How to prove a wave is travelling ?

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In summary: Think of the wave as a 'waveform' (which is a fixed shape) that is traveling left or right. A point in the waveform is uniquely determined by the argument to the sine function. So to fix the point in the waveform, set the phase equal to any constant C. So you have ##C=kx-\omega t##. Now solve for location ##x## as a function of ##t##. That tells you that the location of that point in the waveform is a linear function of time. It's easiest to visualise this if the point is a crest or a trough, where C is ##(2m+\frac12)\pi## or ##(2m-
  • #1
i_hate_math
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Homework Statement


This is not a homework of any form, I am simply interested in proving a wave equation of the form f(x,t)=A•sin(k•x-ω•t) is a traveling wave, preferably an algebraic proof. Thanks heaps!

Homework Equations


f(x,t)=A•sin(k•x-ω•t)

The Attempt at a Solution


I was able to proof this by using graphs, but I need to know the algebra behind it.
 
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  • #2
Think of the wave as a 'waveform' (which is a fixed shape) that is traveling left or right.

Write an equation for the location of a particular point in the waveform, as a function of time.
A point in the waveform is uniquely determined by the argument to the sine function. So to fix the point in the waveform, set the phase equal to any constant C. So you have ##C=kx-\omega t##. Now solve for location ##x## as a function of ##t##. That tells you that the location of that point in the waveform is a linear function of time. It's easiest to visualise this if the point is a crest or a trough, where C is ##(2m+\frac12)\pi## or ##(2m-\frac12)\pi## respectively (##m## being an integer). But it's just as true for any point on the waveform.
 
  • #3
Here's something to get you started. Look at the wave at a single point (for simplicity, look at a zero of the function). What can you deduce about the argument of the sine function? What does this tell you about the motion of the wave?

EDIT. Just got beat to it :frown:
 
  • #4
To be a more accurate what you have here is a function ##f(x,t)## that satisfies the one dimensional wave equation https://en.wikipedia.org/wiki/Wave_equation

Any function that satisfies the wave equation is a traveling wave. (or in some cases a standing wave).

PS. You ll need to know a bit of differential calculus to understand the Wikipedia article.
 
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  • #5
i_hate_math said:

Homework Statement


This is not a homework of any form, I am simply interested in proving a wave equation of the form f(x,t)=A•sin(k•x-ω•t) is a traveling wave, preferably an algebraic proof. Thanks heaps!

Homework Equations


f(x,t)=A•sin(k•x-ω•t)

The Attempt at a Solution


I was able to proof this by using graphs, but I need to know the algebra behind it.
I guess we are talking about waves traveling without changing shape (otherwise we have to discuss dispersion, group velocity vs phase velocity, etc).

In general, if you have a function of the combination ##kx- \omega t## or of ##kx+ \omega t##, then you have a wave traveling to the right (or to the left in the second case) at a speed equal to ##v=\omega/k##. In other words, you can tell by using the following trick: if you set ##kx=\omega t## in the function and magically all dependence on x and t disappears, you have a wave traveling to the right without changing shape. If you set ##kx=-\omega t## in the function and magically all dependence on x and t disappears, you have a wave traveling to the left without changing shape.
 

1. How do you measure the speed of a wave?

The speed of a wave can be measured by dividing the wavelength by the period. The wavelength is the distance between two consecutive peaks or troughs of the wave, while the period is the time it takes for one complete wave cycle to pass a fixed point.

2. Can the direction of a wave be changed?

Yes, the direction of a wave can be changed through the process of reflection, refraction, or diffraction. Reflection occurs when a wave bounces off a barrier, such as a wall or a mirror. Refraction happens when a wave passes through a medium with a different density, causing it to change direction. Diffraction refers to the bending of a wave around an obstacle or through an opening.

3. What is the difference between a transverse and a longitudinal wave?

A transverse wave is a type of wave where the particles of the medium vibrate perpendicular to the direction of the wave's motion. An example of a transverse wave is a wave on a string. On the other hand, a longitudinal wave is a type of wave where the particles of the medium vibrate parallel to the direction of the wave's motion. An example of a longitudinal wave is a sound wave.

4. How can you prove that a wave is travelling?

To prove that a wave is travelling, you can observe the motion of the particles in the medium. In a transverse wave, the particles will move up and down as the wave passes through them, while in a longitudinal wave, the particles will move back and forth in the same direction as the wave's motion. Additionally, you can measure the frequency and wavelength of the wave, which will both change as the wave travels.

5. What are some real-life examples of wave motion?

There are many real-life examples of wave motion, including ocean waves, sound waves, light waves, seismic waves, and electromagnetic waves. Ocean waves are caused by wind and can travel long distances, while sound waves are created by vibrations and can travel through various mediums, such as air, water, and solids. Light waves are a type of electromagnetic wave that allows us to see, and seismic waves are produced by earthquakes. Electromagnetic waves include radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma rays.

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