5.13.1 Rectangular bar If a solid rectangular bar is excited by striking it, energy is supplied that starts the bar vibrating transversely. The bar will vibrate in a number of modes simultaneously since the striking action supplies energy over a range of frequencies. The motion of the bar will be the superposition of the standing-wave patterns of the excited modes. Assume for the moment that the rectangular bar is supported in such a way that both ends are free to vibrate and the effects of the supports can b
5.13 Other primary vibrators You saw in the previous two sections that stringed instruments and wind instruments possess primary vibrators that have harmonically related natural frequencies. As a result, these two classes of instruments produce notes that have a well-defined sense of pitch. In this section, I want to briefly introduce you to some primary vibrators that don't have harmonically related natural frequencies. Specifically we shall take a look at a rectangular bar, a circular membrane and a circular plat
5.10 Vibrating air column: end effects In the previous two sections on standing waves in cylindrical tubes, we assumed that at an open end there must be a pressure node. In fact, the pressure node (and the corresponding displacement antinode) actually lies a small distance outside the tube. The effect is that the air column behaves as though it were a little longer than it really is by an amount called the end correction. Because of this end correction, the resonance frequencies will be a little lower than originally expect
5.9 Vibrating air column: standing waves in a cylindrical tube closed at one end We'll now turn our attention to the setting up of standing waves in an air column contained within a cylindrical tube that is open at one end but closed at the other. Straight away we can say that the closed end must be a displacement node since the air molecules can't move at this boundary. That means it must be a pressure antinode. The open end, as we saw previously, will be a displacement antinode (that is, a pressure node). Now, you may recall that the distance between a node and a
5.7 Vibrating air column: reflection at the end of an air column When a sound wave reaches the end of an air column, it is clear that it will be reflected if the tube end is closed. You only have to imagine yourself standing some distance, let's say 50 metres, away from a flat wall. If you shout, you will hear an echo – the reflection of the sound wave you projected. There is one difference, though, between the reflection of a sound wave and the reflection of the wave on a string that you met previously. When a sound wave is reflected from a closed
5.6 Vibrating air column You learned in the previous section that for standing waves to be set up on a string there must be reflection. A travelling wave reaches the end of the string and is reflected. This results in a second travelling wave, which moves back up the string in the opposite direction to the first wave. The two travelling waves interact to produce a standing wave. Standing waves are set up in an air column enclosed within a tube in a very similar way. Again there must be reflection. In this case,
5.4 Vibrating string: normal modes of vibration The frequencies at which standing waves can be set up on a string are the string's natural frequencies. They can be determined quite easily. The first thing to note is that the end of the string being held by the person is tightly gripped so any pulse or wave that returns to the person's hand will be reflected and inverted. Therefore both ends of the string can be considered to be fixed and so must be at nodes of the standing wave. But you learned earlier that the distance between adjacent no
5.3 Vibrating string: standing waves on a string We still haven't answered the question of how standing waves are set up on a string. To do so we need to return to our string, fixed at one end and held in someone's hand at the other end. Imagine now that instead of sending a single pulse along the string, the person flicks their hand up and down periodically and sends a sinusoidal wave along the string. This wave gets reflected and inverted at the fixed end and travels back towards the person holding the string. There are now two waves of t
5.2 Vibrating string: speed of wave propagation If standing waves are set up when two travelling waves moving in opposite directions interact, then how are standing waves set up on a string and why are they set up only at certain frequencies? To help answer these questions, I want you first to imagine a length of string that is fixed at one end and held in someone's hand at the other. Suppose the person holding the string flicks their end of the string in such a way that an upward pulse is sent along the string. As the pulse pa
4 Excitation For a player to be able to sound a musical instrument, there must be a means of inputting energy to set up the vibration. This energy may be introduced in a short, sharp burst or continuously over a period of time. In the case of brass instruments such as the trumpet and trombone, and woodwind instruments such as the flute and oboe, the player feeds in energy by blowing air into the instrument. The energy can be supplied in a short burst – in which case short-lived ‘staccato’ note
Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit: This extract is taken from TA212. ©
Audio Materials
8.1 The octave sound One feature of pitch that seems to be universal to all cultures is that for musical purposes the pitch range is divided into discrete steps: for instance, the notes of a scale. This is not to say that musicians rigidly adhere to those steps when they play, but the existence of such steps is fundamental to the way music is conceived and organised. Different cultures have different ways of defining the steps in their scale of pitches, but nearly all cultures take the octave as their starting po
7.1 The subjective experience Two of the properties of sound that we have examined from an objective stance, frequency and amplitude, have a fundamental importance to our appreciation of sound and music. In this section I want to look more closely at the subjective interpretation of these two properties of sound. I should stress that I am talking about sine-wave sounds in this section. The complex, non-sinusoidal sounds encountered in music add extra layers of complexity to the relationships I am discussing here. Ke
6.1 Defining amplitude Another important property of a sine wave we need to be able to specify is its amplitude. In essence, the amplitude of a sine wave is its size. Unfortunately there are various ways of defining what is meant by the size of a sine wave, and you are likely to come across many of them in material you look at outside this unit. Before I explain what our definition is, it will help matters if we look at what is meant by the average value of a sine wave. Figure 16 shows a sinusoidally a
4.6 Appreciating some implications for practice I think for most people, the CSA case study would be experienced as a complex situation. If so this would be a good example of perceived complexity. Remember though, if you engaged with it as if it were a difficulty, just as the government minister did in Activity 42, you would not d
3.2 Being aware of the constraints and possibilities of the observer It is often claimed that the essence of a systems approach is that of seeing the world in a special way. This immediately prompts the question of what is meant by the phrase ‘seeing the world’. Because we live so intimately with the world of objects, categories and people and phenomena, we tend to think our own way of seeing the world is the only way, or even of thinking, ‘Well that is my view because the world is like that’. Actually, your view is special in several separate ways.
2 Part 1 Starting the unit Welcome to T306_2 Managing complexity: a systems approach – introduction. As I write, I experience a sense of excitement. For me, as for you, this is the beginning of the unit. These are the first few sentences I'm writing and so, although I have a good idea of how the unit is going to turn out, the details are by no means clear. Nevertheless, the excitement and anticipation I, and maybe you, are experiencing now is an important ingredient in what will become our experiences of the u
13 Part 2: 4 Preparing for innovation Many inventors have said that having the idea for an invention is the easy part. This is often demonstrated by the frequency of examples of simultaneous invention. At one exhibition of inventions I attended there were three separate portable ladders to escape from fires, two systems for using rainwater to flush toilets, two types of portable vehicle wheel clamp, and two methods of reducing red-eye in flash photography. In most cases of technological innovation only one of the competing techno
12.4 Coupling model There are examples where either technology or the market appears to be more significant in stimulating invention but the majority of innovations involve a creative coupling of technological and market factors. In some respects successful innovation is a case of the survival of the fittest. Failure can come both from not getting the technology right and from misjudging the market. Success is more likely if the focus is not too one-dimensional but rather a balance between technology and market
11.5.4 Analogy Analogy draws on similar situations to provide ideas for invention and design. Alexander Graham Bell used the analogy of the human ear when designing telephone apparatus to receive sound. As mentioned above, his first receivers were much better than his transmitters where the analogy with the ear didn't work as well. When devising their flying machine, the Wright brothers used the analogy of soaring birds twisting their wings to restore balance. They designed the wings of their aircraft to be
