One dimensional wave, function of a wave

In summary, on page 20 of 'Optics' by Eugene Hecht, the author discusses the function of a wave and its direction of travel, represented by the equation ##\psi(x)=f(x-t)##. He also mentions an equivalent form of this equation, ##\psi(x-vt)=F(-\frac{x-vt}{v})=F(t-\frac{x}{v})##, which can be used to find the form of a wave at any other time or location. This concept is not referenced again in the chapter or exercises, but can be applied in real-world problems such as in the reflection of electromagnetic waves.
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Taylor_1989
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I am currently reading through 'Optics' by Eugene Hecht chp 2 page 20, he talks about the function of the wave and the direction of travel of the wave i.e ##\psi(x)=f(x-t)## and right at the bottom of the page he say this:

Equation (2.5) is often expressed equivalently as some function of ##t - x/v)##, since,

$$f(x-vt)=F(-\frac{x-vt}{v})=F(t-\frac{x}{v})$$

What I am trying to understand is the relevance of this, because he make no mentioned to this again in the chapter nor dose he refer to it in any of the exercise at then end, and just curious to now when this type of format would be used?

edit: the equation 2.5 he is referring to is: ##\psi(x)=f(x-vt)##
 
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For a wave traveling to the right ## \psi(x,t)=f(x-vt) ##. If we have the form ##\psi(x,0)=f(x) ## at ## t=0 ##, we can find the form at any other time by taking ## \psi(x,t)=\psi(x-vt,0) =f(x-vt) ##. ## \\ ## Alternatively, and this is what he is referring to, if we have ## \psi(x,t) ## as a function of time at ## x=0 ##, which is ## \psi(0,t)=g(t) ##, if it is a traveling wave with velocity ## v ##, it obeys ## \psi(x,t)=g(t-\frac{x}{v}) ##.
 
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And if you want to see an application of this, try working this homework problem that appeared on Physics Forums about a year ago. The solution is in post 2, but you might see if you can work it yourself before looking at the solution. https://www.physicsforums.com/threads/reflection-of-em-wave.934953/ Also notice it fooled a couple of other people who tried to work the problem without using the concept of your post above.
 
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1. What is a one dimensional wave?

A one dimensional wave is a type of wave motion that occurs in only one direction, typically represented by a graph of displacement vs. distance. This means that the wave propagates in one direction, but the particles in the medium through which the wave travels move in both directions perpendicular to the direction of wave propagation.

2. How is a one dimensional wave different from a two or three dimensional wave?

A one dimensional wave only travels in one direction, while a two dimensional wave travels in two directions (such as a surface wave on water) and a three dimensional wave travels in three directions (such as a sound wave in air). Additionally, the amplitude and wavelength of a one dimensional wave are represented by a single value, while in two and three dimensional waves, they are represented by vectors.

3. What is the function of a wave?

The function of a wave is to transfer energy from one point to another without transporting matter. This allows for the communication of information and the propagation of energy through various mediums.

4. How is a wave's function related to its frequency and wavelength?

A wave's function is determined by its frequency and wavelength. The frequency of a wave is the number of complete cycles it completes in one second, while the wavelength is the distance between two consecutive points of equal amplitude. These two values are inversely related - as the frequency increases, the wavelength decreases, and vice versa.

5. What is the mathematical representation of a one dimensional wave?

A one dimensional wave can be represented mathematically by a sine or cosine function. This function describes the displacement of particles in the medium as the wave passes through, and can be used to calculate various properties of the wave, such as its amplitude, wavelength, and velocity.

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