Derivation and Analysis of Some Wave Equations
Derivation and Analysis of Some Wave Equations
Derivation and Analysis of Some Wave Equations
Wave phenomena are ubiquitous in nature. Examples include water waves, sound waves, electro-
magnetic waves (radio waves, light, X-rays, gamma rays etc.), the waves that in quantum mechanics
are found to be an alternative (and often better) description of particles, etc. Some features are
common for most waves, e.g. that they in cases of small amplitude can be well approximated by
a simple trigonometric wave function (Section 4.1) Other features differ. In some cases, all waves
travel with the same speed (e.g. sound waves or light in vacuum) whereas in other cases, the speed
depends strongly on the wave length (e.g. water waves or quantum mechanical particle waves). In
most cases, one can start from basic physical principles and from these derive partial differential
equations (PDEs) that govern the waves. In Section 4.2 we will do this for transverse waves on a
tight string, and for Maxwell’s equations describing electromagnetic waves. In both of these cases,
we obtain linear PDEs that can quite easily be solved numerically. In other cases, such as water
waves, discussed in Section 4.3, the full governing equations are too complex to give here, and we
need to restrict ourselves to a number of general observations. In still other cases, such as the
Schrödinger equation for quantum wave functions, a quite simple set of PDEs are well known and
extremely accurate (often said to describe all of chemistry!) but these are prohibitively difficult to
solve in all but the simplest special cases. We note in Section 4.4 that some important nonlinear
wave equations can be formulated as systems of first order PDEs. Not only are these systems
usually very well suited for numerical solution, they also allow a quite simple analysis regarding
various features, such as types of waves they support and their speeds. In some cases, discussed
in Section 4.5, we find some closed-form analytic solutions. We arrive in Section 4.6 to Hamilton’s
equations. These are fundamental in many applications, such as mechanical and dynamical sys-
tems, and the study of chaotic motions. In the context of this book, their key application is to
provide the governing equations for the freak wave phenomenon that is discussed in Chapter ??.
1
2 CHAPTER 4. DERIVATION AND ANALYSIS OF SOME WAVE EQUATIONS
small, such a progressive wave can be well approximated by a single trigonometric mode
If t is increased by one and x by ω/k, the argument of the cosine in (4.1) is unchanged. Hence,
the wave in (4.1) travels with
We will later come across another speed, group speed, cg . If only ’wave speed’ is mentioned, or no
subscript for c is given, the phase speed cp is assumed.
For almost all waves, ω is not a constant, but a function of k. In some cases, this relation takes
the form ω = c · k, e.g. for sound waves and for light in vacuum. In such cases the wave speed
(according to (4.2)), becomes independent of the wave number. In other cases, the dispersion
relation ω = ω(k) takes different forms. For waves on deep water, the leading order approximation
(when the wave amplitude is small) can be shown [...] to be
ω= gk (4.3)
φ(x, t) = φ0 Re ei (k x− ω t) ,
φ(x, t) = φ0 ei (k x− ω t) . (4.4)
4.2. TWO EXAMPLES OF DERIVATIONS OF WAVE EQUATIONS 3
Figure 4.2: 2-D progressing wave. Wave crests are marked with dotted lines; waves progress in the
direction of the k-vector.
Not all wave forms are sinusoidal. However, by Fourier analysis (cf. Chapter ??), any other shape
(for a linear wave equation) can be viewed as a superposition of sinusoidal waves of different wave
numbers k. Together with knowledge of the dispersion relation ω = ω(k), we can analyze how an
initial wave form evolves in time.
where x= (x1 , x2 ) and k= (k1 , k2 ) are two-component vectors. The wave φ(x, t)
given by (4.5) clearly reduces to (4.4) in case we introduce a (scalar) x-direction parallel
to the k-vector. We can also note that φ(x, t) is unchanged if x moves along
any direction orthogonal to k. From the fist observation follows that the wave length
λ = 2π / |k| and the phase speed cp = ω / |k| .
Two kinds of waves will travel along a string under tension - transverse and longitudinal. Their
speeds are typically vastly different. Transverse (sideways) oscillations are usually fairly slow,
and visible, whereas longitudinal (lengthwise) waves would travel with the speed of sound in the
material, maybe of the order of 1 km/s, while causing no visible deflections. In a loosely stretched
’slinky’, both wave types can be seen traveling at about 10 m/s. The transverse waves in a string
is the simplest case to obtain an equation for, and we will do that next.
A string, with density ρ per unit length, is stretched in the x-direction with a tension force T
(cf. Figure 4.3). At any time, the vertical forces on the small string segment must balance. They
are
∂2u
ρ · ∆x · 2 = T (x + ∆x, t) sin θ(x + ∆x, t) − T (x, t) sin θ(x, t)
∂t
mass accele- difference b etween vertical tension forces at the two ends
ration
∂u
Assuming the deflection angles θ(x, t) are small, sin θ ≈ tan θ = ∂x . Dividing both sides by ∆x
and letting ∆x → 0 then gives
∂2u ∂ ∂u
ρ(x) 2 = T .
∂t ∂x ∂x
Still assuming that the deflection is small, the tension T becomes approximately a constant, and
can be factored out. Introducing c2 = T /ρ, we arrive at the 1-D wave equation in its standard
form
∂2u 2
2∂ u
= c (4.6)
∂t2 ∂x2
We will soon see that this equation supports waves traveling with the velocity c to either left or
right.
4.3. WATER WAVES 5
Here
Ex , Ey , Ez the components of the electric field
Hx , Hy , Hz the components of the magnetic field
µ permeability
ε permettivity
For lossy media, we need to subtract σEx , σEy , σEz , ρHx , ρHy , ρHz , resp. from the six RHSs
(with σ and ρ denoting conductivity and magnetic resistivity).
For waves on a string, we could write down a PDE which describes how any initial state evolves
forward in time. In contrast to this, for deep water gravity waves, there is no single PDE which
describes the evolution of a surface disturbance.
6 CHAPTER 4. DERIVATION AND ANALYSIS OF SOME WAVE EQUATIONS
Figure 4.4: Physical and Fourier space representation of a uniform wave train and of a wave packet.
where the acceleration of gravity g ≈ 9.8 m/s2 . Denoting the wave length by λ = 2π/k, the
velocity of this wave (cf. (4.2)) becomes
ω g gλ
cp = = = , (4.9)
k k 2π
√
and the time period T = 2π/ω. The fact that cp grows proportionally with λ leads to many
notable features of water waves. Some of these are mentioned in Section ....
part of the figure. In Fourier space, the wave packet is a superposition of waves with
very similar frequencies. If the peak (in Fourier space) is getting narrower, the packet
becomes wider in physical space. We can clearly see two different velocities associated
with the wave packet, both of which can be expressed in terms of the dispersion relation:
ω(k)
phase speed speed of individual crests cp =
k
dω(k)
group speed speed of the whole group cg =
dk
Derivation of the formula for cg : The wave function for a single wave number k0 can
be written
ei(k0 x−ω(k0 ) t)
φ(x, t) = φ (4.10)
0
A wave packet with the same main wave number is similarly a superposition of different
waves ∞
φ(x, t) = 0 ei(k x−ω(k) t) dk
φ
−∞
where φ(k) is very near zero everywhere but has a sharp peak at k = k0 . In that small
neighborhood of k0 we have (by Taylor expansion)
dω
ω(k) = ω(k0 ) + (k − k0 ) α where α = .
dk k=k0
Hence
∞
φ(x, t) ≈
φ(k) ei(k x−(ω(k0 )+(k−k0 )α) t) dk
−∞
∞
= ei(k0 x−ω(k 0 )t)
·
φ(k) ei(k−k0 )(x−αt) dk
−∞
Pure harmonic of Factor providing the envelope
wave number k0 of the wave packet
We notice that in the second factor, the variables x and t appear only in the com-
bination x − αt, showing that this expression translates with the speed α = dω
dk k=k0 ,
i.e. this quantity is equal to the group speed.
Table 4.2 summarizes the dispersion relations and the two speeds cp and cg for some different
types of waves (in the case of surface waves, the liquid is assumed to be water, with density ρ = 1).
The results in this table have some notable implications:
• Since cp > cg for gravity waves, a surfer gets the longest ride is he/she can catch a wave at
the end of a wave packet
• Compared to the wave lengths of tsunami waves (hundreds of kilometers), all oceans are
shallow. Timing of the departure and arrivals of such waves offered the first means (in the
19th century) of estimating average ocean depths.
8 CHAPTER 4. DERIVATION AND ANALYSIS OF SOME WAVE EQUATIONS
Gravity wave,
√
gk
g 1
g 1 g = acceleration
deep water
Gravity wave,
√
g k tanh kh
kg 2 k
cp ·(cg /cp )
2
1 kh
of gravity
h = water depth
shallow water tanh kh +
√ √ k √ 2 sinh(2hk)
Capillary wave T k3 T k 3 T k 3 T = surface
2 2 tension
h = Planck’s constant
Quantum mechanical h k2 hk hk m = particle mass
particle wave 4πm 4πm 2πm 2 cg = particle velocity
Light in vacuum c = 299,792,458 m/s
ck c c
1
′ ′
Light in a ck c kn (k) kn (k) n(k) = index of refrac-
transparent medium n(k) n(k) cp 1 − n(k) 1− n(k) tion in the medium
Table 4.2: Dispersion relations and wave speeds for some different types of waves
• If a twig sticks up in a stream, the phase angle of waves gets locked at it. Since capillary waves
have cg > cp , they can be seen as small stationary ripples upstream of the twig. Gravity
waves will instead appear in some relatively narrow sector downstream of the twig.
• The ratio cg /cp = 12 for gravity waves can be shown to imply that the distinct V-shaped
wake left behind a ship will always form the angle arcsin 13 ≈ 19.5o to each side of the center
line of he wake independently of the speed of the ship (intuition might wrongly suggest that
the wake should get narrower with increased speeds, like the case is for shocks generated by
a fast-moving object).
• In quantum mechanics, a particle’s position is undetermined within the width ∆x of its wave
packet. Fourier analysis will show that ∆x and its spread in Fourier space are related by
∆x · ∆k > 1 (or ∆x · ∆k > constant; there is some arbitrariness in how wide one regards a
Gaussian pulse to be). De Broglie’s relation k = 2πmν/h relates wave number k to Planck’s
constant h; here m is the particle mass and ν (= cg ) its velocity. From this, we obtain
Heisenberg’s uncertainty relation ∆x · ∆ν > h/(2πm). In other words, the product of the
uncertainties in a particle’s position and velocity must always exceed h/(2πm).
• The governing equations for surface water waves turn out to be nonlinear. Apart for infini-
tesimal waves, solutions can therefore not be linearly superposed (or multiplied by scalars)
to give other solutions.
4.4. FIRST ORDER SYSTEM FORMULATIONS FOR SOME LINEAR WAVE EQUATIONS9
• In the limit of large amplitude, already Stokes (1880) showed that the wave would have a
top angle of 120o . However, it was noted later that the local wave structure near the top -
in this limit - will feature a complicated fine structure.
• The expansion (4.11), if continued to more terms, will diverge before the highest Stokes’
wave is reached. This is related to the fact that a Taylor expansion will fail to converge at a
radius determined by the nearest singularity - in this case there will arise unphysical complex
singularity points for the real variable a.
• When including further terms in (4.11), the coefficients for the individual modes will no
longer be pure powers of a, but will turn into power series expansions in a.
• In real water, high Stokes’ waves are never seen. Such a wave is (near the top) unstable to
oscillatory disturbances. Another (physically more significant) instability arises already at
low amplitudes. The Benjamin-Feir instability causes uniform periodic wave trains to loose
their periodicity. Wave trains will always exhibit irregularities in amplitude between the
individual waves.
y′′ sin y ′ y
y ′′′ + 2
− t y + 11 arctan =0 (4.12)
1+t y ′ 1+t
The initial conditions for y, y′ and y′′ become the initial values for the three variables u1 , u2 and
u3 . This system, like the more general first order system
′
u = f1 (u1 , . . . , un , t)
1′
u2 = f2 (u1 , . . . , un , t)
,
···
′
un = fn (u1 , . . . , un , t)
can be solved accurately with almost any of the standard numerical techniques for ODEs (such as
Runge-Kutta or linear multistep methods).
Analytical solutions of systems of ODEs are rarely available. However, linear systems
u1 u1 f1 (t)
d . .. + ..
. = A .
dt . .
un un fn (t)
where A is a matrix with constant coefficients, form a notable exception. If the f -vector is absent,
and A has distinct eigenvalues λ1 , . . . , λn with corresponding eigenvectors v1 , . . . , vn , the general
solution becomes
u = c1 eλ1 t v1 + c2 eλ2 t v2 + . . . + cn eλnt vn . (4.13)
The coefficients c1 , c2 , . . . , cn will follow from the initial conditions. Standard ODE text books
will discuss the minor modifications that will need to be done to the form of (4.13) in the (usually
rare) cases of multiple eigenvalues or missing eigenvectors. If the f-vector is present, variation of
parameters can be used to obtain a general solution.
The equation describes for example the vibrations of a string (4.6) or acoustic waves in 1-D
∂2u 2
2∂ u
= c
∂t2 ∂x2
If a 2-D thin plate also possesses elastic properties, the most straightforward derivation of the
governing equations (for motions within the x, y−plane) leaves them in the form
∂u
= ∂f ∂g
+
∂t ∂x ∂y
/ρ
∂v
= ∂g
+ ∂h
∂t ∂x ∂y /ρ
∂f
= (λ + 2µ) ∂u ∂v . (4.15)
∂t ∂x + λ ∂y
∂g ∂v
∂t = µ ∂x + µ ∂u
∂y
∂h
∂t = λ ∂x + (λ + 2µ) ∂v
∂u
∂y
Here,
u, v local displacements in x-and y-directions
f, g, h local x-compression, shear, and y-compression respectively
λ, µ density, and elastic constants (wrt. compression and shear)
The equations (4.15) are directly obtained in first order system form, and it is therefore immediate
to express them in terms of matrices
u 0 0 1/ρ 0 0 u 0 0 0 1/ρ 0 u
v 0 0 0 1/ρ 0 v 0 0 0 0 1/ρ v
∂
=
∂ ∂
f λ + 2µ 0 0 0 0
∂x f
+ 0 λ 0 0 0
∂y f
.
∂t
g
0 µ 0 0 0 g
µ 0 0 0 0 g
h λ 0 0 0 0 h 0 λ + 2µ 0 0 0 h
We obtained these equations directly in the form of a first order system (4.7). In the case that
ε and µ are constants, we can re-arrange them into higher order equations for each individual
field component (However, this is seldom a useful thing to do since, in most applications, material
interfaces, conductors, etc. are present. The main task is often to understand the influence these
will have. Mainly out of mathematical interest, we will do this re-arrangement in two different
ways.
12 CHAPTER 4. DERIVATION AND ANALYSIS OF SOME WAVE EQUATIONS
Differentiation of the governing equations Differentiation of the first and the two last equa-
tions in (??) with respect to t , z and y respectively gives (Ex )tt = 1ε ((Hz )ty − (Hy )tz ) =
1 1
εµ ((Ex )yy + (Ex )zz ) − µε ((Ey )yx + (Ez )zx ) . Since the electrical field is divergence free 0 =
∂
div E = (Ex )x + (Ey )y + (Ez )z , we also have 0 = ∂x div E = (Ex )xx + (Ey )yx + (Ez )zx . With the
last result, the expression for (Ex )tt simplifies to
1
(Ex )tt = ((Ex )xx + (Ex )yy + (Ex )zz ) .
µε
Similarly, all the other components of the electric and magnetic fields will each satisfy the 3-D
acoustic wave equation (to repeat ourselves, on the very restrictive assumption that ε and µ are
constants).
∂ 1 ∂ 1
E = (∇ × H) , H = − (∇ × E) (4.16)
∂t ε ∂t µ
we get
∂2 E
∂t2 = 1ε ∇ × ∂H∂t by (4.16)
1
= − εµ (∇ × (∇ × E)) by (4.16)
= − εµ1
∇ (∇E) − ∇2 E by the vector identity (...)
= εµ1
∇2 E since ∇E = 0
Similarly,
∂2 H 1 2
2
=− ∇ H,
∂t εµ
implying again that each component of both fields will satisfy the 3-D acoustic wave equation.
We recognize this as an eigenvalue problem. Since the matrix has eigenvalues ±1, we get α = ± 1c
and we can conclude that (4.14) admits translating solutions with speeds 1/α1,2 = ±c
When we next turn to more space dimensions, (4.18) will generalize in a way that no longer
allows this interpretation as an eigenvalue problem. That difficulty can be avoided by noting that
(4.17) alternatively can be written as
′
1 0 0 1 u 0
+ cα = .
0 1 1 0 v 0
1 cα
This system has a non-trivial (non-zero) solution if and only if is singular. From
cα 1
1 cα
0 = det = 1 − c2 α2
cα 1
′ ′ ′
u u u u u u
∂ ∂ ∂
v = v , v = −cα v , v = −cβ v
∂t w ∂x ∂y
w w w w w
follows ′ ′ ′
u 0 1 0 u 0 0 1 u
v = −cα 1 0 0 v − cβ 0 0 0 v
w 0 0 0 w 1 0 0 w
with non-trivial solutions only if
1 cα cβ
0 = det cα 1 0 = 1 − c2 (α2 + β 2 ).
cβ 0 1
This shows that there is no ’preferred direction’ in the (x, y)−plane. There are solutions which
translate with speed c in any direction.
14 CHAPTER 4. DERIVATION AND ANALYSIS OF SOME WAVE EQUATIONS
Analogously to the previous case, looking for translating solutions leads us to consider
1 0 α/ρ β/ρ 0
0 1 0 α/ρ β/ρ
0 = det
(λ + 2µ)α λβ 1 0 0
µβ µα 0 1 0
λα (λ + 2µ)β 0 0 1
2
2
= µ(α2 + β ) − ρ · (λ + 2µ)(α2 + β ) − ρ /ρ2 .
There are now two types of possible waves, both with velocities that are direction independent:
In many materials (such as for seismic waves√in the earth), the two material constants λ and µ are
of similar size. It then follows that cp /cs ≈ 3.
Omitting the details, the same approach as above shows that there is exactly one type of wave -
√
again direction independent - and that it travels with the velocity c = 1/ εµ. Since ε and µ can be
found independently by experiments, this relation offers one possibility for determining the speed
of light.
In the three subsections below, we discuss briefly the general solution to (4.19), (4.20) in 1-D, in n-
D, and finally, we show how (4.19) simplifies in the case of radial symmetry. The analytic solutions
all assume constant material properties, and they cannot easily be extended to irregular domains.
They can still be useful for obtaining general insights about the character of wave equations.
However, in most practical situations, numerical solutions are needed.
4.5. ANALYTIC SOLUTIONS OF THE ACOUSTIC WAVE EQUATION 15
1-D: The solution at a point u(x, t) depends on f(x − ct), f(x + ct) and on all the g-values
in-between x − ct and x + ct. As a consequence, the general solution describes two very different
kinds of outgoing wave motions:
1. Cleanly translating pulses (if started by f(x) = 0 within some small area only, and g(x) = 0
everywhere). Such waves leave a zone of perfect silence behind them,
2. Disturbances which spread out throughout a complete interval (whenever g(x) = 0 initially).
Both of these situations can arise on a tight string: Case 1 if the string is locally deformed
to feature a small ’hump’ but is otherwise straight, and is then released, and Case 2 if a straight
string is lightly hit at one point.
2-D: (and also higher even dimensions): The solution u(x, t) depends both on f and g everywhere
within a distance ct from the point x. Both solution types (with f (x) = 0, g(x) = 0 and f (x) =
0, g(x) = 0 resp.) fall in the second category above, i.e. outgoing signals never leave any zone of
silence behind.
3-D: (and also higher odd dimensions): The solution u(x, t) at the point x depends on the values
of f and g only on the surface of a sphere centered at x and with radius ct. As a consequence,
however we initiate a sound signal at a point in 3-D, it will leave perfect silence behind itself as it
travels out. This very remarkable property makes speech possible in 3-D. These outgoing sound
signals attenuate with the distance traveled, but undergo no other changes. The conclusion is that
’clean speech’ is possible only in odd dimensions from three and up.
∂x2i
= ∂xi ∂xi = ∂r ∂r r r = ∂r2 r + ∂u 1
∂r r − ∂r r3
∂u ∂u ∂u
F (x1 , x2 , . . . , xn , u, , ,..., )=0
∂x1 ∂x2 ∂xn
or
F (x1 , x2 , . . . , xn , u, p1 , p2 , . . . , pn ) = 0 (4.31)
∂u ∂u ∂u
where we have used p1 , p2 , . . . , pn to denote ∂x ,
1 ∂x2
, . . . , ∂xn
respectively. The equations for the
characteristic curves (generalizing how (4.30) was obtained from (4.29)) become now
∂xi ∂F
= , i = 1, 2, . . . , n. (4.32)
∂s ∂pi
du
Along these paths, will no longer simplify all the way to zero, but we will nevertheless get a
ds
quite simple ODE for the evolution of u:
n
du $ ∂F
= pj . (4.33)
ds j=1
∂pj
Along the path, we need this time also to know how p1 , p2 , . . . , pn change. There turns out to be
simple ODEs for that as well:
∂pi ∂F ∂F
=− pi − , i = 1, 2, . . . , n. (4.34)
∂s ∂u ∂xi
The equations (4.32)-(4.34) form a set of 2n + 1 coupled ODEs with s as the independent vari-
able. They can be solved with most standard numerical ODE solvers, providing paths through
(x1 , x2 , . . . , xn )-space and, along these paths, values for u and also for its derivatives p1 , p2 , . . . , pn .
Equation (4.32) is a definition, but equations (4.33) and (4.34) need to be proven. This
can be done as follows for (4.33):
du #n ∂u ∂xj
= j=1 = by chain rule on u
ds ∂xj ∂s
# ∂F by the definition of the path (4.32)
= nj=1 pj ∂u
∂pj and of pj = ∂x j
.
H(x, p) = 0 (4.35)
p = ∇φ(x) (4.36)
Calling the parameter t instead of s (since it in applications often turns out to correspond to
physical time), the equations (4.32) and (4.34) show that x(t) and p(t) satisfies the Hamilton’s
system of equations
∂xi ∂H
=
∂t ∂pi
, i = 1, 2, . . . , n. (4.37)
∂p ∂H
i
=−
∂t ∂xi
Since H(x, p) is assumed to be a known function, the RHSs in (4.37) are explicitly known, and
these equations amount to a system of ODEs.
Relations of the form (4.35), (4.36) arise very frequently in areas such as classical mechanics,
nonlinear wave motion, and chaos. Equation (4.35) then express a conservation law, such as
conservation of energy. Assuming we have managed to find a formulation such that (4.36) also
holds, then the equations (4.37) become available, and can be solved to provide the evolution of
the x and p - variables.
The procedure of moving from (4.35) and (4.36) over to the system (4.37) offers a very important
approach for obtaining equations that are suitable for numerical solution. We will next give several
examples of this.
Example 1: ...
Example 2: ...
Example 3: ...
Example 4: ...