Class exercises
Handout 7. Atmospheric Oscillations
Section 7.2 Representation and Properties of Waves
7.1 Consider the wave functions
θ1 = a cos(kx −ν t ) and
θ 2 = a cos(kx + ν t )
where both k and ν are larger than 0.
a.
Show that θ1 represents a wave moving to the right (in the positive xdirection) and θ2 a wave moving to the left (in the negative x-direction).
b. Give a formula for the speed of the wave propagation in these cases.
An additional angle α in the expression θ 2 = a cos(kx + ν t + α ) is called a phase
shift.
c.
Show, with a drawing, that a phase shift does not affect the wavelength λ,
the period τ, or the propagation direction of the wave.
7.2 The phase velocity for light waves is equal to 300 000 km/s. The periods for
1
1
FM-band broadcasting range from
⋅10 −6 s to
⋅10−6 s. What are the
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108
wavelengths of such waves?
Section 7.3 The Wave Equation
7.3 Verify by direct substitution that
a.
θ ( x, t ) = θ 0 cos(kx − ν t ) , and
b. θ (x, t ) = θ 0 e i (kx −ν t )
are solutions of the 1-dimensional wave equation (7.7).
Section 7.6 Linearization of the Basic Set of Equations
7.4 The equation of continuity is given by Holton’s equation (2.31) and can be
written in full as:
∂ρ
+ U ⋅ ∇ρ + ρ ∇ ⋅ U = 0 .
∂t
a.
b.
Use the linearization assumptions (7.22) and rewrite the continuity
equation in terms of the basic state and perturbed state quantities.
Next, remove all terms containing products of perturbation quantities and
also remove the basic state from the equation in a).
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c.
If we finally assume that the basic state is barotropic, i.e. ρ = ρ ( z ) , and
consists of a constant zonal flow, then derive equation (7.30) from the
result in b).
Section 7.7 Applications
7.5 For the propagation of sound waves we use the set of equations (7.35)-(7.37).
a.
Derive (7.38) starting from (7.37).
b. What does the ± sign in equation (7.39) means in this case?
c.
What would you hear if sound waves had a group velocity cg≠c?
7.6 For the derivation of the sound waves in the previous exercise and in section
= 0. However, usually this is not
7.7.1 we assumed an atmosphere at rest: the case. Therefore we now consider an atmosphere with a constant zonal
velocity = (̅ = 0).
a.
For this situation derive alternative expressions for the equations (7.35),
(7.36) and (7.37) for an atmosphere with a constant zonal velocity .
b. Show that the momentum equation is equivalent to Holton Eq. (5.20).
c.
Combine the continuity equation and the energy equation in order to
remove the fluctuating density from the set of equations.
d. Show that the result of equals Holton eq. (5.21).
e.
From Holton Eqs. (5.20) and (5.21) derive Holton Eq. (5.22) in the manner
indicated by Holton.
7.7 We assume that the pressure perturbation for a one-dimensional acoustic wave
is given by
p = Ae ik ( x −ct )
We use (7.35) to find the corresponding solution of the zonal wind perturbation
(u) by assuming that
u = Be ik ( x −ct )
a.
b.
A
ρc
Use (7.39) and the equation of state (7.32) for the basic state to derive
Show that for B we can derive: B =
u= p
c
γp
where
γ =
cp
cv
Section 7.8 Rossby Waves
7.8 For a mean zonal flow of 30 m/s, at what wavelength will a Rossby wave be
stationary? Use β for 45˚N.
7.9 a.
Show that equation (7.58) can be rearranged into equation (7.59) on a
β-plane.
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b.
c.
Use the perturbation method on equation (7.59) and all assumptions
( u ≠ 0 , v = w = 0 , H = const ) to derive equation (7.60)
Derive equation (7.62) from equation (7.60) with the geostrophic
approximation (7.61).
7.10 Short Rossby waves are characterized by k >> f 0 / gH . For H we take the
scale height of the atmosphere: H = Rd T / g , where we choose T = 273 K.
a.
Calculate the wave number k and the corresponding wavelength for a
latitude of 45˚N for which k = f 0 / gH .
b. Use equation (7.65) and derive an expression for the group velocity cg of
Rossby waves in the x-direction.
c.
To make the analysis simpler make the short-wave approximation. Show
that for the group velocity we now find:
cg = u +
d.
e.
β
k2
Show that Rossby waves show anomalous dispersion.
If we assume that > 0 show that:
- the group velocity increases as wavelength increases, and
- the group velocity is always eastward, and
- the phase speed is eastward for short waves but westward for long
waves.
Answers
7.2
λ = 3.41 m and λ = 2.78 m, respectively
7.8
λ = 8553 km
7.10 a. k = 3.68 × 10-7 m-1 and λ = 17.08 × 106 m
b.
,
= − +
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+ 2 +
!
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