3-Dimensional Waves

2Our 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∝A2] 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]       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 [E0] 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 [4πr2] 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 ∝ A2] still stands, but the energy of the 3D wave now reduces by the reciprocal of [r2], while the amplitude would reduce by the reciprocal of [r].

2

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.