# Is there a way to know how the amplitude of a wave will evolve?

• SergioPL
In summary, the conversation discusses the difference between plane waves and spherical waves and how they propagate. It is mentioned that plane waves do not decrease in amplitude as they propagate because all points on the same plane have the same phase, amplitude, and direction. However, spherical waves do decrease in amplitude because the points on the contour have the same amplitude but different directions. The Pointing's vector definition is then mentioned as a way to explain the decrease in amplitude and how it relates to conservation of energy. The J·E term is also discussed as a way to account for energy loss from fields to accelerating charges.
SergioPL
It’s very well know that Maxwell equations bring the solution for plane waves but as far as I know I cannot use them to detect how the field would evolve by only looking the E and B fields on the contour of the point under evaluation. Is that possible?
My question comes because on a plane wave, if we look at some point, all the points on the same plane of propagation that the point under study have the same phase, the same amplitude and they propagate on the same direction so this must be the reason the field doesn’t decrease as it propagates.
On the other hand, on a spherical wave, the contour of a point in the plane parallel to the (local) direction of propagation will have the same amplitude but different direction. I suppose this difference between these two types of waves is what makes spherical waves to decrease whereas plane waves don’t.
Does somebody know if there are some equations that can locally explain this?

Sergio

Ok, the matter gets closed by looking at the Pointing's vector definition :).

The equation is:

∂u/∂t = - $\nabla$ · S - J · E

And looking how u evolves in the direction of propagation you can get how it's power decreases.

Yes, this is essentially a statement of conservation of energy. If a wave spreads out, its energy density must decrease. The J dot E term accounts for the fact that energy can also be lost from the fields to accelerating charges that create currents.

Yes, the J·E is the power done on the charges for a distribution of charge.

J = qv where q is the density of charge and v the velocity, so J·E = qE·v = F·v, the classical definition of power, the decrease on the field's energy density is the (density of) power that is done on the current.

Sergio

## 1. How is the amplitude of a wave measured?

The amplitude of a wave is measured by calculating the maximum displacement of the wave from its equilibrium position. This can be done by measuring the vertical distance between the crest (peak) of the wave and its equilibrium position, or the trough (lowest point) and its equilibrium position.

## 2. Can the amplitude of a wave change over time?

Yes, the amplitude of a wave can change over time. This is known as the evolution of a wave. The amplitude can increase or decrease depending on various factors such as the energy of the source, the medium through which the wave is passing, and any external forces acting on the wave.

## 3. Is the amplitude of a wave related to its frequency?

No, the amplitude and frequency of a wave are independent of each other. The amplitude is a measure of the wave's intensity or energy, while the frequency is a measure of the wave's oscillations per unit time. However, the amplitude can affect the frequency of a wave in certain situations, such as when the amplitude is too high and causes the wave to distort or break.

## 4. How can we predict the future evolution of a wave's amplitude?

Predicting the future evolution of a wave's amplitude can be done through mathematical calculations using principles of wave mechanics. This involves analyzing the initial conditions of the wave, such as its amplitude, frequency, and direction of propagation, and applying equations that describe how these parameters will change over time. Additionally, computer simulations and physical experiments can also help in predicting the evolution of a wave's amplitude.

## 5. Are there any practical applications for understanding the evolution of a wave's amplitude?

Yes, understanding the evolution of a wave's amplitude is crucial in various fields such as engineering, physics, and geology. It is used to design and improve technologies that rely on wave phenomena, such as sonar, radar, and seismic imaging. Additionally, studying the evolution of wave amplitudes can also help in predicting and mitigating the effects of natural disasters such as earthquakes and tsunamis.

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