Basic Wave Physics

1Hopefully, the previous examples of 1, 2 and 3-dimensional waves have illustrated that waves can come in many forms. However, while these examples cover all 3 dimensions and introduce the variants of standing and travelling waves plus traverse and longitudinal waves, all appeared to have an associated sinusoidal wave attribute. Therefore, the idea of a sinusoidal waveform, which can be explained in terms of 'Simple Harmonic Motion (SHM)', seems to be an appropriate starting point for this overview. As such, the goal is simply to introduce some of the basic concepts and the attributes associated with sinusoidal waveforms. When starting to review any text discussing wave physics, you quickly see the appearance of the mathematical symbols (π, 2π) occurring in many wave equations. While, to many, this may be an obvious geometric association, the animation tries to show how a sinusoidal wave motion can be linked to a circular motion, even though this motion does not physically exist in any of the example waves discussed so far. This is because the link between sinusoidal waveforms and circular geometry is simply reflecting a mathematical, rather than physical, relationship between angles and [π]; where angles can be expressed in degrees [0..360] or radians and [π] helps define the ratio of the circumference of a circle to its radius [2πr]:

[1]     1 

Alternatively, we can say that 1 radian equates to the angle subtended by an arc of length [r] on the circumference of a circle of radius [r]. Therefore, as the trackerball rotates on the circumference of the circle it subtends an angle [θ] that can be expressed in terms of degrees or [π], as illustrated by the x-axis of the graph. On this basis, we can also proceed to point out a number of other basic relationships that extend out from these two different perspectives; but first, we need to define the propagation velocity of the trackerball on the sinusoidal waveform:

[2]     2

While equation [2] expresses the variation of the parameters that can be associated with propagation velocity [v], this quantity still remains physically dependent on the media through which the mechanical wave propagates. However, [2] also helps us to see that a relationship exists between the linear propagation velocity [v] along the x-axis and the idea of an angular velocity [ω] around the circle, which can be extended as follows:

[3]      3

In this expression, the wavelength [λ] corresponds to the circumference of the circle, i.e. 2πr or a 360° rotation, which then leads to the angular velocity [ω=2πf]. This expression will also appear in many wave equations, even though there may appear to be little evidence supporting the physical existence of any angular velocity. In most cases, such expressions appear to adopt [ω] as a matter of mathematical convenience in conjunction with the conceptual model being outlined by the animation. One final point that might be useful to highlight is the division of the sinusoidal graph into the ‘time domain’ and the ‘space domain’. These domains are making reference to the description of a wave in terms of its frequency, i.e. time domain, or wavelength, i.e. space domain, where the two are connected via [2], which will ultimately lead to an equation of the form:

[4]      4

At this stage, [a, b, d] can be considered as constants, while [A0, A] represent the initial amplitude of the wave plus its projected amplitude at some point in space [x] or time [t]. As such, [4] is a generalised equation that expresses some form of sinusoidal wave nature and, in the non-relativistic world of mechanical waves, we might assert that space and time remain distinct concepts. As such, it is worth reflecting, in advance of the derivation of [4], as to the implication of the sine function being dependent on both space [x] and time [t]. In practice, we can say that a wave either has a certain amplitude [A] at a given point in space [x] or time [t]. Therefore, the inference of [x-t] in [4] is that if we know the amplitude A(x) at some position [x], then we can calculate its amplitude A(x,t) at some time [t] before or after this point in space. Alternatively, if we know the amplitude A(t) at some time [t], then we can calculate its amplitude A(t,x) at some offset [x] relative to this point in time. However, as we shall see, it is possible to describe a waveform that could be said to only oscillate in time, not space. This waveform conforms to the classical description of Simple Harmonic Motion (SHM), which can be described in terms of the up-down motion of a weight on a spring, which is the subject of the next discussion.

Footnote:

As a somewhat tangential point, the animation above shows the trackerball moving with constant velocity around the circle. However, it is worth noting that to any point in space, the distance to the trackerball is changing at a variable rate, i.e.  it is subject to acceleration. This acceleration is reflected in the rate of change of position of the trackerball on the sinsoidal curve. This implied acceleration is the reason why a charged electron was assumed to radiate energy in the original atomic model and in many ways remains an issue of debate even today. In addition, this type of acceleration to a fixed point in space may have some relevance in connection with the rotation of galaxies and Mach's principle to be discussed in a later section.