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Quantifying and using sound - Physics narrative
Physics Narrative for 11-14
A Physics Narrative presents a storyline, showing a coherent path through a topic. The storyline developed here provides a series of coherent and rigorous explanations, while also providing insights into the teaching and learning challenges. It is aimed at teachers but at a level that could be used with students.
It is constructed from various kinds of nuggets: an introduction to the topic; sequenced expositions (comprehensive descriptions and explanations of an idea within this topic); and, sometimes optional extensions (those providing more information, and those taking you more deeply into the subject).
The ideas outlined within this subtopic include:
- Frequency and amplitude as fundamental
- Pitch and frequency
- Loudness and amplitude
- Choosing representations
Range of sounds
The range of sounds that humans are able to hear is impressively large, taking us from the loud roar of an aircraft as it makes its final approach on landing to the quiet murmur of the wind as it passes through the trees. Alternatively we might think in terms of moving from the low pitched rumble of distant traffic to the high-pitched screech of the dentist's drill.
We can consider these two ranges of our hearing as forming two sides of a
- From very quiet to extremely loud.
- From a low-pitched rumble to a high-pitched squeak.
Pitch and frequency
Hearing different notes
People vary in their ability to detect the pitch of a sound. For example, in tuning a guitar, some are able to distinguish easily between two notes of similar pitch while others can't hear the difference. A more objective measure than pitch is frequency, measured in hertz (Hz).
The frequency of any vibration is defined as the number of complete vibrations made each second: the frequency (hertz) is equal to the number of vibrations per second.
Sometimes a complete vibration is referred to as a cycle: the frequency (hertz) is equal to the number of cycles per second.
You can picture one vibration or cycle of the speaker cone being completed as it starts from its most forward position, moves backwards, and then moves forwards to return to its original position.
The frequency value not only specifies the number of complete vibrations made by the source each second, but also the number of regions of high (or low) density produced in the medium each second. In other words, it gives the frequency of the actual sound.
High pitch corresponds to high frequency, low pitch to low frequency. There is a whole spectrum of sounds arranged along the frequency axis.
More on frequency
Quite often the numbers get large, and the frequency is referred to in terms of kilohertz (kHz): 1000 hertz is the same as 1 kilohertz.
For example, a hi-fi loudspeaker might have a range of 20 hertz to 20 kilohertz and so be capable of producing sounds over our whole range of hearing.
In 1939 musicians moved to an agreement on standard pitch. Middle C (called that because it is a note around the middle of the piano keyboard) is 256 Hz and concert A (the note the orchestra tunes to before the show begins) is 440 Hz.
The frequency doubles for every octave upwards.
Period and frequency
The relationship between period and frequency
To find the frequency you simply count the number of vibrations each second. An alternative way to do the counting is to find out how long one complete vibration takes and then to calculate how many of these you can get in one second.
The time for one complete vibration is called the period (T) and is measured in seconds.
For example, if the period of a vibration is 0.1 second (one vibration takes 0.1 second), the frequency of the vibration is 10 vibrations per second or 10 hertz.
- If the period is large the frequency is low (relatively few vibrations each second).
- If the period is small the frequency is high (lots of vibrations each second).
More formally, the frequency is inversely proportional to the period. If you double the period, the frequency is halved.
frequency = 1period
You can also write this out in symbols:
f = 1T
Loudness and amplitude
What does the loudness of sound depend on?
Sounds vary not only in terms of the pitch or frequency, but also in their relative loudness.
As you might expect, louder sounds are produced by larger vibrations of the source. If the loudspeaker cone moves backwards and forwards over a greater distance, a bigger disturbance is produced in the air and a louder sound results.
As with pitch, the loudness of a sound is a subjective measure – it depends on the person. You may have an elderly relative who listens to the TV at what seems to be ear-shattering volume to you, and yet is comfortable for their ears. The same sound is judged to be of different loudness by two people.
A more objective measurement of loudness is the amplitude of the vibration.
The human ear is sensitive to sounds over a huge range of loudness. For example, the loudest sound that the ear can safely detect without suffering any physical damage is more that one billion times more intense than the threshold of hearing (where sounds can just be heard).
The ear responds to sounds of different loudness in such a way that huge increases in amplitude are registered as small increases in loudness.
The decibel scale
To match this way in which the ear responds, the relative loudness of sounds is usually measured in decibels.
The decibel scale, which is a logarithmic scale, stretches from 0 dB at the threshold of hearing to 140 dB at the threshold of pain (values vary, as this is subjective – pain is not a precisely defined experience). Each time you go up one decibel, the loudness of the sound increases by a constant factor and you can just about hear this change.
Logarithmic intensity scales are also used to report on the perceived brightness of light (the sensitivity of the eye is also logarithmic) and for measuring the strength of earthquakes (the Richter scale).
A complicated explanation of the decibel, included here as it may be hard to find elsewhere
The bel was invented by telephone engineers, and is named after Alexander Graham Bell, inventor of the telephone.
The bel unit (a decibel is one tenth of a bel) is derived by comparing the power of one sound with another:
number of bels = log(P1P0)
where P1 and P0 are powers in watts (for sound, that is the energy arriving at the ear per second) and P0 is a reference power value with which P1 is compared.
The logarithmic scale compresses the range of loudness values. Instead of having a loudness scale that stretches from the threshold of hearing to a value that is one billion times greater, the decibel scale goes from 0 dB to 140 dB.
number of decibels = 10 × log(P1P0)
For sound we need some agreed intensity as a reference value. Intensity is power per square metre: this eliminates concerns over the size of the ear. The agreed reference intensity is 1 × 10-12 watt metre-2 at 1000 hertz. So every other sound level is compared with this using the decibel scale. This reference value comes out at 0 dB.
We started describing the loudness of vibrations in terms of amplitudes and now we are talking in terms of intensities – the energy arriving per square metre per second. There is an intermediate step: the energy associated with a vibration depends on the amplitude.
So the connection between the to and fro of the loudspeaker and the measure of the loudness in decibels involves several steps. Knowing about these steps is for your interest, and certainly not for the great majority of pupils at this age.
There's more to sounds...
Hearing different frequencies, all at once
Frequency and amplitude provide a very good start to describing sounds, but most sounds that we hear in daily life consist of more than one frequency.
This is clearly illustrated when you look at the graphic equaliser display, which actually shows the different ranges of frequencies present in the music.
Just as we recognise familiar faces, so too can we spot familiar voices. You hear a voice much as you see a face – it is instantly recognisable. You can certainly recognise the voice if the loudness is altered, perhaps even if the pitch is changed. The collection of amplitudes and frequencies in the voice, even if distorted, give you many clues.
Speed of travel
Echoes, imaging and measurement
Clap your hands not too far from a big wall and you can hear both the original clap and, shortly afterwards, its echo. What is happening here is that the sound takes one path straight from your clapping hands to your ear (a short distance), while another path involves the sound starting off from your hands, reflecting off the wall and then travelling back to your ears (a somewhat greater distance).
In fact, these are just two of the many paths followed by the sound as it travels out in all directions through the surrounding air.
By measuring the total distance travelled (from your hands to the wall and back) and timing the gap between the original clap and you hearing its echo, it is easy enough to get a rough value of the speed of sound in air.
Depending on the accuracy of your timing, you are likely to find a value of about 300 metre / second. In textbooks the value of the speed of sound in air is often quoted as 340 metre / second.
Under normal conditions, this speed remains pretty well constant. However, since the motion of the sound depends on the movement of air particles, the speed of sound does change with temperature because this determines how fast the particles are moving.
If we assume that the speed of sound is 340 m s-1, we have a way of measuring distances. This is used in sonar measurements.
Timing lots of pulses in a cunningly planned pattern can give an ultrasound scan. With really careful and fast processing of these signals you can get a real-time image, of, for example, a baby in a womb.
Bats use ultra-sonic (too high a frequency for us to hear, but good for the bats) pulses to navigate with. It's possible to imagine that they
see the world very differently from us.
Speeds of sound in different media
Physics Narrative for 5-11 11-14
Sounds in solids, liquids and space
Sound can travel through any medium, so long as there are sufficient particles in each cubic metre to allow the to and fro motion of one block of particles to be passed on to their neighbours.
Solids, liquids and gases differ in the arrangements of their particles and in the forces between the particles. So you might expect variations in both the speed and range of sounds travelling through them, since both depend on how the disturbance passes on from one block of particles to the next.
Sounds travel faster and farther in solids. Presumably this fact must have been well known to all of those who put their ear to a rail track to detect whether or not a train was coming.
In liquids sounds also travel over great distances at considerable speed. This is apparent from listening to whales as they communicate with one another. The complexity of the paths which these vibrations follow is exploited by Tom Clancy in his book
The Hunt for Red October, where he makes much of the difficulties involved in interpreting sonar signals while tracking submarines.
In deep space there are only about 3 particles of hydrogen per cubic metre and this is not enough to sustain a sound. Disappointingly the
booms of space movies are a fiction! All explosions in outer space take place in eerie silence. As the
Alien movie tells us:
In space, no-one can hear you scream.
The speed of sound varies in different media. Here are some typical examples: in air, 340 m s-1 ; in water, 1500 m s-1 ; in steel, 6000 m s-1.
All around you there is plenty of evidence that different frequencies of sound travel at the same speed. The very existence of music depends on this. If the notes of differing frequencies, from different instruments, arrived at your ears all at different points in time, the effects would probably be rather unpleasant!
Working with sounds
Sound consists of audible vibrations, which travel from a source to a detector. These can be of a whole range of frequencies and a whole range of amplitudes. We hear these as sounds of different pitches and different loudnesses. Sometimes one sound will be made up of many of these frequencies at once, each with their own amplitude. All of these contribute to the amplitude of the sound, as well as its characteristic quality.
These vibrations travel at one quite constant speed, about 340 metre / second in air, whatever the frequency or amplitude. Sound travels through solids and liquids at higher speeds.