Wave Propagation

a1

The goal of this discussion is to continue the expansion of the SHM wave function to encompass a wave that propagates in both time [t] and space [x]. In the previous discussion, simple harmonic motion was linked to an oscillating mass [m] on a spring, which had an associated wave function with respect to time only. However, extending the wave function to a mechanical wave that has a propagation velocity [v] in space, as well as time, needs some introduction, which it is hoped the first animation helps to address. This animation is based on the same pebble-in-a-pond model previously used to describe a 2-dimensional wave, which was initially rendered to a 3-D perspective. However, the current animation has reduced the horizontal scale so that the origin of the graph is now set at the centre of the original expanding circular wave, effectively being viewed as a vertical slice through the wave. As such, the point the pebble entered the surface of the water is marked by the red trackerball, which remains stationary at the centre of an expanding wave front. In this context, the stationary ball is trying to highlight that the wave propagation only involves the transport of energy, not any wholesale shift of the water through which the wave propagates. The second reference point, shown as a black dot, simply marks an arbitrary point on the surface, which is effectively offset in both time [t] and space [x] from the origin. What the animation is trying to show is that it takes a finite amount of time for the initial wave displacement to propagate outwards as a mechanical wave. However, it can also be seen that the energy associated with the amplitude of the original wave does not get propagated in just one wave cycle, as the centre continues to oscillate over a number of wave cycles. Therefore, the following equations are only correct in terms of the definition of the amount of energy radiated per cycle [Er], which will be some fraction of the original potential energy [Ep] associated with the volume of water displaced at the centre.

[1]      

As such, the equations in [1] reflects a theoretical maximum energy dissipating over the expanding circumference of the circular wave as the radius [r] increases. However, it also does not account any damping effects due to the nature of the propagating media. While it was felt important to start this discussion with some description of the actual complexity that can exist, even in what appears to be a simple example, it does not serve the purpose of the overview of the basic wave physics to overload the model with too much detail, as outlined, when deriving the basic equations. Therefore, the next animation rationalises the model so that we see an infinite travelling wave moving from left to right with a propagation velocity [v], which loses no energy due the effect of [1] or the damping effect of the media. As such, the model appears to align to the earlier SHM spring model except that we are now assuming that the wave propagates in both the time and space domains.

a2In order to be able to use the previous SHM model as a comparative source of reference for the wave functions to be developed within this discussion, the cosine orientation of the first animation is switched to a sine function. Again, all 3 mathematical components are being shown as reference, i.e. circular rotation, SHM oscillation and sinusoidal propagation. Of course, the main point now under investigation is in connection to the propagation of the sinusoidal wave form in space [x], as well as time [t]. As such, we can start by defining 1 cycle of the wave in terms of its space wavelength [λ=x] or its time period or frequency [P=1/f]. Clearly the sinusoidal oscillations in time and space have to be synchronised, as will be finally shown in [12], but first we need to rationalise the relationship between time and space that links these two perspectives. This relationship can be seen to be dependent on the propagation velocity [v] of the wave, which in-turn is linked to the rate of change of distance [x] in time [t]:

[2]     2

It is stated, yet again, that while the propagation velocity [v] is indeed a function of wavelength [λ] and frequency [f], as shown in [2], it is the attributes of the media that actually determines the velocity of a mechanical wave through any given media. Based on the premise that the sine wave is propagating in both time [t] and space [x], there is an implication that any prototype sinusoidal wave function must also have 2 forms:

[3]      At = A0* sin(t)

[4]      Ax = A0 * sin(x)

Previously, we clarified sin(t) as shorthand for the angular displacement [θ], which in the time domain could be rationalised as follows:

[5]    5    

Therefore, when substituting [5] back into [3], we get a general wave function for the amplitude [A] in the time domain:

[6]      At = A0 * sin(wt)                 Time Domain

Of course, all [5] is doing is specifying the conceptual angular rotation [θ] as a ratio of any given time [t] to the time period [P] for one cycle. As such, we can do the same with distance [x] with respect to wavelength [λ], which is the distance of one cycle in the space domain at a given propagating velocity [v]:

[7]    7

Note: it is recognised that the definition of [k] might cause some confusion at this point because it might be thought to be the elasticity constant [k] previously used in connection with Hooke’s Law. Unfortunately, the definition of [k] in [7] is the accepted symbol referring to the angular wave number [k]. While all further references will be implicitly linked to this new definition, the new definition will be highlighted in bold, i.e. [k], for the rest of this page.

If we substitute the result of [7] back into [4], we get a general wave function for the amplitude [A] in the space domain:

[8]      Ax= A0 sin(kx)                   Space Domain

We have previously determined the 1st and 2nd derivatives for [6], i.e. the time domain; therefore this result can simply be cross-referenced and presented for continuity within this discussion:

[9]     9   

Of course, the same logic can be applied to [8]:

[10]    10   

It can be seen that both [9] and [10] lead to a definition of [A] that can be equated to each other as follows:

[11]    11   

On the basis of [5] and [6], we can substitute for both [k] and [ω], which allows [11] to be rationalised to the form in [12]:

[12]    12

Equation [12] is the 1-dimensional equation of wave motion for a plane wave, i.e. constant frequency, which allows the transposition of wave motion in terms of both time [t] and space [x] via the propagation velocity [v]. As such, the wave equation is a partial differential equation that describes the evolution of a wave over time, in a given medium, where the wave propagates at constant velocity [v] independent of wavelength [λ] and amplitude [A]. However, we can rationalise [12] back to the more simplistic form originally present in [2]:

[13]    13   

However, the implication and interpretations of the various wave equations developed, i.e. [6], [8] and [12], will be discussed in the next section.