# 3-Dimensional Waves

Our last example will try to represent a 3-dimensional outbound spherical wave, i.e. one that is expanding in all 3 [x,y,z] directions. In the case of the previous 2D traverse surface waves, the amplitude was considered to exist outside 2D [x,y] space, i.e. the height of wave was represented in the [z] plane]. As such, it would appear that any physical perception of amplitude associated with a 3D transverse wave would have to exist in a higher dimension, which clearly presents a few problems when it comes to illustrating such a wave. Therefore, the simpler option has been taken to present the 3D example in terms of a longitudinal wave, which also provides the opportunity to highlight the general mechanics associated with longitudinal wave as opposed to transverse waves.

So, the first animation illustrates the basic expansion of a wavefront
expanding in all 3 [x,y,z] planes. While this could be described in
terms of an explosion, it is possibly more intuitive to initially consider
the animation in terms of a sound wave expanding in all directions from
its source. As such, the expanding sphere of sound might be considered to be
reducing in volume with
radial distance [r], as this volume can be related to the
energy [E∝A^{2}] associated with the wave at any point
[x,y,z]. The actual energy at any point is dependent on 2 factors:

**The ability of the mechanical media to transmit sound.**

Sound waves travel well in water, but poorly in something like cotton wool. This aspect relates to the propagation velocity of the media, which can be generally expressed as:

[1]

**The distribution of energy as a function of radius [r]**.

This issue is similar to the energy distribution issue previously described for the 2D surface wave, although the results are slightly different, as explained below.

We can assume that the sound wave was generated at the centre of
the expanding sphere with an initial energy [E_{0}] and that
this energy has to be distributed over an increasing surface area corresponding
to the surface area of the sphere of radius [r]. However, the area of
this sphere increases based on the formula [4πr^{2}] and
therefore given that energy is proportional to the square of the amplitude:

[2]

Again, the reduction in energy [E], as implied by [2a], does
not reflect any energy loss, due to friction or heat loss, simply the
distribution of the original energy over a greater spherical surface
area.
The proportional relationship [E ∝ A^{2}] still stands,
but the energy of the 3D wave now reduces by the reciprocal of [r^{2}],
while the amplitude would reduce by the reciprocal of [r].

So far, we have not really said anything about the nature of a longitudinal
wave in comparison to a traverse wave. Hopefully, the second animation
above helps provide some initial visualisation of this type of wave.
The first point to clarify is that while this animation may appear to
be showing a traversing sine wave, this is only for comparative reference
against which the vertical line are oscillating backwards and forwards.
As such, we might visualise a sound wave as a longitudinal motion of
the particles in the air, i.e. the vertical lines, comprising of regions
where the air particles are compressed together and other regions where
the air particles are spread apart, i.e. as illustrated in the bottom
section of the animation. These regions are known as compressions and
rarefactions, where compressions correspond to regions of high pressure
and rarefactions are regions of low pressure. As such, the wavelength
of a longitudinal wave is the distance of one complete wave cycle of
compression and rarefaction, as indicated by the sine wave propagating
outwards. The second point to note is that this animation is not showing
any decay in amplitude, as it is primarily focused on the propagation
mechanics within a longitudinal wave. However, to correlate the two
animations, imagine that the second animation exists within a cone with
its apex at the centre of the previous expanding sphere, where the energy
at the apex gets spread across the expanding area of the base of the
cone.