# Wave Energy

As
implied by the picture, waves can be complex and somewhat messier than
the simplicity of our basic sinusoidal assumptions. However, while this
section does attempt to take a closer look at the nature of waves, it
is often counter-productive from a learning perspective to make the
models too representative of the actual complexity of nature. However,
there is still the possibility that the complexity of the real world
might make reference to one of the simplest models. Therefore, we will
begin this discussion with an example that fits nicely with theory;
albeit that this example is still somewhat conceptual in that we shall
assume that this system does not lose any energy. It is essentially
the model of a weight-on-a-spring that oscillates up and down as a lossless
SHM pendulum, with its motion described by a sine wave. As we have already
outline the concept of* Simple Harmonic Motion*,
we need not repeat all the supporting arguments, but will restate the
initial information known about this system:

*Mass on the spring: [m]**Elasticity of the spring: [k]**Displacement of mass: [y]*

To start with we might note that the physical motion of the weight
on the end of the spring corresponds to the amplitude of the wave, i.e.
y=A. Therefore, in this case, we might recognise that the oscillating
motion does not propagate as a wave through space, only time. As such,
the oscillating motion does not have a propagation velocity [v=x/t],
only the up-down velocity [u] of the mass [m], which implies a corresponding
kinetic energy of [E_{k}= ½mu^{2}]. Of course, in terms
of the initial energy of this system, there is no kinetic energy component
as the weight is held, displaced by a distance [y], from its natural
equilibrium position [y=0]. Therefore, the initial energy of this wave
is unambiguously described in terms of its potential energy only:

[1]

However, [1] still reflects the axiom of all mechanical waves that
the energy is always proportional to the square of the amplitude, e.g.
[y^{2}], even though this wave does not propagate in space [x].
For when the weight is released, it accelerates under the force of the
spring towards its equilibrium point, i.e. y=0. By this process, the
mass [m] acquires velocity [u] and, under the conservation of energy,
the initial potential energy [E_{p}] is converted to kinetic
energy [E_{k}] and back again. As such, energy is conserved
between two maximum states of potential and kinetic energy:

[2] *E _{T} = E_{p} + E_{k}
*

However, while [1] represents the potential energy [E_{p}]
at any offset [y], it must also represent the total energy [E_{T}],
when [y] is a maximum [y_{0}]. Therefore, we can extend [2]
as follows:

[3]

The last^{ }terms in [3] corresponds to the kinetic energy
[E_{K}], where [u] is the rate of change of the amplitude. We
can re-arrange [3] in terms of the mass velocity [u]:

[4]

As such, we can add [1] through [4] to the information in the following
table, which was previously derived in the earlier discussions relating
to *SHM wave functions*.

Quantity | Symbol | Value |

Angular Velocity | ω | = 2πf = √(k/m) |

Amplitude | y | = y_{0}sin(ωt) |

Velocity | u | = -ωy_{0}cos(ωt) |

Acceleration | a | = -ω^{2}y_{0}sin(ωt) = ω^{2}y |

As such, we have a descriptive model of this system in terms of its wave-like properties, which all have roots in the following time domain amplitude [y] equation:

[5]

So to reiterate, the term `*wave-like`* is being used because
there is no wave propagating in space since the wave amplitude [A] now
corresponds to the offset [y] of the mass [m] from its equilibrium point.
As such, we might be describing the up-down motion of a point in space
with unit mass [m], subject to some form of elasticity [k] linked to
the passage of a mechanical wave. Therefore, it might be possible to
apply some of the basic principles of the SHM model to the propagation
of *‘real’* mechanical waves passing through a physical medium
in both time and space. So, this might be a useful point to remind ourselves
of the original definition of a mechanical wave.

*A mechanical wave is a wave that
propagates through a medium by means of a `mechanical interaction` between
adjacent or neighbouring particles within that medium. *

As such, we might reflect on the fact that a `*mechanical interaction*`
relates to oscillations that take place in a given medium, i.e. gas,
liquid or solid. However, let us try to be a little bit more precise
about how the medium is oscillating by saying that individual particles,
within the medium, must also exhibit a periodic motion, e.g. move up
and down about an average equilibrium position. As such, each particle
in the medium may be subject to its own SHM wave function and, in this
context, our spring model might be considered as a model for an individual
particle within a given medium. So while there are possible problems
with this assumption, we shall pursue the idea for the moment:

Clearly, it is quite difficult to show all the real-time dynamics
of an expanding surface wave in what amounts to a 2-dimensional snapshot
as illustrated above. However, to some extent, the animation that introduces
*2D waves* and another illustrating
a cross-section of the *2D wave propagation
*both reflect how the wave being described acts in real-time. These
animations correspond to a ‘*pebble-in-the-pond*’ model in which
a volume of water is displaced that then generates a succession of expanding
circular waves that decay as the potential energy [E_{p}] of
the original displacement is dissipated across the expanding radius
of a series of waves. However, with reference to the diagram above,
point [P_{0}] corresponds to a small mass of water [m]
within a column of water, which has been lifted to a height
[y=h] above the surface of the surrounding water. As such, this column
of water has potential energy [E_{p}] related to its height
[h], mass [m] and the pull of gravity [g]. When this column of water
'*falls*', it collapses under gravity and, in so doing, creates an
expanding circular wave. However, we need to make a number of clarifications
about the diagram above, because it can give a very misleading perception
of how this mechanical wave really dissipates the original potential
energy represented by [E_{p}=mgh]. The first point is that the
red sinusoidal curve, shown in the diagram above, represents the peaks
and troughs of a wave that only exists as a function of different points
in time, i.e.[ t_{1}, t_{2}..t_{n}]. It is also
assuming that all the original potential energy [E_{p}] is transported
by a single expanding wave, which has been proved not to be the case
in the pebble-in-the-pond animations.

The picture above is essentially a single frame taken from the *
wave propagation* animation, annotated
with a point P_{(x,t)} that might be seen as corresponding to
P_{n} in the earlier diagram. So what we need to clarify is
that the potential energy [E_{p}] does not propagate out in one single
wave, but rather as a series of wave peaks, because the original energy imparts
momentum to individual volume-masses [m] within the media distributing
the wave. Second, contrary to the impression that might have been given
by the first diagram, by the time the wave reaches point P_{(x,t)},
the amplitude at the centre has already changed. In this context, the
main purpose of the first diagram was only to highlight, via the inset
diagrams associated with [P_{1}] and [P_{n}], that both
of these points might be initially considered as localised masses [m_{1},
m_{n}] that move up and down in accordance to a damped SHM model.

At this point, let us stop to consider the energy implication at
some arbitrary point [P_{n}] based on the assumption that this
particle has initially neither potential nor kinetic energy. Clearly,
in order to raise the particle [P_{n}], the propagating wave
must transfer some of its energy to this particle, i.e. it gains kinetic
energy by virtue of its upwards motion and potential energy by virtue
of its increased elevation [h]. At its peak amplitude, the energy of
[P_{n}] can be described solely in terms of potential energy,
which is then converted back in to kinetic energy as [P_{n}]
falls back towards the zero height represented by the water surface.
As such, there appears to be a parallel to our earlier SHM spring model,
which oscillates continuously between potential and kinetic energy,
but with the exception that this energy system is no longer lossless,
as previously assumed. While the actual process of energy propagation
between millions of water molecules is almost impossibly complex, we
might intuitively understand that some of the energy associated with the
volume-mass [P_{n}] is transferred to its neighbours in the
direction of the waves propagation, i.e. potential energy is being
dissipated outwards. As such, we are trying to generalise
a model in which the surface of the pond can be approximated to a series
of oscillating point masses [m] with the decaying displacement described
as a function of time, which in practice will depend on the viscosity
of the medium, e.g. water. However, at another level, we still need
to consider the overall wave motion of the medium, which maintains its
shape as it propagates through the medium. Therefore, let us table a
question:

*What property of the medium maintains the wave
shape during propagation? *

The assumption of the wave models outlined has always been based
on a sinusoidal shape propagating in space [x], which we are now trying
to consider collectively across a 2-dimensional surface [x,z] at some
arbitrary point in time, which might be thought of as ‘*now*’,
which allows us to remove time [t] from the equation:

[6]

The symbol [γ] simply represents a ‘*decay factor’* affecting
the wave in propagation, although we are not trying to specify its form
at this point. Equally, in the generalised 1D wave equation, it was
shown that the second derivative of amplitude [A] with respect to space
[x] was proportional to the second derivative of amplitude [A] with
respect to time [t], i.e.

[7]

In the context of the SHM spring model, the acceleration of amplitude
was linked to the mass [m] attached to the spring. However, in the case
of a surface wave, the second order rate of change of amplitude [A]
is associated with the up-down acceleration [a] of any particle [P_{n}]
along the path of the propagating wave. As such, the second derivative
of a mass particle [m_{p}] in time must have a corresponding
force [F] acting on it, which is proportional to the second derivative
of [x], i.e. a small section of the medium. If so, we might combine
[7] with Newton’s 2^{nd} law of motion:

[8]

We might wish to do a quick sanity check of [8] by comparing the units:

[9]

Based on the perceived consistency of units in [9], it might be suggested that a wave has to maintain its shape in propagation because it is the only way to balance the forces associated with the wave’s amplitude and propagation velocity [v]. This might also be a useful point to introduce the basic idea of dispersion within a physical medium, for when the propagation velocity [v] of a mechanical wave does not depend on the frequency of the disturbance, the propagation medium itself is said to be non-dispersive. In a non-dispersive medium, the superposition of the different wave frequencies within a pulse-wave will travel through the medium at the same velocity [v=fλ]. Therefore, in non-dispersive media, higher frequencies [f] have to result in shorter wavelengths [λ] so as to maintain a constant propagation velocity [v] in any given medium. To illustrate this point, if air were not a non-dispersive medium, music at the back of the hall would sound very different than that heard at the front of the hall because sound of different frequencies would arrive at different times.

*So how might we apply this insight to other mechanical
waves? *

While we have shown that reference can still be made to the SHM spring
model, the energy and force associated with large-scale mechanical waves
cannot, in practice, be easily modelled by millions of individual particles
[m_{p}], although some reasonable approximation might be made
with this approach. Of course, we can still consider the physics of
the aggregated mass [m] of the water displaced by the wave as a whole.
For example, let us consider a generalised model of a pebble being dropped
into a pond, but then scale this example to that of a relatively small
meteor strike in the middle of the Pacific Ocean, so that we might try
to cross-reference some of the known wave physics of
*ocean and tsunami
waves*. While this undoubtedly changes the complexity of the problem
beyond the initial scope of this discussion, it also requires us to,
at least, consider the energy of this type of wave in a much wider context.
For example,

*Does the meteor create a central displacement
of water that radiates outwards? Does the cross-section of this
wave look more like a pulse than a sine wave? *

To address the last question first, a Fourier series might still allow the pulse-wave to be modelled as a fundamental sine wave in superposition with its harmonic components. However, such a model might require us to consider the implications on the propagation velocity of a wave comprised of harmonics of different frequencies. If so, we would need to understand whether the medium in question is dispersive or non-dispersive. As indicated earlier, air as a medium is non-dispersive and therefore all sound waves of different frequencies travel at the same propagation velocity. However, in general, water is a dispersive medium, therefore waves of a different frequency, and even amplitude, travel at different propagation velocities, such that long wavelength propagate faster than shorter wavelengths. This said, we will start by making some sort of estimate of the initial potential energy of this type of wave in terms of the pulse-wave as a whole, i.e. it has height, depth and width.

*On what assumption might we define the energy
of a tsunami wave? *

Like the SHM spring model, we could begin by assuming the energy of the wave is initially associated with the potential energy of a volume of water displaced by the meteor. Subsequently, a wave propagates via a process involving the conversion of potential energy to kinetic energy and back again.

*So how does kinetic energy fit into the wave
propagation description?*

With reference to the earlier diagrams and the SHM spring model, when a point on the wave has zero potential energy, it has acquired maximum kinetic energy and the associated momentum of a mass [m] moving with velocity [u]. This momentum is then the effective source of energy that pushes the SHM model into its next cycle of oscillation, i.e. the kinetic energy is converted back into potential energy. Of course, if all the energy was retained within this cycle, then there would be no energy to propagate within the wave itself. As such, we might realise that some fraction of the energy is retained by the local oscillating system, while another fraction is imparted to its neighbours within the media that ultimately contributes to the wave propagating outwards.

In many ways, the animation right is built on a model that assumes the wave medium consists of a series of SHM oscillators that retains some fraction of the energy, linked to the viscosity of the medium, to maintain the next cycle of oscillation, albeit at a reduced offset, while still imparting some of its energy to its neighbours. The net result being the somewhat idealised 2D sinusoidal wave radiating outwards, which may not reflect the true nature of a tsunami wave. Therefore, it might be useful to complement this conceptual model against some of the analysis done on tsunami waves before proceeding to draw any further conclusions. In a paper in the American Journal of Applied Sciences (2005), some analysis was presented of the physics associated with the tsunami wave that caused the disaster in the Indian Ocean in 2004. It is highlighted that this analysis was predicated on the fact that most tsunami waves are caused by seismic events in which the wave generated was essentially aligned and orthogonal to the direction of a seismic fault line. Wave of this nature do not necessarily lose energy within an ever-increasing circular wave, as assumed by our earlier model and, for this reason, the energy of this wave was approximated to a volume of displaced water using the following equation:

[10]

The shape of this wave is assumed to be a 3-dimensional [x,y,z] rectangular
block called a `*waterberg*`. Where [a] corresponds to the height
of the wave, [λ] is the wavelength within the waterberg, [L]
is the length or width of the wave, [ρ] is the density of the sea-water
and [g] is the acceleration due to gravity. However, this equation can
still be reduced to the form [E_{p}=mgh], if we assume that
[x=λ], [y=a] and [z=L]:

[11]

Clearly, basing the volume of water displaced on the cubic shape of the waterberg has to be a very rough approximation. However, we might use a similar logic to construct another approximation of the potential energy of the initial central wave in our animation. However, this wave would have to be closer to the volume of a cone with a base radius [r] with a centre of mass at one-quarter the height [h] of the cone:

[12]

Even so, we can see that [11] and [12] still reflect a potential energy that remains proportional to the square of the amplitude or height of the wave, i.e. [a=h]. Although it is possibly more important to recognise that the energy of the wave, as a whole, depends on the mass-volume of the water displaced, i.e.

[13]

So while it is true to say that the initial potential energy being
put into this wave system is proportional to the square of the amplitude,
it is more instructive, in this case, to recognise the link to the basic
equation [E_{p}=mgh]. Given the circular nature of our pebble-in-the-pond
analogy, we may also predict the decay of the amplitude of the wave
as a function of the expanding radius [r], as previously discussed:

[14]

In essence, the initial potential energy input into the wave system,
as a whole, represents the total energy [E_{t}], which in the
case of a lossless system is dissipated over time by the expanding wave
system. However, we might intuitively understand that the initial central
displacement of water cannot collapse and come to rest instantly and,
as a consequence, a succession of ever-smaller expanding waves may also
be created. We might also recognise that this undulating motion must
correspond to some component of the total input energy [E_{t}],
which now exists as both kinetic energy [E_{k}] and potential
energy [E_{p}] being distributed over an ever-expanding area
through a succession of waves over time. By way of a visualisation of
the total energy [E_{t}] of the wave, the diagram below focuses
on a single particle [P_{n}] that acquires both potential and
kinetic energy from the expanding wave:

For simplicity, the diagram above only shows a single lossless wave
cycle at an arbitrary point [P_{n}] that is lifted by the expanding
wave. At time [t=1], the particle [P_{n}] has maximum potential
energy [E_{p}]. As the wave peak passes, the particle [P_{n}]
falls back down, but in so doing gains maximum kinetic energy [E_{k}]
at [t=2]. As such, some fraction of the total energy is retained by
particle [P_{n}] after the initial wave continues to propagate
outwards. So while there is no net radial movement of water, the total
energy within any complete wave cycle exists as the sum of both potential
[E_{p}] and kinetic [E_{k}] energy. Therefore, the total
energy does not oscillate, as per its potential and kinetic components,
although it will decay in any practical wave system. Each successive
wave, generated from the collective momentum imparted to the particle
via its kinetic energy, will continue to carry both potential and kinetic
energy away from the initial centre until the water surface returns
to a state of equilibrium, i.e. point [P_{n}] has zero energy,
as illustrated by the previous animation. The wider considerations of
this energy model will be discussed in the
*final page** of* this section.