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Part A Fluid Dynamics & Waves Draft date: 9 March 2010 5–
5 Water waves
5.1 Equations and boundary conditions
5.1.1 Setup
In this Section we will analyse so-called Stokes waves, namely small-amplitude waves on the free surface of an inviscid fluid, for example small ripples on a container of water. Consider fluid filling the half-space y < 0 with a free surface at y = 0, gravity acting in the −y-direction. Now suppose that the fluid is disturbed by small-amplitude waves, so that the free surface is displaced to y = η(x, t), as shown schematically in Figure 5.1. We assume that the flow is irrotational and incompressible, so that it may be de- scribed by a velocity potential φ such that u = ∇φ and φ satisfies Laplace’s equation. We will restrict our attention to purely two-dimensional disturbances, so that φ is a function of x, y and t and hence
∇^2 φ = ∂^2 φ ∂x^2
∂^2 φ ∂y^2
5.1.2 Boundary conditions
Far from the free surface, as the depth tends to infinity, we expect the velocity to tend to zero, that is ∇φ → 0 as y → −∞. (5.2)
At the free surface, there are two boundary conditions, and we will treat each separately in detail.
Dynamic boundary condition A force balance on the interface y = η(x, t) implies that the pressure must be continuous there; otherwise there would be a finite force acting on a surface with zero mass, which contradicts Newton’s Second Law. We therefore impose the dynamic boundary condition
p = Patm at y = η, (5.3)
where Patm denotes the atmospheric pressure above the fluid, which we assume to be constant. We can write the boundary condition (5.3) in terms of the velocity potential by using Bernoulli’s Theorem. For unsteady irrotational flow, we recall from Section 1 the equation ∂φ ∂t
|u|^2 + p ρ
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y y = η(x, t)
(i) (ii)
y
x x
Figure 5.1: (i) Fluid at rest in the half-space y < 0. (ii) The fluid following a disturbance that displaces the free upper surface to y = η(x, t).
where the gravitational potential χ = gy for gravity acting in the −y-direction. The integration function F (t) may be chosen arbitrarily by absorbing a suitable function of t into φ. Evaluating (5.4) at the free surface y = η and using (5.3), we find that
∂φ ∂t
|∇φ|^2 + Patm ρ
- gη = F (t) on y = η. (5.5)
It is convenient to choose the arbitrary function F (t) = Patm/ρ to cancel the constant term on the left-hand side of (5.5), and thus we obtain the dynamic boundary condition in the form ∂φ ∂t
|∇φ|^2 gη = 0 at y = η. (5.6)
Kinematic boundary condition We recall that the normal velocity of the fluid is required to be zero at a fixed impermeable wall. The corresponding condition at a moving boundary such as the free surface of a fluid is that the velocity of the fluid normal to the boundary must equal the velocity of the boundary normal to itself. If this were not true, the fluid would either be flowing through the boundary or separating from it, leaving behind a vacuum, neither of which is acceptable. It may be shown that this condition is equivalent to the requirement that material fluid elements on the free surface must remain on the free surface. Hence, if y = η for some particular fluid particle at time t, then y = η for the same particle for all time. It follows that D Dt
(y − η) = 0 when y − η = 0, (5.7)
and, by expanding out the convective derivative, we obtain the kinematic boundary condition in the form
v =
∂η ∂t
∂η ∂x at y = η. (5.8)
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y
x
A c
Figure 5.2: Schematic of a harmonic travelling wave, showing the amplitude A, wave- length λ and wave-speed c.
The far-field condition (5.2) and the free-surface conditions (5.11) imply that f (y) must satisfy the boundary conditions
f (y) → 0 as y → −∞, (5.16) f ′(0) = ωA, −ωf (0) + gA = 0. (5.17)
Without loss of generality, we assume that k is positive, so the solution of (5.15) that satisfies the far-field condition (5.16) is
f (y) = Beky^ (5.18)
for some constant B. The boundary conditions (5.17) at y = 0 thus give us a system of linear equations for the two constants A and B, which may be written in the form ( ω −k g −ω
A
B
The homogeneous linear system (5.19) admits the trivial solution A = B = 0, corresponding to η and φ both being identically zero. A nontrivial solution can only exist if the determinant of the left-hand side is zero, that is if
ω^2 = gk. (5.20)
This equation for the frequency in terms of the wavenumber is called the dispersion relation. The corresponding wave-speed c satisfies
c^2 = g k
Part A Fluid Dynamics & Waves Draft date: 9 March 2010 5–
which depends on the wavenumber k, so that waves with different wavenumbers move at different speeds. Such waves are called dispersive, in contrast with waves on a string or sound waves, for example, which have a constant wave speed. We see from (5.21) that the wave-speed is a decreasing function of the wavenumber, so that longer waves propagate more quickly. In principle, the wave-speed may be arbitrarily large for very long waves. We will see below that this is an artefact of our assumption that the fluid has infinite depth.
5.3 Generalisations
5.3.1 Finite depth
The analysis performed above is easily generalised to describe waves on a fluid of finite depth h. Suppose fluid occupies the region −h < y < η(x, t) between a rigid base at y = −h and a free surface at y = η(x, t). We recall that the normal velocity at the base must be zero, and hence φ must satisfy the boundary condition
∂φ ∂y
= 0 at y = −h. (5.22)
This replaces the far-field condition (5.2); otherwise the problem is identical to that solved in §5.2. We again seek a solution in the form of a harmonic travelling wave, so that
η(x, t) = A cos(kx − ωt − β), φ(x, y, t) = f (y) sin(kx − ωt − β), (5.23)
for some function f (y). By substituting this expression for φ into Laplace’s equation, we again find that f (y) satisfies the differential equation
d^2 f dy^2 − k^2 f = 0 (5.24)
and the boundary conditions
f ′(0) = ωA, −ωf (0) + gA = 0. (5.25)
However, the condition (5.22) on the base now leads to the boundary condition
f ′(−h) = 0. (5.26)
Clearly the general solution of (5.24) is a linear combination of eky^ and e−ky. Al- ternatively, we can write f (y) as a combination of cosh(ky) and sinh(ky). However, the neatest approach is to note that the boundary condition (5.26) is satisfied identically by setting f (y) = B cosh
k(y + h)
Part A Fluid Dynamics & Waves Draft date: 9 March 2010 5–
At the free surface, the kinematic boundary condition (5.8) now reads
∂φ ∂y
∂η ∂t
U +
∂φ ∂x
∂η ∂x
at y = η(x, t). (5.34)
When we linearise, as in §5.1.2, this is simplified to
∂φ ∂y
∂η ∂t
+ U
∂η ∂x at y = 0. (5.35)
Next we turn to the dynamic boundary condition. With the velocity given by (5.32), Bernoulli’s equation (5.4) is modified to
∂φ ∂t
|U i + ∇φ|^2 + p ρ
Setting p equal to the atmospheric pressure Patm at the free surface y = η, we therefore obtain the boundary condition
∂φ ∂t
U +
∂φ ∂x
∂φ ∂y
Patm ρ
- gη = F (t) at y = η(x, t). (5.37)
It is convenient to choose the arbitrary function F (t) to cancel the constant terms on the left-hand side, that is
F (t) =
U 2 +
Patm ρ
Then linearisation of (5.37) leads to the condition
∂φ ∂t
+ U
∂φ ∂x
Again, we can seek travelling-wave solutions of the form (5.23). The modified bound- ary conditions (5.35) and (5.39) imply that f (y) must satisfy
f ′(0) = (ω − U k)A, −(ω − U k)f (0) + gA = 0. (5.40)
The boundary condition (5.33) at the base again implies that f (y) should take the form
f (y) = B cosh
k(y + h)
and substitution into (5.40) leads to the homogeneous linear system
( ω − U k −k sinh(kh) g −(ω − U k) cosh(kh)
A
B
For there to exist nontrivial solutions, ω must satisfy the dispersion
(ω − U k)^2 = gk tanh(kh). (5.43)
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p 1
x
y
a b
n
T t
−T t
p 2
Figure 5.4: Schematic showing the surface tension T acting at a fluid interface.
We deduce that there are two possible wave-speeds, namely
c± = U ±
( (^) g k
tanh(kh)
We recall that the bracketed term in this equation is bounded by gh, as shown in Figure 5.3. Hence we can identify two possible cases in (5.44). If the flow speed U is less than
gh, then waves may propagate both upstream and downstream. Such a flow is termed subcritical. On the other hand, if U >
gh, then all waves are carried downstream and the flow is said to be supercritical.
5.3.3 Two fluids
Now suppose the interface y = η separates two fluids with different densities, say ρ = ρ 1 in y < 0 and ρ = ρ 2 in y > 0. We denote the velocity potentials and pressures on either side by φ 1 , φ 2 and p 1 , p 2 respectively. The kinematic condition (5.8) applies to both sides of the interface, and leads to the linearised boundary conditions
∂η ∂t
∂φ 1 ∂y
∂φ 2 ∂y
at y = 0. (5.45)
The dynamic boundary condition (5.3) is replaced by the pressure continuity condition p 1 = p 2 at y = η. After use of Bernoulli’s equation (5.4) and linearisation, this leads to the boundary condition
ρ 1
∂φ 1 ∂t
= ρ 2
∂φ 2 ∂t
at y = 0. (5.46)
Notice that (5.11) is recovered if we let the density ratio ρ 2 /ρ 1 tend to zero.
5.3.4 Surface tension
Real fluid interfaces exhibit a phenomenon called surface tension, which acts like a membrane stretched over the interface to a tension T. In Figure 5.4 we show the forces
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x
y
ρ 2 y^ =^ η(x, t)
ρ 1
U
Figure 5.5: Schematic of a fluid of density ρ 2 flowing at speed U over a fluid of density ρ 1.
Example 5.1 One fluid flowing over another We illustrate the effects outlined above by analysing the situation shown in Figure 5.5, where an infinite layer of fluid with density ρ 2 flows at speed U over an infinite layer of density ρ 1. We include a surface tension T at the interface between the two fluids, so that the disturbance potentials φ 1 , φ 2 and the free-surface deflection η satisfy
∇^2 φ 1 = 0 in y < 0 , ∇^2 φ 2 = 0 in y > 0 , (5.56)
∂φ 1 ∂y = ∂η ∂t , ∂φ 2 ∂y = ∂η ∂t
ρ 1
( ∂φ 1 ∂t
) − ρ 2
( ∂φ 2 ∂t
) = T ∂^2 η ∂x^2
at y = 0. (5.57)
As usual, we look for harmonic travelling waves with η(x, t) = A cos(kx − ωt − β), where the wavenumber k is assumed to be positive, without loss of generality. The corresponding solutions φ 1 , φ 2 of Laplace’s equation that decay as y → −∞ and y → +∞ respectively are then easily found to be
φ 1 (x, y, t) = Beky^ sin(kx − ωt − β), φ 2 (x, y, t) = Ce−ky^ sin(kx − ωt − β), (5.58)
where B and C are arbitrary constants. On substituting these into the boundary conditions (5.57), we obtain a system of three ho- mogenous linear equations for A, B and C, which can be written in the form
ω −k 0 ω − U k 0 k (ρ 1 − ρ 2 ) g + T k^2 −ρ 1 ω ρ 2 (ω − U k)
A B C
(^) =
0 0 0
(^). (5.59)
For nontrivial solutions, the determinant of the system must be zero, and hence we obtain the dispersion relation
(ρ 1 + ρ 2 )ω^2 − 2(ρ 2 U k)ω + ρ 2 U 2 k^2 − (ρ 1 − ρ 2 )gk − T k^3 = 0. (5.60)
Part A Fluid Dynamics & Waves Draft date: 9 March 2010 5–
5.4 Instability
We have always assumed thus far that the dispersion relation gives rise to real values of the frequency ω. However, it may well arise that ω is complex, for example when the dispersion relation is a quadratic equation such as (5.60). If we write the real and imaginary parts of ω as ω = ωR ±iωI, then a harmonic travelling wave like (5.12) becomes
η = A cos(kx − ωt − β) = A cos (kx − ωRt − β) cosh (ωIt) + iA sin (kx − ωRt − β) sinh (ωIt). (5.61)
We infer that a complex value of ω corresponds to an exponentially growing amplitude, and implies that the corresponding wave is unstable.
Example 5.2 Rayleigh–Taylor instability We return to the problem of one fluid flowing above another, analysed above in Example 5.1. If there is no relative flow, that is U = 0, then the dispersion relation (5.60) reduces to
ω^2 =
( (ρ 1 − ρ 2 )g + T k^2
) k ρ 1 + ρ 2
. (5.62)
If ρ 1 > ρ 2 then the right-hand side of (5.62) is positive, so there are two equal and opposite values of ω, corresponding to waves propagating at speed c = ω/k in either direction. However, if ρ 1 < ρ 2 , ω^2 is negative for some values of k, namely
k <
√ (ρ 2 − ρ 1 )g T.^ (5.63)
For these wavenumbers, ω is pure imaginary, so the disturbance grows exponentially. Hence the situation with the denser fluid above the lighter fluid is (not surprisingly) unstable; this is known as the Rayleigh–Taylor instability.
Example 5.2 reminds us that the frequency ω is a function of the wavenumber k so that, in general, ω may be complex only for certain values of the wavenumber. This implies that the system is unstable only to waves of certain wavelengths. In Example 5.2, equation (5.63) implies that only waves with wavelength λ such that
λ > 2 π
T
(ρ 2 − ρ 1 )g
are unstable. At a water-air interface, we would have T ≈ 0 .07 N m−^1 , ρair ≈ 1 .2 kg m−^3 , ρwater ≈ 1000 kg m−^3 , g ≈ 9 .8 N kg−^1 , so that only waves longer than roughly 1.7 cm are unstable. If the system is too narrow to allow waves this long, then the instability will be eliminated. This explains why a glass of water may be tipped upside-down without the water spilling out if a sufficiently fine mesh is stretched over the end.
Example 5.3 Kelvin–Helmholtz instability When U is nonzero, the solution of the quadratic equation (5.60) is given by
ω = ρ 2 U k ±
√ ∆ ρ 1 + ρ 2 , (5.65)
Part A Fluid Dynamics & Waves Draft date: 9 March 2010 5–
η
x
Figure 5.7: Schematic of a spreading train of ripples caused by a localised disturbance.
is out by a factor of 2. This occurs because the wave-speed c is the speed at which the crests (or troughs) propagate in a pure sinusoidal wave, not in a realistic free surface profile containing a combination of many different wavenumbers. A profile like that illustrated in Figure 5.7 can be represented as
η(x, t) = A(x, t) cos
α(x, t)
in terms of a rapidly-varying phase α and a slowly-varying amplitude A. In the vicinity of a fixed position x = x 0 and time t = t 0 , we can Taylor-expand the phase to get
α(x, t) ≈ α (x 0 , t 0 ) + (x − x 0 ) ∂α ∂x
(x 0 , t 0 ) + (t − t 0 ) ∂α ∂t
(x 0 , t 0 ). (5.70)
Hence, the profile (5.69) is locally approximated by the harmonic travelling wave
η(x, t) ≈ A 0 cos
k 0 x − ω 0 t − β 0
where
A 0 = A (x 0 , t 0 ) , β 0 = α (x 0 , t 0 ) − k 0 x 0 + ω 0 t 0 , (5.72a)
k 0 =
∂α ∂x (x 0 , t 0 ) , ω 0 = −
∂α ∂t (x 0 , t 0 ). (5.72b)
It is therefore natural to define the local wavenumber and frequency at position x and time t by
k(x, t) = ∂α ∂x
, ω(x, t) = − ∂α ∂t
It immediately follows from (5.73) that k and ω satisfy the equation
∂k ∂t
∂ω ∂x
For a pure harmonic wave, there is a dispersion relation ω = ω(k) specifying the frequency as a function of the wavenumber. We assume that the same relation holds here,
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η η
x x
(a) (b)
increasing t
increasing t
Figure 5.8: Schematic of a moving wave packet with (a) cg < c, (b) cg > c. One wave crest is highlighted to illustrate how it moves relative to the packet.
since the free surface profile is locally approximately sinusoidal. Thus (5.74) becomes a partial differential equation for the wavenumber k, namely
∂k ∂t
∂k ∂x
where
cg(k) = dω dk
is called the group velocity. We deduce from equation (5.75) that k is constant along straight lines in the (x, t)- plane satisfying dx/dt = cg. Indeed, the general solution of (5.75) is
k = F
x − cg(k)t
It follows that waves with wavenumber k propagate with speed cg(k), and not at the wave-speed c(k) as might have been expected. For waves on deep water, with the dispersion relation (5.21), the wave-speed and group velocity are given respectively by
c = ω k
g k
, cg = dω dk
g k
c 2
At first glance, this may appear to be a contradiction: how can the wave crests propagate twice as quickly as the waves themselves? The answer is that the waves separate into wave packets corresponding to different wavenumbers. Within each wave packet, the waves move at speed c, but the packet as a whole moves at speed cg. This phenomenon is illustrated in Figure 5.8(a) for a single wave packet travelling from left to right at speed cg. The wave crests move through the packet at speed c = 2cg, seeming to appear
