Geography Reference
In-Depth Information
(a)
(b)
ct = 0
ct = 0
ct = 2
π
ct =
π
ct = 2
π
ct = 4
π
ct = 6
π
ct = 3
π
Fig. 7.4
Schematic showing propagation of wave groups: (a) group velocity less than phase speed
and (b) group velocity greater than phase speed. Heavy lines show group velocity, and light
lines show phase speed.
where for brevity the Re[ ] notation in (7.4) is omitted, and it is understood that
only the real part of the right-hand side has physical meaning. Rearranging terms
and applying the Euler formula gives
e
i(δkx
−
δνt)
e
−
i(δkx
−
δνt)
e
i(kx
−
νt)
=
+
(7.5)
δνt) e
i(kx
−
νt)
=
2 cos (δkx
−
The disturbance (7.5) is the product of a high-frequency
carrier wave
of wave-
length 2π/k whose phase speed, ν/k, is the average for the two Fourier compo-
nents, and a low-frequency
envelope
of wavelength 2π/δk that travels at the speed
δν/δk. Thus, in the limit as δk
→
0, the horizontal velocity of the envelope, or
group velocity
, is just
c
gx
=
∂ν/∂k
Thus, the wave energy propagates at the group velocity. This result applies gener-
ally to arbitrary wave envelopes provided that the wavelength of the wave group,
2π/δk, is large compared to the wavelength of the dominant component, 2π/k.
7.3
SIMPLE WAVE TYPES
Waves in fluids result from the action of restoring forces on fluid parcels that
have been displaced from their equilibrium positions. The restoring forces may be
due to compressibility, gravity, rotation, or electromagnetic effects. This section
considers the two simplest examples of linear waves in fluids: acoustic waves and
shallow water gravity waves.
