in any case need to have one and the same
value, as in our example with the elastic rod. The general dependence of a wavelength
on the frequency is called the dispersion relation. We will come back to this in Chap. 4
when discussing the linear chain. If the angular velocity ω is proportional to the wave
number k with the unchangeable constant c for the velocity of propagation, one states
that the medium (or space) is free of dispersion,
c = λ ν =
ω
k
.
Dispersion free medium (5)
The wave’s velocity of propagation c is independent of its frequency. This has been
fulfilled in near approximation for the field of acoustics and is especially valid for
electromagnetic waves in a vacuum. When light travels through a medium, e.g.
light through glass or water, it gets dispersed or broken. This can be observed in a
spectrum of a glass prism, or in drops of water in a rainbow. The front velocity of
an electromagnetic wave in a medium is dependent on its frequency (its colour). In
this case, more complicated equations are at work, but we will not take them into
consideration.
Taking the dispersion relation ω = c k, we can describe the general solution
of the wave equation as a superposition of harmonic waves with a variable wave
number k,
s(x, t) =
k
[A(k) sin(k(x − c t) + B(k) cos(k(x − c t))] . General solution
(6)