# Basic Wave Physics

Hopefully,
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]

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]

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]

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]

At this stage, [a, b, d] can be considered as constants, while [A_{0,
}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 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.