Speed wave in a spring (Hooke's Law)

[Maurice]

First sorry for my English.

If I know the tension T in a elastic string, the initial length L , the length after stretching (L+l),the mass density, how could I find the speed wave in the string by using Hooke's Law and considering the string as a spring,

Thanks.

 

[ajb]

You know that Hooke's law gives rise to simple harmonic motion and from there you know the speed of the mass, or can calculate it easy enough using calculus. I don't want to give it all away here...

[JonathanApps]

It's a bit fiddly to derive: the wave equation on the string is

 

[math]

\frac{\partial^{2}\psi}{\partial t^{2}} = \frac{T}{\mu}\frac{\partial^{2}\psi}{\partial x^{2}}

[/math]

 

where [math]\mu[/math] is the mass density and [math]\psi(x,t)[/math] is the displacement, either along the

string or perpendicular to it, depending on what kind of wave it is (longitudinal or transverse).

 

You might be able to work out the speed from that

[elfmotat]

You may find this helpful:

 

Edited October 24, 2014 by elfmotat

[JonathanApps]

I'm not sure what it has to do with Hooke's Law though. That is about a mass on the end of a spring: T = l x constant. This is a more complicated situation, although in the example elfmotat gave above, I guess you're applying a version of Hooke's law to the infinitesimal string elements.

[elfmotat]

I'm not sure what it has to do with Hooke's Law though. That is about a mass on the end of a spring: T = l x constant. This is a more complicated situation, although in the example elfmotat gave above, I guess you're applying a version of Hooke's law to the infinitesimal string elements.

 

I guess it depends on what the OP means by "wave." I don't know whether he's talking about transverse waves (like in the video I posted) or about longitudinal waves.

 

EDIT: I re-read the question I'm still unsure about which type of wave he's considering.

Edited October 24, 2014 by elfmotat

[Maurice]

In fact I found the tension, the mass density and the wave speed in the string but what I don't understand is if I add a thin elastic string and if I stitched the thin elastic string at a normal string (the 2 are considered as a continuum), how can I find the wave speed in the thin elastic string by using Hooke's Law and considering the thin string as A SPRING and without knowing the linear elasticity

[studiot]

You need to tell us if you understand calculus, and the calculus of the wave equation.

 

Or did your professor give you an equation using sine or cos?

 

Can you obtain an equation using a restoring force? you need this for your question if you want to use Hooke's law

Edited October 24, 2014 by studiot

[Maurice]

Indeed I have some difficulties to understand clearly the wave equation and the solutions. One of the solutions is a sinusoid and take the first derivative to find the wave speed but I don't understand how to use the amplitude A, the angular frequency w and t?

Thanks.

[studiot]

Maurice, you really do need to help us here to find the correct level to answer you.

 

I asked about calculus because you said the stretched length is (L+l), which is not calculus.

 

Also you need to answer elfmotat's question

 

Do you mean a transverse wave or do you mean a longitudinal wave.

 

For a wave proceeding down a spring I would guess you mean a longitudinal wave, which is a pressure wave.

 

For such a wave you should be converting the tension to a stress (=pressure) by dividing by the cross sectional area and.

 

Also you should be considering displacement not length for the spring.

[Maurice]

In fact there is a thin elastic string stitched to a normal string (to be considered as a continuum) and I must find the speed wave in this new thicker string.

Don't the mass of the thin elastic string, all what is found is the tension T, the linear density and also the speed wave in the normal string (v=sqrt(T/mu).

But the speed wave in the new thicker spring by using Hooke's Law F=-k.x?

At equilibrium T=-k.x, I know T and k(new length - original length),so I can find k(spring constant) but then to find the new wave speed,for me it is the same consider the same new length and the negligible mass of the thin elastic string?