Unravelling Spherical Waves: Intensity Behaviour Explained

In summary, there is a confusion about the different expressions used to describe a spherical wave. While one is a complex exponential function and has an intensity that decreases as 1/r^2, the other is a cosine function and has an intensity that is an oscillating function of r and t. However, when considering time averages, the intensity of the cosine function is 1/2 of the intensity of the complex exponential function, as <cos^2> = 1/2. This is not surprising since the cosine function is the real part of the complex exponential function.
  • #1
Repetit
128
2
Hey!

Im quite confused about spherical waves. I mean, I understand that a spherical wave can be described by

[tex]
\Psi = \frac{1}{r} e^{i r},
[/tex]

because the intensity of such a wave decreases as [tex]1/r^2[/tex]. The intensity of such a wave is given by [tex] I = 1/r^2 [/tex] which makes sense to me. But a spherical wave can also be described by

[tex]
\Psi = \frac{1}{r} \cos r,
[/tex]

which gives a much different behaviour of the intensity because the intensity of such a wave is [tex] 1/r^2 cos^2(r) [/tex]. If these two expressions both describe a spherical wave, how come they don't have the same intensity?
 
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  • #2
At a guess, I'd say that they're the same thing if you take time averages.

<cos^2(kx-wt)>=1/2
 
  • #3
Those are spherical functions as in that they are angle independent. As they have no time dependence, in what sense are they waves?
 
  • #4
Okay, so if they both had time dependence [tex]-i \omega t[/tex] so that

[tex]
\Psi = \frac{1}{r} e^{i ( k r - \omega t)}
[/tex]

and

[tex]
\Psi = \frac{1}{r} \cos( k r - \omega t)},
[/tex]

but they still don't have the same intensity, since the intensity of the second one is an oscillating function of r and t whereas the first one takes off as 1/r^2 and is therefore not oscillating.
 
  • #5
Again, <cos^2(kr-wt)>=1/2 at any particular value of r and averaging over time.
 
  • #6
Repetit said:
Okay, so if they both had time dependence [tex]-i \omega t[/tex] so that

[tex]
\Psi = \frac{1}{r} e^{i ( k r - \omega t)}
[/tex]

and

[tex]
\Psi = \frac{1}{r} \cos( k r - \omega t)},
[/tex]

but they still don't have the same intensity, since the intensity of the second one is an oscillating function of r and t whereas the first one takes off as 1/r^2 and is therefore not oscillating.

? Not oscillating? The second equation is the real part of the first.
 
  • #7
But isn't the intensity given by [tex]\Psi \Psi^*[/tex]? This gives an intensity equal to 1/r^2 for the wave described by a complex exponential function but an intensity equal to [tex]1/r^2 cos^2(k r - w t)[/tex] for the other one.
 
  • #8
Ah, that's what you're trying to say. Yes, the cos one has a time-dependent 'intensity' and the other doesn't. But as christianjb pointed out, <cos^2>=1/2. So in an average sense one is 1/2 of the other. Not surprising since it's also the 'real half'.
 

1. What are spherical waves?

Spherical waves are a type of wave that radiates outwards from a central point in all directions, creating a spherical pattern. They are commonly observed in physics, such as in sound or light waves.

2. How is intensity related to spherical waves?

Intensity is a measure of the amount of energy that passes through a unit area in a given time. In the case of spherical waves, intensity decreases as the distance from the source increases due to the spreading out of the wave over a larger surface area.

3. What factors affect the intensity of spherical waves?

The intensity of spherical waves is affected by the distance from the source, the amplitude of the wave, and the size and shape of the source. Additionally, obstacles and mediums that the wave passes through can also affect its intensity.

4. How is the inverse square law related to spherical waves?

The inverse square law states that the intensity of a wave is inversely proportional to the square of the distance from the source. This means that as the distance from the source increases, the intensity decreases at a rate of 1/distance^2. This law applies to spherical waves, as they spread out in a spherical pattern from a central point.

5. Can spherical waves interfere with each other?

Yes, spherical waves can interfere with each other when they overlap. Constructive interference occurs when the waves have the same frequency and are in phase, resulting in an increase in intensity. Destructive interference occurs when the waves have opposite phases, resulting in a decrease in intensity. This interference can be observed in phenomena such as sound cancelling headphones or the colors of soap bubbles.

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