Two hours
04 June 2014
09:45 – 11:45
Answer ALL six questions.
Electronic calculators are permitted, provided they cannot store text.
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1. You are given that a harmonic signal (in the usual notation)
y(x, t) = A exp i(kx − ωt) ,
satisfies the equation
= 0.
Here a and b are positive constants, x is a spatial coordinate and t is time.
Obtain the dispersion relation and sketch the graph of c2 as a function of k, where c is the wave
speed. Obtain the group velocity cg and show that there is a critical wavenumber k = kc at which
(cg /c − 1) changes sign. Find the value kc in terms of a and b.
[8 marks]
2. A plane wave signal, f , takes the form:
f (x, t) = A exp i Kx − ω(K)t ,
where A is a constant amplitude, K > 0 is the wavenumber, ω(K) is the frequency, x is a spatial
coordinate and t is time.
Consider two such plane waves of equal amplitude, but with slightly different wavenumbers K =
k + ∆ and K = k − ∆ , where ∆ is small.
(i) Show that the sum of these two waves can be viewed as a slow amplitude modulation of a
plane wave of wavenumber k. Define the ‘envelope’ and ‘carrier’ waves in the resulting sum.
(ii) Give the speed of propagation of the envelope of this amplitude modulated wave.
(iii) Sketch the (real part of the) signal at a fixed instant in time, indicating the direction of
propagation for ω > 0.
[8 marks]
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3. Two-dimensional surface-wave motions are taking place on a semi-infinite layer (z > 0) of incompressible fluid. When surface tension T is introduced, the appropriate linearised conditions at the
free surface (z = 0) are
and − ρ
= −ρgη + T 2 .
Here ρ is the constant fluid density, g is the local gravitational acceleration, φ(x, z, t) is the velocity
potential and z = η describes the free surface displacement during the motion.
(i) Show that the free surface condition, including the effects of surface tension, is
∂φ T ∂ 3 φ
ρ ∂ 2 x∂z
on z = 0 .
(ii) State the equation satisfied by φ(x, z, t) in the fluid region 0 < z < ∞.
(iii) A travelling surface wave exists on this fluid layer, and has a velocity potential of the form
φ(x, z, t) = Φ(z) cos(Kx − ωt) ,
where K and ω are positive constants. Determine Φ(z) such that Φ decays as z → ∞. Obtain the
dispersion relation for such a wave when surface tension effects are included.
(iv) If such a wave has a phase speed of U, obtain the possible wavelength(s) of the wave as a
function of g, T , ρ and U.
[12 marks]
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A schematic diagram for question 4.
4. Two monopoles (A, B), of equal and opposite sign, are separated by a distance 2l and placed in a
compressible fluid of sound speed c, as shown in figure 1. A solution of the three-dimensional wave
equation exists in the form of a velocity potential:
φ(r, θ, t) =
m(t − r1 /c) −
m(t − r2 /c) ,
for a given function m, where θ, r, r1 and r2 are as shown in the figure.
By considering the limit of l/r → 0, where 2l m(t) → d(t), give a simplified expression for φ in
terms of d(t).
Hint: The cosine rule for a triangle of side lengths a, b, c, gives
c2 = a2 + b2 − 2ab cos α ,
where α is the angle between the sides of length a and b.
[12 marks]
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5. A compressible fluid of sound speed c fills a pipe with rigid impermeable walls.
(i) State the equation to be satisfied in the fluid and the boundary condition to be imposed at
the walls for a velocity potential φ. Hence verify that, for a pipe with a constant cross section, a
velocity potential solution exists in the form
φ(x, t) = B exp iω(t − x/c) ,
for any constant B, where ω is a frequency, t is time and x is measured along the axis of the pipe.
(ii) The pressure perturbation associated with this velocity potential is
for a gas of mean density ρ0 . Give a corresponding expression (in terms of the velocity potential) for
the downstream volume flux Q = Au, where u is the velocity component along the pipe and A the
cross sectional area of the pipe.
(iii) Now consider a pipe that has a piecewise constant cross sectional area, being A1 for x ≤ 0,
A2 for x > 0, with a velocity potential of the form
p = −ρ0
φ(x, t) = φI (x, t) + φR (x, t) ,
for x ≤ 0 and
φ(x, t) = φT (x, t) ,
for x > 0. For an incoming wave φI (x, t) = BI exp{iω(t − x/c)}, give similar forms for φT and φR .
Hence determine the amplitudes of φT and φR relative to φI , given that p and Q must be continuous
at x = 0.
(iv) Interpret your results in the special cases A2 ≪ A1 and A2 ≈ A1 .
[20 marks]
6. An incompressible, inviscid fluid layer of depth h is in (linear) irrotational motion. The fluid
occupies the space x ≥ 0, and 0 < z < h, with z measured downwards into the fluid. The free
surface has a mean position z = 0 and the linearised condition to be imposed there is
where φ is the velocity potential, g is the local gravitational acceleration and t denotes time.
(i) If z = h is a rigid, impermeable base, state the appropriate boundary condition to be applied
there. State the governing equation to be satisfied in the fluid.
(ii) By looking for a solution in the form φ(x, z, t) = Z(z) cos(Kx − ωt) where K, ω > 0, obtain
the appropriate solution for Z(z). Give the corresponding dispersion relation. How many possible
wavelengths can satisfy this dispersion relation for a fixed wave frequency ω?
(iii) By seeking a more general separable form of φ(x, z, t) = X(x)Z(z) cos(ωt), find alternative
solutions to X(x) and Z(z) such that φ remains bounded in the fluid. Give the corresponding
dispersion relation. How many possible wavelengths can satisfy this dispersion relation for a fixed
wave frequency ω?
(iv) Hence give the most general form of solution as a linear combination.
[20 marks]
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