The convolution of two functions of x, f(x) and g(x), is defined by the definite integral Convolution is defined mathematically but it is possible to understand what it means in pictures. So, if you don’t like mathematics, ignore the next paragraph and the appendices. The definition of convolution can be extended into two, or more… Continue reading 22.20 Convolution

Before you read this, I suggest you read post 16.33. Stability is a more complicated idea than we often suppose. For example, a mechanical system can be stable but not in equilibrium because it is moving (with respect to an observer – see movement) on a stable path – it has dynamic stability. Similarly, a… Continue reading 22.19 Kinetic stability

Before you read this, I suggest you read post 18.11 The picture below shows an orthogonal Cartesian coordinate system in which the z-axis is vertical. We are going to think of two pendulums: one oscillates in the xz plane and the other oscillates in the yz plane. For small oscillations, we can consider that the… Continue reading 22.18 Coupled oscillators – Lissajou’s figures

Before you read this, I suggest you read post 19.10. Foxes eat other animals (they are predators), including rabbits: rabbits are eaten by other animals (they are prey), including foxes, but rabbits eat only plants. Now let’s imagine an island that contains foxes and rabbits; there are no other animals for the foxes to eat… Continue reading 22.17 Model for a simple ecosystem – coupled differential equations

This post is about an obscure mathematical technique that will not interest many people. So why have I written it? There are two reasons. One: I mentioned this topic in appendix 2 of post 22.15 so some readers may want to know more about it. Two: many books on physics and engineering change the mapping… Continue reading 22.16 Jacobians – changing the mapping of an integral

Before you read this, I suggest you read post 22.4. The electrons in an atom can scatter x-rays because they move up and down as the electric vector of the x-rays oscillates, as describe in post 22.4. So, the moving electron is a simple harmonic oscillator if the x-ray wave is a sine wave. The… Continue reading 22.15 X-ray scattering by an atom

Before you read this, I suggest you read post 19.20. In post 22.13, we saw that we couldn’t use a microscope to find the positions of atoms in molecules. But, in principle, we should be able to obtain some information from an x-ray diffraction pattern. This post gives more details. X-rays are scattered by the… Continue reading 22.14 X-ray diffraction

Before you read this, I suggest you read post 22.12. In post 22.12, we saw that, in the microscope, the first stage of image formation was formation of a diffraction pattern in a plane that I have labelled the diffraction plane. Waves then continued to form an image, so that the image of point A… Continue reading 22.13 Resolution of the microscope

Before you read this, I suggest you read posts 19.20 and 22.11. In post 22.11, we saw that a microscope forms an intermediate image which is then magnified to give a final image. The picture above uses ray-tracing to show how the intermediate image, A’B’, is formed from the object AB. But I now want… Continue reading 22.12 Diffraction, Fourier transforms and image formation

Before you read this, I suggest that you read posts 22.1, 22.2 and 22.3. The picture shows a microscope. Light passes through a bi-convex lens called the condenser and then through a thin specimen that scatters the light. This scattered light then passes through another lens called the objective. The objective acts in the same… Continue reading 22.11 The microscope