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 [Ek= ½mu2]. 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:
However,  still reflects the axiom of all mechanical waves that the energy is always proportional to the square of the amplitude, e.g. [y2], 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 [Ep] is converted to kinetic energy [Ek] and back again. As such, energy is conserved between two maximum states of potential and kinetic energy:
 ET = Ep + Ek
However, while  represents the potential energy [Ep] at any offset [y], it must also represent the total energy [ET], when [y] is a maximum [y0]. Therefore, we can extend  as follows:
The last terms in  corresponds to the kinetic energy [EK], where [u] is the rate of change of the amplitude. We can re-arrange  in terms of the mass velocity [u]:
As such, we can add  through  to the information in the following table, which was previously derived in the earlier discussions relating to SHM wave functions.
|Angular Velocity||ω||= 2πf = √(k/m)|
|Acceleration||a||= -ω2y0sin(ωt) = ω2y|
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:
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 [Ep] of the original displacement is dissipated across the expanding radius of a series of waves. However, with reference to the diagram above, point [P0] 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 [Ep] 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 [Ep=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.[ t1, t2..tn]. It is also assuming that all the original potential energy [Ep] 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 Pn in the earlier diagram. So what we need to clarify is that the potential energy [Ep] 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 [P1] and [Pn], that both of these points might be initially considered as localised masses [m1, mn] 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 [Pn] based on the assumption that this particle has initially neither potential nor kinetic energy. Clearly, in order to raise the particle [Pn], 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 [Pn] can be described solely in terms of potential energy, which is then converted back in to kinetic energy as [Pn] 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 [Pn] 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:
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.
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 [Pn] along the path of the propagating wave. As such, the second derivative of a mass particle [mp] 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  with Newton’s 2nd law of motion:
We might wish to do a quick sanity check of  by comparing the units:
Based on the perceived consistency of units in , 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 [mp], 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:
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 [Ep=mgh], if we assume that [x=λ], [y=a] and [z=L]:
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:
Even so, we can see that  and  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.
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 [Ep=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:
In essence, the initial potential energy input into the wave system, as a whole, represents the total energy [Et], 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 [Et], which now exists as both kinetic energy [Ek] and potential energy [Ep] being distributed over an ever-expanding area through a succession of waves over time. By way of a visualisation of the total energy [Et] of the wave, the diagram below focuses on a single particle [Pn] 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 [Pn] that is lifted by the expanding wave. At time [t=1], the particle [Pn] has maximum potential energy [Ep]. As the wave peak passes, the particle [Pn] falls back down, but in so doing gains maximum kinetic energy [Ek] at [t=2]. As such, some fraction of the total energy is retained by particle [Pn] 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 [Ep] and kinetic [Ek] 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 [Pn] 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.