Longitudinal Waves

On page 1, we looked at transverse waves using a slinky animation. In contrast, some waves (such as sound) are longitudinal. In a longitudinal wave the particles vibrate parallel to the direction the wave is travelling in...not at 90 degrees as for waves on a string.

A longitudinal wave can be shown using a slinky spring; have a look at the animation below:

The information content in this wave comes down to the question 'how close together are the coils of the spring?'. At rest, the coils are a certain distance apart, and when the end of the slinky is shaken (along its length) then a compressed region travels along. The red line shows graphically 'how compressed' the slinky is - compression is on our y-axis, and distance on our x-axis.

Like waves on a string, it helps to slow things right down when looking at longitudinal waves on a slinky. Have a look at the video here:

The wave loses power as it travels from left to right, partly because the strings we suspended the slinky from absorb some of the wave energy. However, you shold still be able to see the wave reach the end of the spring, and if you look carefully you'll notice a small reflection where the wave bounces from the end and starts to travel from right to left. If you're really patient and continue to watch, you'll see that after some time things get quite complicated, and that different parts of the spring appear to be streched (science-speak: rarefied) or compressed at different points in time.

The animation below shows a particle model of a longitudinal wave - it could for instance represent a bunch of air molecules in the presence of a sound wave. The air molecules vibrate in the direction of wave travel and form a series of compressions (high pressure) and rarefactions (low pressure), where the molecules are squashed together and pulled apart respectively.

Longitudinal wave

Sticking with the sound-wave example, we might be interested in 'how loud' such a wave is. Loudness has to do with ampltitude or intensity - 'how compressed / how rarefied' the air molecules are. This is often measured using the decibel scale.

Decibel Scale

The human ear in capable of hearing very quiet (low intensity) sounds and extremely loud (high intensity) sounds. The ratio of intensities between 'silence' and 'oooow, that hurts my ears' is about 1:100 million million. To make a sound 'twice as loud', you need to mulitply its intensity by about 10...so an intensity of 1,000 is twice as loud as an intensity of 100, but half as loud as an intensity of 10,000. (The units of acoustic intensity are watts per square meter or W m-2).

It makes things easier if a logarithmic scale is used, called the decibel scale. In decibel terms, a doubling in loudness corresponds to an increase in 10dB - it doesn't matter whether that increase is from 10dB to 20dB or 100dB to 110dB. How does this work? Let's see...

The logarithmic scale

Linear and Log Graphs

In the graphs above, imagine the x-axis represents the percieved loudness of a sound, and the y-axis represents the acoustic intensity needed to create that loudness. Our '10x' rule means that as the overall level increases, we need more and more intensity to get small changes in loudness. On the left-hand graph, where intensity is plotted on a linear ( W m-2) scale, this relationship is clear. On the right-hand graph, where intensity is plotted on a logarithmic (dB) scale, the curve becomes a straight line.

To see why this is, we need to get familiar with the idea of a logarithm. Just about every piece of audio equipment (microphones, loudspeakers, sound cards, amplifiers, mixers, etc) will have specifications expressed logarithmically (i.e. in dBs).

The idea of logarithms is fairly straightforward (even though at first they can look like a mind-bender!); they are simply a way of describing numbers which vary by very large amounts, as numbers which vary by relatively small amounts. Have a look at this:

The quietest sound the average person can hear has an intensity of about 1 picowatt per square metre (1x10 -12 W m-2), and this is defined as the reference intensity level which is equivalent to 0 decibels (0dB).

The decibel scale

Intensity levels (IL) are measured in bels (B)

IL = log10 intensity of sound / intensity at threshold of hearing (1pW m-2) = log10I / I0

1 decibel (dB) = 10 bels (B)

IL (dB) = 10log10I / I0


A sound has intensity 1 W m-2. What is the intensity level in dB?

IL (dB) = 10log10 (1 W m-2 / 10-12 W m-2) = 120dB

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