Spherical wave in far field is a plane wave ?

In summary, the conversation discusses the difference between a spherical wave and a plane wave in the far field, specifically in terms of the electric field equation. While a plane wave follows the formula E(s) = E(0) * exp(-jks), a spherical wave has a 1/s decay factor, as shown in the equation E(s) = E(0) * 1/s * exp(-jks). The question is then raised about simulating a plane wave with a spherical wave in the far field and which formula should be used. The conversation also mentions a calculation involving a 2 foot long piece of a circle with a radius of 20000 feet and the difference between the ends of the arc and the tangent line at
  • #1
broli86
4
0
Is it true that we can consider a spherical wave in the far field (i.e. away from antenna, at a large distance) as a plane wave ? For a plane wave the electric field at a distance s is usually given as:

E(s) = E(0) * exp(-jks) ----- (1)

where k is the wave vector, s is the distance traveled and E(0) is the electric field at a reference point.

But if I'm not wrong then in case of a spherical wave, there is a 1/s decay so for spherical wave:

E(s) = E(0) * 1/s * exp(-jks) --------(2)

Now my question is: If I am trying to simulate a plane wave with a spherical wave in far field, then should I use formula (1) or (2).
 
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  • #2
Suppose you are look at a 2 foot long piece of a circle with radius 20000 feet. What does it look like? As an interesting calculation, you might calculate how much the ends of that arc differ from the tangent line to the circle at the center of the arc.
 
  • #3


I would say that it is not entirely accurate to consider a spherical wave in the far field as a plane wave. While the electric field may exhibit some characteristics of a plane wave, such as having a constant magnitude and phase over a large distance, there are still important differences between the two.

In the case of a spherical wave, the electric field does indeed decay with distance according to the 1/s factor in equation (2). This is due to the spreading out of the wave as it propagates through space. In contrast, a plane wave maintains a constant magnitude and phase over all distances, as shown in equation (1).

Therefore, if you are trying to simulate a plane wave, it would be more accurate to use equation (1) rather than (2). However, if you are dealing with a spherical wave, then equation (2) would be more appropriate.

It is important to note that in reality, there are no perfectly plane waves or spherical waves. All waves exhibit some degree of spreading over distance, and the distinction between a spherical wave and a plane wave is a matter of the degree of spreading. So while it may be convenient to use the simplified equations (1) and (2) in certain scenarios, it is important to keep in mind the limitations and differences between the two types of waves.
 

1) What is a spherical wave in far field?

A spherical wave in far field is a type of wave that propagates from a point source in all directions equally. It is also known as a radial wave because it expands outward in a circular pattern. This type of wave is commonly seen in applications such as sound waves, light waves, and electromagnetic waves.

2) How is a spherical wave different from a plane wave?

A spherical wave and a plane wave are fundamentally different in their wavefront characteristics. A spherical wave has a curved wavefront, whereas a plane wave has a flat wavefront. Additionally, a spherical wave spreads out in all directions, while a plane wave travels in a specific direction.

3) What is meant by "far field" in the context of a spherical wave?

The far field is the region in which the wave has traveled a significant distance from its source and has reached a point where the wavefront can be considered to be plane. In the case of a spherical wave, the far field is the region where the wavefront is approximately flat and the wave appears to be a plane wave.

4) How is a spherical wave in far field useful in practical applications?

A spherical wave in far field has many practical applications, such as in wireless communication systems, where it is used to transmit and receive signals. It is also used in medical imaging techniques, such as ultrasound, to create 3D images of structures inside the body. Additionally, it is used in astronomy to study the properties of distant objects.

5) What are the mathematical equations that describe a spherical wave in far field?

The mathematical equations that describe a spherical wave in far field include the wave equation, which describes the propagation of the wave, and the Huygens-Fresnel principle, which explains how the wavefront of a spherical wave evolves over time. These equations can be used to calculate various properties of the wave, such as its amplitude and phase, at a given point in the far field.

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