A sigle wave
Superposition of waves
© The scientific sentence. 2010

Wave packets
1. Superposition of two waves
Supperposing two waves in phase of equal frequency and amplitude just
doubles the resultant amplitude; when they are out of phase, the two combined waves just cancel each other.
Supperposing two waves in phase of equal frequency and different amplitudes just
enlarge the resultant amplitude.
The most interesting is the supperposition of two or more waves of almost the same frequency
that corresponds to the beats phenomenon.
Let's consider the two following wave functions:
ψ_{1}(x,t) = A sin(k_{1}x  ω_{1}t)
ψ_{2}(x,t) = A sin(k_{2}x  ω_{2}t)
Which are waves traveling in the same direction with the same
amplitude A, different wave number k and different angular frequency ω .
We use the principle of superposition to find the resultant wave
ψ(x,t) = ψ_{1}(x,t) + ψ_{2}(x,t).
Then:
ψ(x,t) = A[sin(k_{1}x  ω_{1}t) + sin(k_{2}x  ω_{2}t)]
Using the trigonometric identity:
sin (a + b) + sin (a  b) = 2 sin a cos b;
With: a + b = x and a  b = y ,
we get: a = (x + y )/2 and b = (x  y)/2
Then:
sin (x) + sin (y) = 2 sin [(x + y )/2] cos [(x  y )/2]
It follows that:
ψ(x,t) = 2A sin ([k_{1}x  ω_{1}t + k_{2}x  ω_{2}t]/2)
cos([k_{1}x  ω_{1}t  k_{2}x + ω_{2}t]]/2)
= 2A sin [(1/2)(k_{1} + k_{2})x  (1/2)(ω_{1} + ω_{2})t]
cos[(1/2)(k_{1}  k_{2})x  (1/2)(ω_{1}  ω_{2})t]
Thus:
ψ(x,t) = 2A cos[(Δk/2)x  (Δω/2)t] sin[k_{av}x  ω_{av}t]
(1)
With:
Δk = k_{1}  k_{2}
Δω = ω_{1}  ω_{2}
k_{av} = (k_{1} + k_{2})/2
ω_{av} = (ω_{1} + ω_{2})/2
The factor cos[(Δk/2)x  (Δω/2)t] represents the envelope of the
combined waves. The resultant wave expressed by the factor sin[k_{av}x  ω_{av}t]
oscillates within the envelope with the wave number k_{av} and angular frequency ω_{av}.
Each individual wave moves with its own phase volocity ω_{1}/k_{1} and
ω_{2}/k_{2}. The combined wave moves with the phase velocity ω_{av}/k_{av}.
The envelope which represents the pulse or the wave packet moves at the group velocity Δω/Δk.
2. Superposition of many waves: The wave packet
Combining many waves leads to a pulse or wave packet as for combining two waves. But the
more we add waves the more the distance between the pulses becomes larger; and for a given time
we can see less wavepacket pass through the direction of the propagation. In other words, a
finite number of different monochromatic waves always produce infinite sequence of repeating
pulses (wave groups). As we want to localize the packet, we need to stabilize
the packet, we will then add an infinite number of waves with a particular well chosen amplitude.
The effective idea is to fix a wave number k_{o} and let the envelope takes effect
about this centred wave number k_{o} (or wavelength λ_{o}). The Gaussian
is the appropriate example. It plays the role of having constructive interference over the spacetime
and destructive interference elsewhere.
The wave function of the wavepacket can then be written as:
ψ(x,t) = ∑ ψ_{i}(x,t) = ∑ A sin(k_{i}x  ω_{i}t)
Or, with different amplitudes:
ψ(x,t) = ∑ A_{i} sin(k_{i}x  ω_{i}t) (2)
Wich is called Fourier series.
When dealing with a continuous spectrum, the Fourier series bocome the
Fourier integral:
ψ(x,t) = ∫A(k) sin(kx  ωt) dk (3)
k:  ∞ → + ∞
If Δx is the space where we can localize the packetwave, ψ(x,t)^{2}dx is
the probability of finding the wavepacket within Δx. If Δk is the range around k_{o}
that is set to form a wave packet, and Δx the region between the two consecutive
points where the envelope is zero (Δx = x_{2}  x_{1}); we can write from the relationship (1):
cos[(Δk/2)x_{1}  (Δω/2)t] = 0; and cos[(Δk/2)x_{2}  (Δω/2)t] = 0.
We can write: (Δk/2)x_{1}  (Δω/2)t = π/2,
and (Δk/2)x_{1}  (Δω/2)t = 3π/2.
We have then:
(Δk/2)x_{2}  (Δω/2)t  (Δk/2)x_{1} + (Δω/2)t = π.
That is: (Δk/2)(x_{2}  x_{1}) = π, which is called:
The uncertainty principle:
Δk Δx = 2π (4)
Contrary to the Classical Mechanics that allows both the measure of the position and the momentum
of a particle at a certain same time, Wave Mechanics allows just one, the other is uncertain.
The above relation is set for the combination of two waves. It is not very precise.
It tells us just in order to know precisely the position of the wave packet (or to
localize a particle), then Δx small, we need a large range of wave numbers (Δk large).
With ω = 2π ν, and k = 2π/λ = 2πν/v, and x = vt(ν is the frequency,
v the velocity, and x the displacement of the particle), the above relation becomes
Δω Δt = 2π. This relationship tells us also that to know precily (at a
fixed small time) when the packet is located at a given point, we need a large range of frequencies ω.
It has an important role in Electronis. It shows that the bandwidth must be large of a circuit to send
a signal in a short time (a dialup Internet connection works with a very narrow bandwidth about 50 Kbps
(Kilo bits per second), whereas, a broadband connection can let move a large amount of data at about
some Mbps (Mega bits per second).
3. Matterwave concept
The de Broglie relationship set the waveparticle duality. Further, the description of matter
particle by a wave packet is justifed by the fact they move at the same velocity. Indeed, the
the wavepacket moves at the speed of the envelope called the group velocity v_{g}
equal to Δω/Δk = dω/dk. This derivative is set about the most probable value
of k that is k_{o}.
If a particle (relativistic) has the total energy E, E^{2}
= p^{2}c^{2} + m^{2}c^{4}, its mass m = m(v) = γm with
γ = 1/[1  v^{2}/c^{2}]^{1/2}, its impulsion p = γmv. With
E = m(v)c^{2} = γmc^{2}, we have E^{2}v^{2}/c^{2} = p^{2}c^{2},
then v = pc^{2}/E.
But the derivative of the relationship E^{2} = p^{2}c^{2} + m^{2}c^{4}
gives: 2EdE = 2c^{2}pdp, that is pc^{2}/E = dE/dP.
Using dE/dP = d(hω/2π)/d(hk/2π) = dω/dk. We get then the group velocity is equal to
the spped of the particle.
v_{particle} = pc^{2}/E = dE/dp = dω/dk = v_{group}
Since the phase velocity is v_{ph} = ω/k, then
v_{group} = dω/dk = v_{ph} + k dv_{ph}/dk
If v_{ph} depends on k, v_{ph} = v_{ph}(k) is called dispersion relation. In this
case the medium where the wave propagates is called dispersive such as water. the group velocity is greater thanj
the phase velocity. If v_{ph} does not depends on k, v_{group} = v_{ph}, and
the medium where the wave propagates is called nondispersive such as vacuum for
electromagnetic waves.
©: The scientificsentence.net. 2007.


