# Faster than the speed of light?

jbriggs444
Homework Helper
2019 Award
[...]as the sine waves nears the X axis the movement/fluctuation in the Y direction was greater than in the X. And since X represented the speed of light, the Y-component movement/fluctuation would have thus been exceeding c -- which prompted my question.
To nitpick, if you look at the graph of f(x) = sin x where x is in radians then the "movement of y" never exceeds the "movement of x". In other words, the slope of a sine wave never exceeds 1. It only reaches 1 (or -1) at the crossings of the x axis.

On the other hand, if you are allowed to increase the amplitude of a sine wave of a fixed frequency, its slope can become arbitrarily large. That could lead to the same difficulty [were it not for the fact that the slope does not amount to a velocity].

Mister T
Gold Member
In other words, the slope of a sine wave never exceeds 1. It only reaches 1 (or -1) at the crossings of the x axis.
Slope never exceeds 1 what? That statement has meaning only if the rise and run are measured in the same units!

And in this case, they aren't!

But in the case of a transverse wave where the rise and run are both measured in the same units, such as meters or the like, that would apply. Interesting! So, for a wave on a string books often misrepresent this situation and get it wrong?

Of course, the amplitude must be small compared to the wavelength for the usual analysis to apply. This is the approximation used when deriving the relationship between wave speed ##v##, string tension ##\tau## and string density ##\mu##. $$v=\displaystyle \sqrt \frac{\tau}{\mu}.$$

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jbriggs444
Homework Helper
2019 Award
Slope never exceeds 1 what?
In the case of a mathematical sine wave, just 1.
That statement has meaning only if the rise and run are measured in the same units!
And in this case, they aren't!
Yes, we are in agreement on that point.
But in the case of a transverse wave where the rise and run are both measured in the same units, such as meters or the like, that would apply. Interesting! So, for a wave on a string books often misrepresent this situation and get it wrong?
Yes, for a wave on a string, one could conceivably have a transverse velocity greater than the propagation velocity. Without thinking it through carefully, I expect things would get pretty non-linear before that happened. [A string with a wave whose amplitude is a significant fraction of its wavelength is going to be under more tension than a string with a lower amplitude wave].

Mister T
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