Click here to purchase the entire book in PDF format. Amplitude vs. Distance There is an obvious relationship between amplitude and distance - they are inversly proportional. That is to say, the farther away you get, the lower the amplitude. Why? Let's go back to throwing rocks into a lake. You throw in the rock and it produces a wave in the water. This wave can be considered as a manifestation of an energy transfer from the rock to the water. All of the energy is given to the wave from the rock at the moment of impact - after that, the wave maintains (theoretically) that energy. The important thing to notice, though, is that the wave expands as it travels out into the lake. Its circumference gets bigger as it travels horizontally (as its radius gets bigger...) Therefore the wave is ``longer'' (if you're measuring around the circumference). The total amount of energy in the wave, however, has not changed (actually it has gotten a little smaller due to friction, but we're ignoring that effect...) therefore the same amount of energy has to be shared across a longer wavefront. This causes the height (and depth) of the wave to shrink as it expands. What's the mathematical relationship between the increasing circumference and the increasing radius? Well, the radius is travelling at the constant speed, determined by the density of the water and gravity and other things like the colour of your left shoe... We know from high school that the circumference is equal to the radius multiplied by about 6.28 (also known as $2\pi$). The graph in Figure 3.14 shows the relationship between the radius and the circumference. You can see that the latter grows much more quickly than the former. What this means is that as the radius slowly expands out from the point of impact, the energy is getting shared between a ``length'' of the wave that is growing far faster (note that, if we double the radius, we double the circumference). Figure 3.14: The relationship between the circumference of a circle and its radius. The same holds true with pressure waves expanding from a loudspeaker into a room. The only real difference is that the energy is expanding into 3 dimensions rather than 2, so the surface area of the spherical wavefront (the 3-D version of the circumference of the circular wave on the lake...) increases much more rapidly than the 2-dimensional counterpart. The equation used to find the surface of a sphere is $4 \pi R^2$ where $R$ is the radius. As you can see in Figure 3.15, the surface area of the sphere is already at 1200 units squared when the radius has only expanded to 10 units. The result of this in real life is that the energy appears to be dissipating at a rate of 6.02 dB per doubling of distance. (When we double the radius, we increase the surface area of the sphere fourfold so the intensity drops to 25% of the original level.) Of course, all of this assumes that the wavefront doesn't hit anything like a wall or the floor or you... Figure 3.15: The relationship between the surface area of a sphere and its radius.