Plane Wave Propogation (help understanding steps in book)

In summary, the problem requires that \vec H vanishes at t = 3ms and the solution provided uses the values of k_0 and \lambda to determine the argument of cosine at t = 3ms. The last step involves dropping a term, which is a multiple of pi, to simplify the expression and still maintain a cosine of zero. This may seem tricky, but it is a common technique used when evaluating expressions involving trigonometric functions.
  • #1
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Plane Wave Propagation (help understanding steps in book)

Basically this problem requires that [itex] \vec H [/itex] vanishes at [itex] t = 3ms [/itex]. I don't understand how they did the last step, here is the solution:

[tex] \vec H = \hat z 4 \times 10^{-6} \cos \left(10^7 \pi t - k_0 y +\frac{\pi}{4} \right) [/tex]

[tex] k_0 = \omega \sqrt{\mu_0 \epsilon_0} = \frac{10^7 \pi}{3 \times 10^8} = 0.105 \,\,\,\, (rad/m) [/tex]
[tex] \lambda = \frac{2\pi}{k_0} = 60 \,\,\,\, (m) [/tex]

At [itex] t = 3 \times 10^{-3} \,\,\,\, (s) [/itex] we require the argument of cosine in [itex] \vec H [/itex]:

[tex] 10^7 \pi (3 \times 10^{-3}) - \frac{\pi}{30}y +\frac{\pi}{4} = \pm n \pi + \frac{\pi}{2}, \,\,\,\,n=0,1,2,\ldots [/tex]

[tex] y = \pm 30n - 7.5 \,\,\,\, (m) [/tex]

Now how the hell do they get the last line? The... [itex] y = \pm 30n - 7.5 \,\,\,\, (m) [/itex] (Lets call this (1)) line.
They drop [itex] 10^7 \pi (3 \times 10^{-3}) [/itex] why?

My problem is that if we plug in (1) into,
[tex] \vec H = \hat z 4 \times 10^{-6} \cos \left(10^7 \pi t - k_0 y +\frac{\pi}{4} \right) [/tex]

and set t = 0.003, then if we feed the function values of n then the cosine doesn't drop to 0.

Thanks in advance!
 
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  • #2
The term they dropped is a multiple of pi (and in fact a multiple of 2pi). Adding a multiple of pi to an angle at which the cosine is zero is still going to give you a cosine of zero.
 
  • #3
OlderDan said:
The term they dropped is a multiple of pi (and in fact a multiple of 2pi). Adding a multiple of pi to an angle at which the cosine is zero is still going to give you a cosine of zero.

Ahh... tricky tricky. (I mean, yeah, it's not too tricky but still, I guess I will add this to my list of things to look for subconsciously while evaluating expressions)
 

Related to Plane Wave Propogation (help understanding steps in book)

1. What is plane wave propagation?

Plane wave propagation is the movement of a plane wave through a medium, such as air or water. A plane wave is a type of electromagnetic wave that has a constant amplitude and phase, and travels in a straight line. It is a fundamental concept in the field of optics and is used to describe the behavior of light waves and other electromagnetic waves.

2. How does a plane wave propagate?

A plane wave propagates by oscillating perpendicular to its direction of travel. This means that the electric and magnetic fields of the wave are perpendicular to each other and to the direction of propagation. As the wave moves through the medium, it transfers energy and momentum along its path.

3. What are the steps involved in understanding plane wave propagation?

The steps involved in understanding plane wave propagation include understanding the concept of a plane wave, knowing the properties of the medium through which the wave is traveling, understanding the equations that describe the behavior of the wave, and applying these equations to specific scenarios to analyze and predict the propagation of the wave.

4. What are some examples of plane wave propagation?

Some examples of plane wave propagation include the propagation of light through air, the propagation of radio waves through space, and the propagation of sound waves through water. Plane wave propagation is also used in various engineering applications, such as in the design of antennas and communication systems.

5. How is plane wave propagation related to other types of wave propagation?

Plane wave propagation is a special case of wave propagation, where the wavefronts are flat and parallel to each other. Other types of wave propagation include spherical wave propagation, where the wavefronts are curved and expand outward from a point source, and cylindrical wave propagation, where the wavefronts are curved and expand outward from a line source. These different types of wave propagation can be described using similar equations, but have different characteristics and applications.

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