In this introductory section, the goal is simply to provide some examples of mechanical waves that propagate in 1, 2 and 3-dimensions. While we live in a 3-D world, we probably have a better intuitive understanding of the idea of 2-D waves as seen expanding out across the surface of a pond. However, 1-D and 3-D waves also highlight important concepts that we will need to reference in later sections. So, to start, we shall repeat the basic definition of a mechanical wave:
A mechanical wave is a mechanism that transports energy ‘independent` of matter
However, this somewhat general definition needs to be clarified because
while there is no wholesale movement of the particles in the medium,
a mechanical wave still depends on localised vibration of particles,
such that the energy of the wave can be transferred through a media
of some description. As a consequence
of this dependency on the vibration of particles within a given media,
there are many forms of mechanical waves:
- 1, 2 and 3-dimensional configurations
- Traverse and longitudinal waves
- Travelling or standing waves
We might initially start with a visual image of a surface wave moving across the ocean and crashing on some distant shore. However, it is not so obvious how the energy has been transported without the wholesale movement of the ocean or even the water within the wave. Equally, we need to recognise that this example of the movement of a mechanical wave is essentially restricted just 2-dimensions. Therefore, we should try to quickly extend this visualisation to encompass examples covering all 3 dimensions, e.g.
- A 1D wave on a guitar string
- A 2D wave caused by dropping a pebble in a pond
- A 3D wave radiating a sound wave in all directions
While this might provide some initial visual examples of waves moving spatially in 1, 2 or 3-dimensions, with respect to time, we might also like to note the additional complexity of theses waves moving in different modes, i.e.
- Traverse Waves:
In this mode, the amplitude of the wave is perpendicular to the direction of wave propagation.
- Longitudinal Waves:
In this mode, the amplitude of the wave is parallel to the direction of wave propagation.
We could also add to this complexity by saying that waves can be a composite of both transverse and longitudinal forms of propagation, which may be inwards or outwards from a nodal centre. Waves can also be confined within a bounded system, e.g. a finite length of a guitar string, or essentially unbounded, e.g. an ocean wave. As such, there is much scope for visualising an almost infinite number of examples of mechanical waves. However, for the purposes of this introduction, we shall have to restrict the scope to a limited number of examples of waves moving spatially in 1, 2 and 3-dimensions with time.
Difference between EM and Mechanical Waves
This sub-section tries to summarise some key differences between EM waves and mechanical waves. Earlier discussions of waves have been primarily focused on the wave-particle duality of photons, but it is important to realise that most EM wave descriptions do not apply to mechanical waves:
- The energy equation for the photon [E=hf] is not subject to
any decay with distance, which is not the case for mechanical waves
that can fade like the ripples on a pond. The exact reasons will
be addressed in the next section.
- The equation [E=hf] also suggests that frequency [f] is directly
proportional to the energy of the wave. While this is true for an
EM wave, this direct equivalence does not necessarily apply to mechanical
- We have assumed that the propagation velocity [v] for a photon
is always [c], at least in vacuum, from which we can calculate the
wavelength [λ=v/f]. Again, caveats apply to mechanical waves.
- While [λ=v/f] can be transposed to the form [v=fλ], it doesn't explain the physical dependencies between these parameters for reasons outlined below.
Having established these points of difference between EM and mechanical waves, let us try to first summarise some of the key parameters in connection with mechanical waves:
- Energy [E] is proportional to the square of the amplitude.
- Frequency [f] is described in terms of its own function of energy.
- Velocity [v] is a function of the propagating media
- Wavelength [λ] equals velocity divided by frequency
As indicated, the energy of a mechanical wave is a function of its amplitude squared. While the frequency [f] of the mechanical wave may still be linked to the energy, the exact relationship is dependent on the nature of the wave, i.e. it cannot be expressed in terms of E=hf. Likewise, the velocity [v] of a mechanical wave depends on the physical characteristics of the media through which it is propagating, i.e. a wave on a piece of string depends on the string, while a water wave depends on the water. Therefore, the wavelength always results from the determination of frequency [f=1/t] and velocity [v=x/t] such that:
Therefore, wavelength is simply the result of the distance [x] traversed
in one cycle at a given frequency, while propagating at velocity [v],
as determined by the media. Again, the specific details associated with
these generalisations will be discussed further in the context of specific
examples to be outlined throughout this section.