If you're wondered whether a rainbow's on it rains, ask yourself why you can surf on the wave when it's right behind the boat, but no one gets further away. The answer is not only that the waves are attenuated, but that they're also dispersed, or in simpler terms, spread out. So let's have a look into what makes this dispersion happen. When we talk about waves, we generally focus on the description of the wave itself. However, we mustn't forget that any mechanical wave can never be separated from its medium, because it's the medium that determines the properties of the wave. With regards to dispersion, we can talk about two types of mediums. In the first type, the speed of the wave only depends on the medium, not the wave length. These are also called non-dispersive waves. All waves in them will travel at the same speed, no matter what. Now in the second type, the speed of the wave will depend on the wave length. This means, for example, that a wave on the surface of water with a wave length of 1 millimeter, which are faster than a wave of 2 millimeter. Such a medium would then be a dispersive medium. Another way to put this would be to say that in dispersive mediums, the wave length is not inversely proportional to frequency. How so? Let's have a look at the equation for the speed of a wave. We know that the velocity is constant, and the wavelength changes. The frequency changes inversely. However, in dispersive mediums, the velocity also changes. So the ratio between frequency and wavelength has to change too. So we have two definitions of a dispersive medium. A medium where the wave changes speed depending on the frequency or wavelength, and a medium in which the frequency and wavelength are not linearly dependent. Waves in such a medium would then be called a dispersive wave. So how does this explain why the waves further away from a boat or smaller? First, let's note that in this case, we don't have to worry about the wave amplitude decreasing due to propagation in all direction. Since the waves are nice and linear, now let's see what happens when a boat passes on a river. Or when a stone is dropped into water. What is created would not be described as a continuous wave, but a wave packet. A single burst or propagating wave, just as you can see on the surface of water. Now for the wave packet to form, it has to be a combination of multiple waves with different wavelengths. How does this work? Let's start with a simple example. When you take two sine waves with slightly different wavelengths, they interfere with each other, producing a wave with varying low and high amplitudes, just as you can see in this example. The region between the two points of zero amplitude is called the wave packet. As you can see, by adding more frequencies, the length of the wave packet decreases and the amplitude increases. In the case of a continuous wave, contrary to a single wave packet, you could also describe this in terms of the resulting wave being composed of an envelope and a modulated wave, which is referred to as beats. The individual wave packets would then be set to move a group velocity, whereas the single peaks would be moving at phase. When a stone is dropped into water, the water oscillates on multiple frequencies, and a wave packet is produced. However, since the speed of a wave in water is dependent on the wavelength, different components over the wave packet move at different speeds. The relative position or phase of the individual components is going to change, and the waveform is going to spread out. This is what it looks like in theory. You can see the length of the wave packet has changed, and the amplitude is decreased as the energy is spread out. This same thing can be observed in water. You can see the wavefront is getting further and further apart, just as the length of the total packet increases and amplitude decreases, a prime example of dispersion. So what have we learned? We know that in some mediums, the speed of the wave is dependent on the wavelength. This gives rise to dispersion, or the separation out due to the varying speeds of the waves. We've seen this in an example of a theoretical wave packet, and to disturbances on the surface of water, both small and large. All the material was used with the permission of the author. If you'd like to learn more about dispersion, water waves, and the math behind them, here are some useful terms to Google. Resources are also listed in the description.