Why is the term w(t-x/c) used in the cosine representation of a traveling wave?

In summary, the conversation discusses wave reflection and transmission. The general form of an incident wave in an infinite string with a density change is W = A cos(w(t-x/c)+θ). The question is raised about understanding the term w(t-x/c) and why x/c is negative. The discussion then delves into the physical meaning and derivation of this term, with some confusion about the conventional signs for space and time. Ultimately, the question remains unclear and more feedback is needed for further understanding.
  • #1
mcheung4
22
0
This is about wave reflection and transmission.

For an infinite string with a density change at x=0, consider an incident wave propagating to the right from x = -∞. The most general form is W = A cos(w(t-x/c)+θ), with amplitude A, angular freuqency w, time t, distance x (from origin), wave speed c and phase θ.

I do not understand the term w(t-x/c). what is x/c negative?


Update : http://www.animations.physics.unsw.edu.au/jw/travelling_sine_wave.htm

I understand this derivation apart from x' = x-vt. Shouldnt it be x' = x-vt since x' is moving relative to x?
 
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  • #2
c is the speed of the wave.

If you distort a string so that the height forms a wave of arbitrary shape y(x,t=0)=f(x), and the wave subsequently travels to the right (+x direction) without changing shape, then for t>0, y(x,t)=f(x-ct), where c is the speed of the wave.

If ##f(x)=A\cos( kx + \theta)##

What is ##y(x,t)=##?

Ignore the derivation you linked to for now: it is unnecessarily convoluted.
 
  • #3
The question is not clear to me. Are you trying to understand why there is a negative sign in the factor w(t-x/c)? Or are you trying to understand the physical meaning of those variables?
 
  • #4
or why it is the "x/c" that is negative as opposed to the "t"
i.e. why w(t-x/c) and not w[(x/c)-t] ... which is what you get off the standard derivation.

... or myriad other possibilities - which is why I wanted to see the derivation done first.
 
  • #5
mcheung4 said:
I do not understand the term w(t-x/c). what is x/c negative?

This is confusing at first. However, if you think that, as you get further away from the source (increasing x) the phase of the wave is 'earlier' (because the 'later' bit hasn't got there yet). So increasing x has to decrease the phase, if you are using the conventional signs for everything else.
 
  • #6
When you do the derivation for the traveling wave, though, the sine ends up on the time rather than the space contribution to the overall phase. A waveform f(x) traveling to the right with speed c changes as f(x-ct) ... see what I mean?

But I suppose of x is the observer position, the the observer is looking at an earlier part of the wave.
... I just don't think that's the context. Really need feedback.
i.e. the second question makes no sense.
 

1. What is the cosine representation of wave?

The cosine representation of wave is a mathematical method used to describe a wave in terms of its amplitude, frequency, and phase. It is a representation of a periodic function that is composed of cosine functions with different amplitudes, frequencies, and phases.

2. How is cosine representation of wave different from other representations?

The cosine representation of wave differs from other representations, such as the Fourier series or Fourier transform, in that it only uses cosine functions instead of a combination of cosine and sine functions. This makes it more suitable for representing real-world signals and simplifies the calculation process.

3. What are the advantages of using cosine representation of wave?

One of the main advantages of using cosine representation of wave is its ability to accurately represent periodic signals with sharp changes or discontinuities. It also allows for easy manipulation and analysis of signals, making it useful in various applications such as signal processing and data compression.

4. Can all waves be represented using cosine functions?

No, not all waves can be represented using cosine functions. Cosine representation is limited to periodic signals, meaning that the wave must repeat itself over a certain period of time. Non-periodic signals, such as a single pulse, cannot be accurately represented using cosine functions.

5. How is the cosine representation of wave used in practical applications?

The cosine representation of wave is used in various practical applications, such as audio and image processing, data compression, and telecommunications. It is also used in engineering and physics to analyze and describe different types of waves, including sound waves, electromagnetic waves, and mechanical waves.

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