Standing Waves

Clearly, one of the concepts that appears to require further consideration within the WSM model is the idea of a standing wave. Therefore, within this initial review of general concepts, it might be worth establishing some of the basic attributes of a standing wave. From a historical aspect, the idea of a standing wave has been recognised for a long time, as it is key property of many musical instruments, although the concept was possibly more understood in terms of the practical skill of an instrument maker.  In this context, standing waves may have only been 'studied' in terms of the harmonics of a violin string, e.g.

In the diagram above, each of the various harmonics relates to a standing wave formed between fixed end-points or its ‘boundary conditions’. The points of zero amplitude are called ‘nodes’, while the points of maximum amplitude are called ‘anti-nodes’. If we define a length [L] to be the distance between the endpoints, we can then establish the relationship between the length [L] and the wavelength [λ] of the standing waves:

[1]     

For example, in the case of the 3rd harmonic, we can see that the wavelength [λ] must be a fraction of the length [L] as it contains 3 antinodes, such that the wavelength is [λ=2/3L]. While this description does not explain how standing waves are formed, it is normally introduced in terms of the superposition of two waves, with the same amplitude [A], frequency [f] and propagation velocity [c], but travelling in the opposite directions. As before, we might visualise this situation as follows:

As the two waves pass through the same point in space-time, the amplitudes of these waves are combined, i.e. in superposition, such that they create amplitudes with a maximum and minimum, i.e. the antinodes and node.

In the context of the current description, it is the reflecting surface in the diagram above that causes the wave to reverse direction, which is then ‘out-of-phase’  by 180° to the incident wave. In the resulting standing wave, the node amplitude is zero, i.e. A=0, while the anti-node amplitude is doubled, i.e. A=2A0. While having outline the attributes of a standing wave in terms of its superposition amplitude [A] and its wavelength [λ], we have not yet mentioned its frequency [f]:

[2]     

In some respects, [2] requires a little interpretation because it suggests that the frequency [f] of a ‘static’ standing wave is a function of a propagation velocity [c]. Clearly, in this case, the relationship [c=fλ] is linked to the travelling waves that form the standing waves into which [1] is substituted. However, we might also appreciate that the frequency [f] of any wave is a reflection of an associated energy [E], which in-turn may tell us something about the medium in which the wave exists. As such, this initial description of standing waves may need to be expanded to consider a ‘system’ in relative motion with respect to the wave propagation medium.

Note: At this point, the discussion is orientated towards mechanical waves, which propagate through a medium of some sort, e.g. sound waves through air.

In a mechanical wave system, the waves might be described as propagating through the medium at a constant velocity [c], which is a property of the medium itself. For the purpose of this discussion, we shall also assume this medium is non-dispersive, i.e. all waves of different frequencies propagate at the same velocity [c]. However, within this description, we might realise that the sources of the travelling waves, both IN and OUT, might have a relative velocity [v] with respect to the medium. In this case, the perceived velocity of the IN and OUT waves may depend on the reference frame [RF] of the observer determining the wavelength. We might initially visualise this situation in terms of the Doppler effect, where the frequency of a moving train whistle is heard to change pitch from the perspective of somebody standing on the station platform.  However, the following diagram may be a better basis on  which to anchor the next part of the discussion. 

What the diagram is illustrating is that the waves are compressed in the direction of motion and expanded in the opposite direction, where [β=v/c] is the ratio of the relative velocity [v] with respect to the propagation velocity [c] of the wave through some given media. Using the diagram above, we might then characterise the  case of two travelling waves propagating with velocity [c] in opposite directions along the x-axis, where the [±] sign reflects the direction of the relative velocity [v]:

[3]       

However, the equations in [3] are representative of the travelling waves, not the standing wave formed in superposition of these waves. The formula used to determine the distance [L] between standing wave nodes can be characterise as follows:

[4]       

So, based on this brief introduction, we might summarise the effect as the result of the ‘sum’ or ‘superposition’ of two waves propagating in opposite directions, although this description does not necessarily explain a resonant standing wave, which can persist in the absence of travelling waves. Standing waves are also often describe in terms of an interference pattern. However, simply assigning a ‘name’ to this effect does not really adequately explain its cause or scope within a wider description of what we see as the ‘natural world’ . So let us try to outline some of the wider implications of a ‘standing waves ’ in nature, which may typically be formed when a wave is reflected. For when an incident wave is met by a reflected wave, standing waves will typically be created, which is true of hydro-waves, acoustic waves and electromagnetic waves. So, in this wider context, the description of a standing wave also emerges in the field of radio and electrical engineering, while chemistry can frequently describe an atomic structure in terms of a lattice of standing waves, where the nodes of these waves align to the perceived positions of the atoms and molecules. If so, then the idea of a standing wave may require some further consideration within the review of the WSM model, but for the moment we shall simply try to summarise some of the key features:

  • Standing waves can be created when incident and reflected waves superimpose.

  • The amplitude of standing waves can remain static in space, e.g. the x-axis, while its amplitude oscillates in time.

  • The point in space where the net amplitude is always zero is called a standing wave node.
     
  • The areas between the nodes where the net amplitude is above zero are called the antinodes of the standing wave.

  • The standing wave length [L] is a distance between the two nodes, which is a function of the incident and reflected wavelengths.

  • Typically, a standing wave propagates no energy, although it can contain energy between its nodes. This condition occurs when the incident and reflected wave transport equal energy in opposite directions.

  • However, it is highlighted that, within the WSM model, the standing wave structure itself can move in space and time, although the details need further clarification.   

So, these and other attributes of standing waves will need to be expanded throughout the review of the WSM model. For the boundary conditions within the WSM model that define the length [L] will not necessarily correspond to two fixed points in space, but rather become defined by the position of two ‘particles’ or ‘wave-centres’ that can move in space and time. As the description of standing wave structure is expanded, two other attributes will need further considerations and questions:

What energy is associated with a standing wave?
Can a standing wave be self-sustaining in the absence of any input waves?

For while it has been implied that standing waves do not transport energy, they may contain energy within two nodes. In terms of mechanical waves that propagate through some form of physical medium, the energy associated with the standing wave might be said to be ‘trapped’ between two nodes, as defined by simple harmonic motion. Within this description, the energy oscillates between maxima and minima potential and kinetic energy and, in this general context, we might describe the amplitude of the wave being proportional to its potential energy, while the rate of change of the amplitude is reflective of its kinetic energy. We might also recognise that within a lossless system, the energy within the standing wave nodes could conceptually oscillate between potential and kinetic energy forever in a self-sustaining manner. Such standing wave oscillations can also be described in terms of a ‘resonance’, which  is linked to the natural frequency of a system. These systems support a natural resonance frequency, including its harmonic frequencies, because these frequencies correspond to an equilibrium energy state of the system.