The Nature of Waves
In contrast to the simplicity of the framework that has outlined some of the basic equations associated with a broad range of mechanical waves, the real nature of waves can be very complex, as indicated in the picture right. Therefore, it is obvious that a lot of simplifying assumptions have been made about the nature of waves, which do not always consist of a single frequency, i.e. a plane wave. Equally, most waves do not conform to the simplicity of a sinusoidal shape and, as such, we need to expand the discussion to include the composition of waves in terms of superposition and harmonic waves. We also need to be a little more explicit about how energy is associated with a mechanical wave and how it transports this energy without the wholesale movement of matter along with the wave. However, let us start with a statement about energy and the general nature of a mechanical wave in the form of a pulse:
“Putting a lot of energy into a transverse pulse will not affect the wavelength, the frequency or the speed of the pulse. The energy imparted to a pulse will only affect the amplitude of that pulse"
One of the interesting things about wave mechanics is reconciling the theory to real-world examples. So far we have mainly discussed the basic physics of mechanical waves in terms of Simple Harmonic Motion, but even so, we have still been able to introduce the key ideas of wave physics in terms of the following basic wave attributes:
- Energy [E] and amplitude [A]
- Frequency [f], wavelength [λ] and propagation velocity [v]
As such, we have developed the idea that a wave has a frequency and wavelength, which defines the propagation velocity of a travelling wave in time [t] and space [xyz], i.e. [v= fλ]. We have also touched on the idea of energy, which in the case of a mechanical waves is said to be always proportional to the square of its amplitude. However, we have not really considered what determines these parameters in the physical world, i.e.
What defines the initial wave energy?
What is the compositional structure of a wave?
How does this affect frequency [f], wavelength [λ] and propagation velocity [v]?
And what is the importance of the propagating medium in all this?
In practice, mechanical waves can be very complex, because the nature of these waves is so heavily dependent on the media through which they propagate. For example, the media can be as diverse as a length of string or the surface of a pond or a volume of air and while we might align these examples to our earlier introduction; each may come in a number of permutations:
- The string can be made of elastic or wire,
- The surface wave may be associated with deep or shallow water
- The spherical wave may depend on the actual gas mixture.
In this respect, the nature of electromagnetic waves, e.g. light,
can often appear to be much more straightforward in certain respects,
at least, initially. For example, a light wave in the form of a photon
can be sourced by an atomic transition, i.e.
the Bohr model, which imparts a defined
quantum of energy [E]. This energy, unlike a mechanical wave, determines
the frequency [f] via the simple relationship [E=hf]. Equally, we known
that the propagation velocity of light, in vacuum, is always denoted
by the value [c] and therefore we would seemingly have 2 out of the
3 parameters required by our basic wave equation [v=fλ]. We could also
be a little more technical and say that the propagation speed of light
is actually determined by the permittivity and permeability of the vacuum,
but such details are the topic of another discussion. So, clearly, we
still have a number of general issues to try and resolve in respect
to mechanical waves, but first we will try to introduce some more details
regarding wave superposition, as we have touched on it at several points,
but have not, as yet, explained its real nature. In many ways, the process
of superposition is one of the most fundamental differences between
the perceived nature of matter and waves.