A common bear trap is to mix up pitch and amplitude.
It is possible to have a high pitched sound with low volume.
The Sound of Music |
1. Making Sounds |
This unit is concerned with the physics of music. Music gives most people a great deal of pleasure, even though tastes may vary considerably. (At about age 35, something dreadful happens to pop music.)
At their most basic level instruments are made to sound by vibrating a string, a skin, a lump of metal, or a column of air. Musical notes are determined by particular frequencies that are especially pleasing to the human ear.
An oscillation is a complete to-and-fro movement. If the movement repeats itself at regular intervals, it's called a periodic oscillation. Since the motion is often associated with sounds it is called harmonic motion.
Oscillations make sound through vibrations, which can travel through solids, liquids, and gases. The vibrations cannot travel in a vacuum.
Some definitions:
A common bear trap is to mix up pitch and amplitude.
It is possible to have a high pitched sound with low volume.
Frequency and period are related by a simple formula:
f = 1
T
We can draw a graph of any oscillation. The graph is often called a waveform. The simplest oscillation is sinusoidal, showing a shape identical to a graph of the sine or cosine function.
From this graph we can see that the period of this oscillation is 25 seconds. Two complete cycles have a period of 50 s.
The cathode ray oscilloscope (CRO) connected to a microphone displays a voltage-time graph of a sinusoidal sound wave. The displacement of the diaphragm of the microphone gives out a voltage, so we can say the display is a displacement-time graph.
The CRO is a valuable instrument with which to investigate all sorts of waveforms. The distinctive quality of the sound of a musical instrument is determined by its more complex waveform.
The word phase is used to describe the stage that is reached in an oscillation. If two waves are analysed at the same time, we describe their phase difference. If two waves are perfectly in step they are said to be in phase. The two waves have different amplitudes, but that does not affect their being in phase.
These two waves are out of phase by 1/4 of a cycle:
Since one complete cycle equates to 360 o or 2p radians, these two waves are 90 o or p/2 radians out of phase. When talking of phase, we can use degrees, but radians (rad) are more common. 360 o = 2p rad, so 1 rad ≈ 57 o. In calculations radians are dimensionless and can be omitted. However in these notes, they will be included.
In this graph, the two waves are 180 o or p rad out of phase:
These two waves would cancel each other out, as their amplitudes are the same.
Waves that travel, or propagate, from a source are called progressive waves. An obvious example is the ripples on a pond when you throw a stone into the water.
We can see easily that the wave pattern is circular. Radio waves propagate in a similar way from a transmitter, but since we are talking of 3 dimensions, the propagation is spherical.
Often for simplicity, we look at a tiny portion of the circular propagation, which enables us to say that the wave crests are straight and parallel.
We call these plane wave-fronts.
A transverse wave is like this:
In a transverse wave the motion of the particles (displacement) is perpendicular to the direction of travel. Note that the wavelength is the distance between two crests (or troughs). The picture above is really a displacement-distance graph.
The distance between crests on a wave displayed on the
CRO is the period, NOT the wavelength.
Examples of transverse waves include:
electromagnetic waves;
shear waves (s-waves) from earthquakes;
waves along a rope.
Water waves are not a good example of transverse waves. This is because the particles tend to go round in circles because of surface tension rather than simply moving up and down. When the amplitude is small compared to the wave length they do look like transverse waves, but when the amplitude increases the crests get more pointed, then break.
Sound is an example of a longitudinal wave. The displacement of the particles is parallel to the direction of travel.
The features of a longitudinal wave are:
Compression - area of high pressure;
Rarefaction - area of low pressure;
Wavelength - distance between two compressions.
We can look at the displacement of individual particles at time intervals:
The green arrows show the particle moving from left to right; the red arrow shows the particle moving from right to left. The displacement to the right is positive; to the left it's negative. You can see that if we join the dots together, the line looks sinusoidal. Remember that the displacement is parallel to the direction of travel.
We can show the longitudinal wave as a displacement distance graph and as a pressure-distance graph.
We can see that the two graphs are p/2 rad out of phase. Since the displacement peaks come first, we say that the displacement graph leads the pressure graph by p/2 rad.
All longitudinal waves are mechanical waves. They need a material to travel in; they cannot travel in a vacuum. As well as sound, p-waves (pressure waves) from an earthquake are longitudinal.
All waves, whether longitudinal or transverse obey the wave equation:
wave speed (m/s) = frequency (Hz) × wavelength (m)
In Physics code:
c = fl
The strange looking symbol l is lambda, a Greek letter 'l'.
Here are a few wave speeds:
|
Wave |
Speed |
| Electromagnetic waves in a vacuum | 3 × 108 m/s |
| Electromagnetic waves in glass | 2 × 108 m/s |
| Sound in steel | 6000 m/s |
| Sound in water | 1500 m/s |
| Sound in air | 340 m/s |
It is therefore possible for particles to travel faster than the speed of light in a material. The little flashes of light, the equivalent of a sonic boom, are called Cherenkov Radiation. Nothing can travel faster than light in a vacuum.
When dealing with the speed of sound, many students
blindly write the speed of light. Read the question carefully!
When two waves come together, they superpose. The directions are very important when describing the resultant wave.
If the waves are in phase they behave like this:
See how the resultant is the red wave and the blue wave added together. The amplitude of the resultant wave is the sum of the amplitudes of the two waves. The waves have reinforced. This is called constructive interference.
We can see that when two waves are p radians out of phase, the amplitude of the resultant is much smaller:
You can see that the larger (red) wave has an amplitude of 0.3 units while the smaller (blue) has an amplitude of 0.2 units. The resultant (green) wave has an amplitude of 0.1 units. The direction of the resultant wave is in the direction of the larger wave. This is called destructive interference.
If both waves have the same amplitude, they cancel out completely and the amplitude is zero.
These two waves are p/2 rad out of phase:
The maximum amplitude of the resultant is where the two waves cross.
Stationary waves are formed when two progressive waves are superposed. To form a standing wave, the following are needed:
Equal frequency
Nearly the same amplitude
Same speed
Travelling in opposite directions.
If we send an incident wave down a string, which is fixed at the end, the wave is reflected at the fixed end and undergoes a phase change of p radians or 180o. There is no phase change at the free end.
If we send a continuous stream of waves down the string, they are reflected and a standing wave gets set up. The frequency will be the same, the amplitude very nearly the same, and the speed will be the same. The directions are opposite. The phase change of p radians causes cancellation at the fixed end. This region of zero displacement is called a node.
In a progressive wave, points X and Y would be in antiphase, p radians out of phase. However, because the wave is reflected, the phase is changed by p radians. So they are now 2p radians out of phase, which means that they are in phase. Superposition is constructive. The amplitude is now at a maximum, and this is called an antinode.
Notice:
All particles between nodes are in phase.
All particles either side of a node are in antiphase.
Each “loop” is half a wavelength.
If we start the frequency of the vibration at a low level, increasing it slowly, we see little of significance until at a certain value, a single large vibration loop is seen. This is due to resonance and is called the fundamental frequency or the first harmonic. The second harmonic has two vibration loops. It is twice the frequency of the fundamental frequency.
The frequency at which resonance happens depends on:
The tension
The length
The mass per unit length (how thick the string is).
We can investigate how these quantities are related.
At the fundamental frequency, length of the loop is 1/2 wavelength, so we can write:
l = 2l
Using the wave equation:
c = fl
So we can combine these two to relate the frequency to the speed of the wave in the string.
f = v = v
l 2l
Since the speed is related to tension and mass per unit length by:
T is the physics code for tension (N), while m is the mass per unit length (kg m-1). The symbol m is 'mu' a Greek letter 'm'. Now we can combine this relationship with the one above to give:
The mass per unit length is often given in grams per
metre. Convert this to kilograms per metre!
In stringed instruments, the standing waves are set up by plucking, hitting, or scraping a string. Before any performance, the artists tune their instruments to a standard tone (usually A at 440 Hz) by altering the tension in the strings.
The musical notes from wind instruments result from setting up standing waves in a pipe. Standing sound waves can be shown with a Kundst Tube as in the picture.
We should note that:
The wave is longitudinal so that all the particles vibrate parallel to the tube.
The amplitude is at a maximum at the open end of the tube, so there must be an antinode.
The amplitude at the closed end is zero; there is a node.
All molecules between nodes vibrate in phase.
All molecules either side of a node vibrate in antiphase.
Adjacent nodes are half a wavelength apart.
This has important implications for wind instruments. A brass instrument like a trombone has an antinode made by the player pursing his lips and vibrating them (an embouchure). There is an antinode at the bell of the instrument, and a node half way down.
The fundamental frequency can be changed by altering the length of the tube. In a trombone, the player moves part of the tube in and out. A trumpeter changes the length by pressing in valves.
This arrangement shows half a wavelength. So the standing wave at fundamental frequency in an open pipe is half a wavelength.
At fundamental frequency, a closed organ pipe has half a vibration loop. The antinode is formed by air passing a whistle arrangement.
Since this is ¼ of a wavelength, the organ pipe sounds a note whose wavelength is 4 times its length.
If we blow hard into a whistle, we find that the frequency of the note is higher than the fundamental frequency. The standing wave pattern has altered. The altered patterns are called overtones or harmonics. It is the harmonics that give a musical instrument its distinctive and pleasing sound. Harmonics are always whole number multiples of the fundamental frequency.
fn = n × f0
We can most easily see harmonics in a string.
From this diagram we can see:
At the fundamental frequency, there is one loop of ½ wavelength.
At the 2nd harmonic, there are two loops each of which is ½ wavelength. Therefore the length of the string is 1 wavelength. This is twice the frequency.
At the 3rd harmonic, there are three loops. The length of the string is 1 ½ wavelengths, i.e. three time the fundamental frequency.
For an open pipe we see the second harmonic like this:
There are two half-loops and a whole loop. Since a half loop is ¼ wavelength, and the whole loop is ½ wavelength, the total is one wavelength. Therefore the frequency is twice the fundamental.
At three times the fundamental frequency we see:
We have 1 ½ wavelengths or three times the fundamental frequency.
For a closed pipe, it's a bit different.
In this case we have 1 ½ loops, which is ¾ wavelength. Since the fundamental was ¼ wavelength the frequency of the second harmonic is three times the fundamental. The third harmonic will give a frequency five times the fundamental.
A sine wave is incredibly dull to listen to. However we have seen that musical instruments have a wide range of harmonics that give them the vital quality that makes them so pleasant to listen to. You can look at the patterns with a CRO, and the computer can display the waveform of music. You can see:
There is a periodicity.
The waveform is quite complex, as you would expect from an orchestral piece.
If you break the waveform down into sufficiently small pieces, you end up with sine waves. A complex waveform is really lots of sine waves superposed.
The display above is not calibrated, so you could not tell what the frequency actually is. A dedicated sound analysis program would put scales onto the display.
We can also use the computer to analyse the sound spectrum, just like you would use a prism to look at the spectrum of visible light. The picture below shows the spectrum of an orchestral piece.
Again this is a crude spectrum analysis of an orchestral piece:
The low frequencies are to the left, and the high frequencies are to the right.
The mid range frequencies are the most prominent.
The base notes are also quite marked.
The highest frequencies are well above the range of the musical instrument, but are important for giving a sense of "life" to the sound.
Everyone has a unique voice pattern, so the waveforms of different people saying the same words are quite different. This is why speech recognition software has had limited success.
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