Why do we use the sin x = x approximation for calculating waves on a rope?

[bob45]

hi, when we try to find the speed of a wave on a rope v = (F/u)^1/2, we use the fact that if the angles are small then sin x = x. I understand the approximation but not WHY we use the approximation. We say delta(Theta) is small (and then amplitude is small) then ... . So the proof is only correct for small value of theta. OK. But why don't we just use d(Theta) (infinitesimal), the result must also be true, and then the proof is correct for all value of amplitude of the wave (i mean we can just zoom in it and the angles will be small ..)

 

[Andy Resnick]

hi, when we try to find the speed of a wave on a rope v = (F/u)^1/2, we use the fact that if the angles are small then sin x = x. I understand the approximation but not WHY we use the approximation. We say delta(Theta) is small (and then amplitude is small) then ... . So the proof is only correct for small value of theta. OK. But why don't we just use d(Theta) (infinitesimal), the result must also be true, and then the proof is correct for all value of amplitude of the wave (i mean we can just zoom in it and the angles will be small ..)

That approximation (sin x = x) called a 'linear approximation' and it simplifies the underlying differential equations. The same approximation is used in, for example, optics (the paraxial approximation). It's certainly possible to work with the full nonlinear equations, but since there are not analytical solutions everything has to be done numerically, making it inappropriate for introductory coursework.

 

[bob45]

"OK. But why don't we just use d(Theta) (infinitesimal), the result must also be true, and then the proof is correct for all value of amplitude of the wave (i mean we can just zoom in it and the angles will be small ..) " ... please

 

[Mark44]

"OK. But why don't we just use d(Theta) (infinitesimal),

What does "d(Theta)(infinitesimal)" mean?

 

[bob45]

if i note x the angle theta, i mean we can use dx instead of Δx, we can look at a infinitesimal angle instead of a little angle.

 

[Ibix]

"OK. But why don't we just use d(Theta) (infinitesimal), the result must also be true, and then the proof is correct for all value of amplitude of the wave (i mean we can just zoom in it and the angles will be small ..) " ... please

Do you mean that the difference in $\theta$ between a point on the rope at $x$ and one at $x+dx$ is an infinitesimal angle $d\theta$? This is true, but the restoring force depends on $\theta$, not $d\theta$, so it's $\theta$ that we need to be small.

 

[bob45]

nope i am not speaking about wave equation for a rope, just for the speed of the wave, so the drawing is symetrical and there is no x + dx, just two forces symetrical, so the interval of the rope i want to make smaller and smaller so i want to work with dx or (dl the length of the part of the rope)

 

[Nugatory]

nope i am not speaking about wave equation for a rope, just for the speed of the wave

But that speed is calculated from the solutions to the wave equation so one way or another you're starting with a correct solution to the wave equation.

so the interval of the rope i want to make smaller and smaller so i want to work with dx or (dl the length of the part of the rope)

You can consider a arbitrarily small section of the rope, but its arbitrarily small mass will still be $\rho{dl}$ where $\rho$ is the linear density of the rope and the net force acting on it depends on the value of $\sin\theta$; in fact that's pretty much what we're doing when we solve the differential equation to derive the formula for the speed of the wave. If you could post or link to the symmetrical drawing you mention in your previous post we may be able to spot your misunderstanding... but if I were to be placing a bet, I would bet that the two forces are only symmetrical if you neglect a term that goes to zero along with $dl$ so cannot be correctly neglected.

 

[Vanadium 50]

OK. But why don't we just use d(Theta) (infinitesimal), the result must also be true,

Why don't you go through the derivation of that here?

 

[Mister T]

(i mean we can just zoom in it and the angles will be small ..)

The amplitude is small compared to the wavelength. No amount of zooming in or out will change that.

 

[suremarc]

If we’re going to do away with the small-angle approximation, by the same logic we might as well use general relativity to predict projectile motion; it has the added benefit of working in extreme gravitational fields. Or why not use the Navier-Stokes equations to analyze the sound waves generated by a handheld tuning fork?

I do not intend to be mean; I am only trying to point out the absurdity of your suggestion. I am no physicist, but it is my understanding that simplifying assumptions are often made in physics. Of course, the theories generated by these assumptions are only accurate in certain regimes, but for experimental purposes they are often good enough. Most importantly, these simplifications allow physicists to save time they would have spent wrestling with the (sometimes intractable) general theory, instead spending that time probing the implications of the simpler-but-still-accurate theory within its domain of validity. I imagine there are entire fields of physics based on making these assumptions in the context of a more general theory.

 

[vco]

Or why not use the Navier-Stokes equations to analyze the sound waves generated by a handheld tuning fork?

Actually, it appears someone has done exactly this.
http://persson.berkeley.edu/pub/froehle14tuningfork.pdf