After studying this unit you should be able to:

• understand how the wave and diffusion partial differential equations can be used to model certain systems;

• determine appropriate simple boundary and initial conditions for such models;

• find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth
Author(s): The Open University

The content acknowledged below is Proprietary (see terms and conditions) and is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

## Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

<
Author(s): The Open University

The displacement from a point P to a point Q is the change of position between the two points, as described by the displacement vector

If P and Q represent places on the ground, then it is natural to use a bearing to describe the direct
Author(s): The Open University

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector a = a 1 i + a 2 j, what are its magnitude |a| and direction θ?

The first part of this question is dealt with using Pythagoras’ Theorem: the magnitude of a v
Author(s): The Open University

Communities such as Findhorn already behave as if natural resources need careful management: they work hard to reduce fossil fuel use. A central assumption of this way of thinking is that people need to root economies more locally (Figure 15). To see the same impulse spread through the mainstream economy would require that th
Author(s): The Open University

People who demand a radical break with the business-dominated path of economic globalisation believe that the claims of the mainstream business community are at best hopelessly inadequate, and at worst deceitful. However, they know they have to come up with some answers of their own. This section outlines ideas that seek to underpin a transition to green economies owned and run at grassroots level. Sounds ambitious? Author(s): The Open University

Business can profit from taking the environment into account (generally called eco-efficiency). Poor environmental performance is seen as a reflection of poor business practice in general. Eco-efficiency promotes the economic benefits of energy and materials savings, at the same time being first to market with new technologies or products. Since business sustainability lobbies promoted eco-efficiency in the early 1990s, the creed has gained rapid acceptance, and with good cause. There
Author(s): The Open University

Business can learn to integrate ecological thinking into the core of its thinking and become the hub of a sustainable society. This is the claim of the business learns position. As environmentalists have spent over 30 years portraying business as the arch-villain of the piece, this is a grand claim. One of the people who have stated it most clearly is Ray Anderson, head of the US carpet giant Interface (Author(s): The Open University

How many ordinary people know that sustainability is the concept that is meant to save the world? How many people who believe in the concept are convinced that it can capture the public imagination? The answer to both questions is ‘not many’. It is easy to lay the charge that the idea has been much talked about in some closed circles, but has no purchase on the public imagination and is little practised. This section takes the three different approaches to global environmental change desc
Author(s): The Open University

It has long been held that we conduct all citizenship, and the obligations it implies, in the public sphere (i.e. outside the private sphere of the home). However, it has been argued that there are other potential sources of obligation. Andrew Dobson argues that the principal duties of the ecological citizen are to act with care and compassion to strangers, both human and non-human – not just in the present, but also those distant in space and time (Dobson, 2000). These virtues of care and
Author(s): The Open University

Citizenship is generally held to be based on a contractual view, where rights and obligations are balanced. In other words, you get various rights in return for your commitment to live by your society's rules and expectations. Political philosopher Andrew Dobson suggests that ecological citizenship is based in a non-reciprocal sense of justice or compassion. The discussion of our relationships with past and future generations in Section 5.2 establishes that our obligations to future generatio
Author(s): The Open University

The term ‘good governance’ implies that ‘ordinary people’ will be involved in deciding what to do, trying to make it happen, and deciding whether it has happened (debate, implementation, monitoring). But what, in practical terms, might citizen involvement in the governance of an issue such as climate change mean? Citizen involvement in decisions and actions can mean anything from filling in a questionnaire to joining a demonstration to sitting on a committee. One helpful approach is A
Author(s): The Open University

It would be a serious error to imagine that ‘government’ has evaporated: it still shapes many aspects of our lives from beginning to end (welfare, taxation, transport and, of course, the recording of births and deaths). Governments are the central negotiators of environmental-change policies at international level, and of their implementation at national and local level. Nevertheless, for many areas of life, governance is undeniably a better description both of new processes that are alre
Author(s): The Open University

The polar bear has become an international climate change icon. But how much is known about this bear, its habitat and life? This unit will talk about the role of language, but by way of introduction how about the name of this bear? To me it is the polar bear; to a German it is an Eisbär (ice bear) and to a French person it is an ours blanc (white bear). In these three examples the bear is referred to as polar, white, or an ice bear – eminently sensible. The Latin name for th
Author(s): The Open University

As a dam built across a river causes a local deepening of the stream, so our atmosphere, thrown as a barrier across the terrestrial rays, produces a local heightening of the temperature at the Earth's surface.

(Tyndall, 1862, quoted in Weart, 2004)

Thus, writing in 1862, John Tyndall (Figure 6) described the key to our modern understanding of why the Earth's surface is so much warmer than t
Author(s): The Open University

One approach to BS 8800 follows the ISO 14001 model, and the ISO 14001 system itself was closely modelled on the previous ISO 9000, with the 2000 revision of ISO 9000 following ISO 14001 principles. As a result, you may imagine that there are similarities between the standards. Many of the elements are similar, and some are nearly identical. Management systems share common elements, including developing and documenting procedures, training, record keeping, auditing, and corrective action. Fig
Author(s): The Open University

A small digression will be made here to question the use of the word ‘cause’ and to reflect on the view expressed by Kletz (1988, p. 2). He argues that the word has an air of finality about it, and is concerned that finding the cause discourages further investigation. He cites an example that the cause of a pipe failure was corrosion – which suggests that we know why the failure occurred. He draws the analogy of the cause of a fall being gravity – suggesting that nothing more can be d
Author(s): The Open University

Figure 6 Cover from the Kew Gardens Millennium Seed Bank Appeal leaflet

Author(s): The Open University

The material acknowledged below is Proprietary, used under licence and not subject to Creative Commons licensing (see terms and conditions). Grateful acknowledgement is made to the following:

Figure 1/Unit Image © John Isaac/Still Pictures