Topic 4: Waves

4.1 Oscillations

Simple harmonic motion. What is it? Well, it’s a specific type of jiggling, where the magnitude of the acceleration is directly proportional to the distance displaced from equilibrium. That is,

a ∝ x

Think of a pendulum. The acceleration and displacement are in opposite directions. That’s sensible. If the acceleration and displacement were in the same direction, it would just fly off. Not much in the way of jiggling. Hence, there’s a negative sign in the equation. Now, we’re going to let the constant of proportionality be -ω2. Why square? It’s convenient. You’ll see in a bit.

Now, let’s say we want to find out where the pendulum will be at any given time. We need to find the displacement as a function of time. So we solve the differential equation above (you can look it up, it’s not covered in this syllabus), and we get

x = x0 sin ωt,

where x0 is the amplitude.

So now we’ve got the displacement. If we differentiate that with respect to time once, we get the velocity. If we do it again, we get the acceleration as a function of time. Earlier we had acceleration as a function of displacement, which may not be so useful in some situations.

Displacement time graphs vs displacement distance graphs. Displacement-time graphs track the motion of a single particle on the wave; that is, how the position of a given particle changes with time. Displacement-distance graphs, on the other hand, are more like photographs. You get to see all the particles and their positions, the entire wave, all at once. Now, the only way we can do that is by taking a snapshot, ie, when time is set still.

Forced oscillations and resonance?

4.2 Wave characteristics and behaviour

In general, there are two types of waves: transverse and longitudinal waves. The oscillations of a transverse wave are perpendicular to its direction of travel (electromagnetic waves, vibration on a guitar string). On the other hand, oscillations of longitudinal waves are parallel to their direction of travel (sound waves). Some waves, like waves in water, are a combination of both.

Wavefronts and rays

Amplitude and intensity (inverse square law)

Lets consider two waves that are passing through the same point. What is the displacement at that point? You simply add the displacement of one wave to that of the other. You can do this for as many waves as you want, as long as they are all passing through that point. This is the principle of superposition, that you can simply add the effects up and get the total effect. Here’s an analogy. This principle works with apples. If someone gave you 3 apples, and someone else gave you 4, then you have 7 in total. It does not work with falling off a building. If you fall from the 5th floor, you may sprain an ankle, but if you fall from the 10th floor, you will probably break your legs, rather than just spraining two ankles. The former is when superposition applies, the latter is when it does not.

Polarisation (methods, maybe types. Malus law)

4.3 Wave behaviour

When propagating from one medium to another, waves can reflect or refract. The reflected wave travels in the same medium as the incident wave, while the refracted wave is in the second medium. Usually, some of the wave is reflected and some of it is refracted.

When reflected, the angle of incidence is equals to the angle of reflection. When refracted, the angle of refraction depends on the angle of incidence, as well as the refractive indices of both media. Their quantitative relationship is given by Snell’s law, which is

n1 sin i = n2 sin r

Reflection and refraction

single slit

double slit

4.4 Travelling waves

As the name suggests, travelling waves travel. They transport energy. Visually, they come across as moving (formally, the phase travels, but you need not worry about that). The speed at which these waves move can be calculated from their frequency, f, and wavelength, λ. The speed, v, is c = fλ. The best way to understand this is by thinking of walking. Your speed is the length of your stride (wavelength) multiplied by the number of strides per unit time (proportional to frequency).

sound and EM waves

4.5 Standing waves

Standing waves are formed from two travelling waves with the same amplitudes, frequencies, and wavelengths, but with opposite directions. Standing waves do not transport energy. They have fixed nodes (points where the are no oscillations) and antinodes (points where the amplitude of oscillation is maximal).

Strings and pipes (shape, frequency, wavelength and L, fixed and free boundaries)