Question

A line source emits a cylindrical expanding wave. Assuming that the medium absorbs no energy, find how ( \(a\) ) the intensity and \((b)\) the amplitude of the wave depend on the distance from the source.

Short answer

The intensity of the cylindrical expanding wave decreases as the inverse of the distance from the source, following a \(I = 1/r\) relationship. The amplitude of the wave decreases according to the square root of the distance, following a \(A = 1/\sqrt{r}\) relationship.

Step-by-step solution

Understanding Wave Intensity and Amplitude

The wave emitted from the source is a cylindrical wave that expands away from the the line source. This means the wave propagates cylindrically and the energy spread out in a cylindrical surface. With energy conservation, the total energy emitted by the source is the same at any distance from the source. The intensity of the wave is the energy passing per unit time across unit area. As the wave distances from the source, the energy is distributed over a larger area—specifically, over the lateral surface area of the cylinder which is \(2\pi rL\), where \(r\) is the radius from source to the point where the intensity is to be determined and \(L\) is length of line source. For wave amplitude, it's directly related to the square root of the wave intensity.

Calculating Wave Intensity

Because intensity \(I\) is power divided by area, and in this case the area is the lateral surface of the cylinder, we have \(I = \frac{P}{2\pi rL}\). Since the power \(P\) is a constant (as the medium absorbs no energy), this tells us that the intensity is inversely proportional to the distance from the source. Therefore, the intensity of the wave diminishes as \(1/r\), with \(r\) being the distance from the source.

Calculating Wave Amplitude

The amplitude \(A\) of the wave is proportional to the square root of the wave intensity. Given that the intensity varies as \(1/r\), the amplitude should vary as \(1/\sqrt{r}\). Thus as the distance \(r\) from the source increases, the amplitude of the cylindrical wave decreases according to \(1/\sqrt{r}\).

Key concepts

Wave Intensity

When we talk about the wave intensity, we're delving into the amount of energy a wave carries per unit area per unit time. In the context of a cylindrical wave, which can imagine as spreading out from a string that's been plucked, this intensity diminishes with distance. This makes sense, right?

The key idea is that as the wave propagates, its energy must cover more ground – or more accurately, more cylinder. Think of throwing a stone into a pond. The ripples spread out, and the further they go, the more spaced out they seem. This is because the energy of the wave is spread over a larger surface area.

It's crucial for students to grasp that intensity decreases as you move away from the source. Why? Because the same energy is dispersed across a bigger area. Mathematically, if we denote intensity by the symbol I, and it is the power P (think of it as the wave's energy per time) divided by the surface area (in our case, the circumference of the circle times the length of the line source, or 2πrL), it becomes clear that intensity inversely depends on r, the radius of the cylinder.

Wave Amplitude

Stepping into the world of wave amplitude, we're looking at the height of the wave or its maximum displacement from its rest position. Remember when you last spoke into a microphone and saw the audio waves on the screen? The taller the wave, the louder the sound – that's amplitude in action!

Relating this to our cylindrical wave, the amplitude directly influences how 'strong' or 'powerful' the wave feels. An important takeaway for students is that while intensity changes with the square of the amplitude, amplitude itself changes more gently with distance from the source.

To put it technically, if our wave's intensity varies inversely with r, then the amplitude will vary inversely with the square root of r. This explains why even when a sound seems to fade as you walk away from its source, the drop in volume isn't as rapid as one might initially think. It's a square root relationship, not a direct one-to-one - which is a lighter, softer kind of drop-off.

Energy Conservation in Waves

Energy conservation in waves is another fascinating topic. When we say energy conservation, we mean that in an ideal situation, no energy is lost. If we were to shout into an open field, conservation of energy tells us that all the sound energy we put into that shout stays within the sound waves moving away from us. However, in reality, different factors like air resistance and absorption by objects would cause some of that energy to be lost.

But let's stick to our ideal scenario with a cylindrical wave. The total energy you pumped into creating that wave, say by plucking a string, doesn't just disappear. As the wave travels, that energy is maintained but spread over an ever-increasing cylindrical surface.

To make this a bit clearer for students, one could think of it like slicing a cake. The total amount of cake doesn't change, but the size of each slice gets smaller as more and more people join the party. Similarly, our wave's energy remains the same, but as the 'party' gets bigger, with our cylindrical surface expanding, the visible effects (intensity and amplitude) decrease proportionately.