Wave Equation: Exploring Motion w/ Angles Near 90°

[quasar987]

If you remember, when in textbooks* they derive the wave equation by considering a small element of string and applying Newton's 2nd law on it, they make the assumption that the angles the tension makes at the two ends of the element with the horizontal is smallish, such that sin ~ tan. Without this assumption, the wave equation is NOT a good approximation of the equation of motion. The motion of the rope is only well described by the wave equ. when the angles are small, i.e. when the shape of the wave is flat-looking.

How can we ally the fact that this analysis shows that the motion of a wave on a rope/string is described by the wave equation only for small angles, but that in reality, a wave that looks like this is observable and its motion is a solution of the wave equation [y(x,t)=g(x-vt)] even though the angles are obviously not small (they're near 90° at some points).

*See Griffiths E&M pp.365 for exemple.

 

[whozum]

Either you or I are confused. The animation in that picture doesn't say much about the angle of oscillation. There doesn't need to be an angle of oscillation for SHM to occur, consider a spring. The wave equation applies to any instance of SHM.

 

[quasar987]

The angles I'm referring to are the thetas in

http://www.math.ubc.ca/~feldman/apps/wave.pdf

(There's no doubt that I'm confused though.)

 

[SpaceTiger]

How can we ally the fact that this analysis shows that the motion of a wave on a rope/string is described by the wave equation only for small angles, but that in reality, a wave that looks like this is observable and its motion is a solution of the wave equation [y(x,t)=g(x-vt)] even though the angles are obviously not small (they're near 90° at some points).

Where did you see that large-amplitude waves on a tense string satisfy the equation of simple harmonic motion? As whozum said, there are solutions to the wave equation for large angles, but they shouldn't apply to a string under tension.

One can certainly get "wave-like" motion from other equations and it would appear that a more precise treatment of the string under tension would yield a non-linear equation with solutions similar to the wave equation. Another example of such non-linear oscillations would be a pendulum. At small angles, it satisfies simple harmonic motion, but things get much more complicated at larger angles.

 

[jtbell]

How can we ally the fact that this analysis shows that the motion of a wave on a rope/string is described by the wave equation only for small angles, but that in reality, a wave that looks like this is observable and its motion is a solution of the wave equation [y(x,t)=g(x-vt)] even though the angles are obviously not small (they're near 90° at some points).

You mean that animation at the top of the page? I'll be astonished if you can make a real string or rope vibrate like that, with an amplitude that big compared to the width of the pulse. If you can do it, please take a picture of it and post it!

 

[quasar987]

You mean that animation at the top of the page? I'll be astonished if you can make a real string or rope vibrate like that, with an amplitude that big compared to the width of the pulse. If you can do it, please take a picture of it and post it!

Haha, I'll try it tomorrow! Seriously, I was under the impression that I could give any shape whatesoever to an impulse on a string and it would travel like a wave.

 

[jdavel]

Haha, I'll try it tomorrow! Seriously, I was under the impression that I could give any shape whatesoever to an impulse on a string and it would travel like a wave.

quasar,

That's partly true. You could give the string the shape of the NYC skyline, but then for it to satisfy the condition of the stretched string approximation you'd have to squish (by a constant multiple) the whole thing down to a very low amplitude. Then that shape would travel like a wave.

 

[quasar987]

I haven't been able to give it a reasonable shot. The wave travels too fast, it's already at the end of the rope before I'm done doing the motion that was suposed to give it it's shape.

But what would have hapened had I succeeded? In which specific ways would the resulting deformation have traveled diffirently than a deformation governed by the wave equation?

 

[quasar987]

I did it. Not with a rope though but will a flexible long rectangular stripe of plastic. I could apparently give the wave the amplitude I wanted and the deformation would seemngly travel at constant speed.

 

[whozum]

How accurately did the shape propagate? Did it keep its form? I would imagine the amplitude would shrink considerably sa the wave travelled. Thats pretty cool. How did you launch the initial shape?

 

[quasar987]

Amplitude is conserved as far as I could tell. If anything, I'd say that when the wave gets to the free end, the free end go higher up in the air than the initial amplitude I gave!


I initiated the wave with my hand. So yes that means I didn't had a very good point of view to evaluate amplitude and shape. But the wave in this material travels very (relatively) slowly. About 1 meter per second. Against ~5m/s for a wave on a rope.

 

[jdavel]

...But the wave in this material travels very (relatively) slowly. About 1 meter per second. Against ~5m/s for a wave on a rope.

1 meter/second? Are you sure this is even an elastic wave?