If you are not familiar with measuring angles in radians, I suggest you read post 17.11 before this.
The picture shows what happens when we drop a stone into a pond: the stone disturbs the water and waves spread out from the point of disturbance – the source of the waves.
Eventually the waves will die down because the viscosity (post 17.17) of the water will dissipate the kinetic energy of motion (post 16.21) of the water. When the waves have died down, you can see that no water has been moved away from the source of the waves. Despite a wave being movement of water (upwards and downwards), no water moves in the direction in which the waves spread out – the direction of propagation.
The wave does not transport water – it transports energy. When the stone drops into the water, it falls more slowly – it loses kinetic energy (post 16.21). This kinetic energy is not stored as potential energy – if it were, the stone would bounce upwards (post 17.30). Some of it may be dissipated as heat but much of it becomes kinetic energy of motion of the water. What happens is very similar to a collision (post 17.30) but in a collision two solid objects collide; here a solid collides with a liquid. As the waves spread out from the source, the kinetic energy of liquid movement is transported away from the source. This process is analogous to heat (which is kinetic energy of molecular motion, post 16.35) spontaneously spreading from its source (see post 16.34).
In the first picture the waves spread out over the surface – they are two-dimensional waves and their peaks and, when we look down on them, we see a series of circles. If we look along a radial direction, at an instant in time, we see the water displaced as shown in the picture above. This is also what we would see if our waves were travelling along a canal – they would then be one-dimensional waves. The distance between peaks is called the wavelength, λ, of the waves. The height of the peak above the average water level is called the amplitude, A, of the wave.
Now let’s suppose we look at a fixed point, along the direction of propagation of the wave, and watch how the water level varies with time – the result is shown in the picture above. The time between the appearance of peaks is called the time period or periodic time, T, of the wave. We call 1/T, the number of times a peak appears in unit time, the frequency, f, of the wave (see also post 16.14).
In the first picture we saw that, in space, the separation between peaks was λ and, in the second picture, we saw that, in time, the separation between peaks was T. In the same way as we define frequency f = 1/T, we can define spatial frequency k = 1/λ. The spatial frequency is also called the wavenumber.
Since the wave travels a distance λ in time T, its speed is given by
v = λ/T = fλ.
Since our waves look like sine waves (see post 16.50 appendix 6), we can represent the dependence of the wave on distance, x, from the source, at an instant in time, t, by
ψ = Asin(2πx/λ) = Asin(2πkx).
Here ψ represents the height of the wave above the average water level; but there are lots of other types of waves and, in general, it means the magnitude of whatever disturbance has an amplitude A – we call ψ a wavefunction. If you don’t understand what I have done here, note that our wave has peaks when x = λ/4, 5λ/4, 9λ/4… and that sinθ has peaks when θ = π/2, 5π/2, 7π/2…(post 16.50 appendix 6 when θ is measured in radians, as described in post 17); similarly, our wave has zero values when x = 0, λ/2, λ, λ/2… and sinθ has zero values when θ = 0, π, 2π…(post 16.50 appendix 6), Finally, our wave has troughs when x = 3λ/4, 7λ/4, 11λ/4… and sinθ has troughs when θ = 3π/2, 7π/2, 11π/2…(post 16.50 appendix 6). So our equation mimics the shape of the wave; for example, when x = λ/4, 2πx/λ = π/2 and ψ= Asin(π/2) = A which represents a peak. As a further example, when x = 3λ/4, 2πx/λ = 3π/2 and ψ= Asin(π/2) = – A which represents a trough.
Similarly, we can represent the dependence of the wave on time, t, from the source, at a fixed value of x, by
ψ = Asin(2πt/T) = Asin(2πft) = = Asin(ωt).
Now let’s think about a red wave that is shifted a distance L along the direction of propagation, with respect to our blue wave as shown in the picture above. We say that there is a path difference of L between the two waves. However, it is more usual to describe the shift in terms of a phase difference defined by θ = 2πL/λ. Why is θ defined in this way? The reason is that L/λ is the difference expressed as a fraction of the wavelength and θ/2π is the difference expressed as a fraction the angular repeat of a sine wave; since these fractions must be identical L/λ = θ/2π so that θ = 2πL/λ.
What happens when two waves meet each other? If the phase difference between them is zero, we say that they are in phase and they simply add together, like the blue and green waves in the picture above. The result is called constructive interference.
If the phase difference between them is π radians, then the peaks of one coincide with the troughs of the other. We say that they are out of phase by π radians. The picture above shows a red and a green wave with the same amplitude When they meet each other they cancel each other out – the result is called destructive interference.
The blue and red waves, in the picture above, are out of phase by an angle between zero and π/2. When they meet their amplitudes at each point add together as shown.
We sometimes call the adding of waves together, as in the three previous examples, the superposition of waves or interference.
Often we are not concerned about the position of the source of a wave. If we are considering the peaks and troughs of one wave, in isolation, the value assigned to θ is then arbitrary and depends only on the origin we use to measure x (see appendix 2 of post 16.50). However, the relative phases of two waves become important when we consider how they interact with each other, as shown in the three examples above.
The graph in appendix 6 of post 16.50, shows that sin(α + π/2) = cosα, where α represents any angle. So sine waves and cosine waves are identical but have a phase difference of π/2 between them. This means that we could choose to represent a wave by the equation
ψ = Acos(2πx/λ) = Acos(2πkx).
ψ = Acos(2πt/T) = Acos(2πft) = = Acos(ωt).
We might choose to represent a wave by this equation if it had a value of ψ equal to A at x = 0 at time t = 0.
In his post I have illustrated the behaviour of waves by waves on water. In these waves, the water moves up and down – perpendicular to the direction of propagation of the wave. Waves like this are called transverse waves. Water waves move in their direction of propagation; waves like this are called travelling waves. In later posts, we will meet waves that don’t move and waves that are not transverse.
18.8 Natural frequency and resonance
18.7 The simple pendulum
18.6 The pendulum: a simple harmonic oscillator