vibrating string with variable density

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Equations

I am fairly sure your equation is wrong. It should be \partial_x c^2 \partial_x w not \partial_x^2 w.

Energy form etc. should give you this

Re: Equations

>I am *fairly sure* your equation is wrong. It should be \partial_x c^2 \partial_x w not \partial_x^2 w.
Fairly sure? Are you sure? As I see there neither any explanation to do so contundent claim nor I agree with your strange notation. On the contrary my notation is crystal. We're talking about the linear differential operator
L[u] = (\partial^2/\partial{t^2} - c^2(x)\partial^2/\partial{x^2})[u],
where L is applied to the transverse response u = u(x,t) which give us the *place* occuped for any particule of the vibrating string at any time.
As far as c^2(x) = T_0/\rho(x), it is interpreted as the ratio between the initial string tension and the linear variable density of the string, which is assumed an unidimensional, flexible continuum media as well as small strains, i.e. e<<1. I don't want to do my explanation so large, as this PDE appears in any book about elementary continuum mechanics. (as well as the inherent assumed assumptions)
>Energy form etc. should give you this
Energy form? What is that? Energy equation? I don't like it! I prefer use Euler-Poisson equation(a.ka. specialized Newton's Eq. for a continuum) since the little string cross section, (l>>d) enables us to disregard any shear force against the longitudinal tension of the string.

So that the problem about initial and boundary values

\partial^2{u}/\partial{t^2} - c^2(x)\partial^2{u}/\partial{x^2}= f(x,t), x\in (0,l), t>0,
u(x,0) = f(x),
\partial{u}/\partial{t}(x,0) = g(x),
and
u(0,t) = u(l,t) = 0,
1.- It is well-posed.
2.- Enjoys existence and uniqueness.
So my dear meylan: I am fairly sure my equation *is not wrong*.
Regards,
perucho

Re: Equations

I now understand that your problem has variable density but constant elasticity. I missed this point and was thinking of the more general string. It might be worthwhile adding this equation in as I may not be the only one who makes this mistake. Also, it might be good to point out that you have divided by density so the forcing function is dependent on this variable density.

Re: Equations

Hi meylan,
>I now understand that your problem has variable density but constant elasticity.

-Yes, in my entry I'm discussing a perfectly alastic string. But also it is possible to study another mathematical models like for instance a Maxwellian (viscoelastic) model (I wrote a paper about this matter). Indeed all depends about the assumed *constitutive equation* which is an experimental-empirical relation or an equation developed by the rational observation about the behavior of the string's material properties. You may assume constitutive functionals so general as

T(s,t) = L(e(s,t),s),
where T(s,t) is the generic string's tension, e(s,t) the strain (deformation) function and "s" is a parameter (usually the arc length). However, such general cases imply advanced numerical methods, a lot of programmation and the adoption of finite elements, perturbations, potential boundary value or discretisation.
In my paper I could get an exact solution for a non-Hookean model under certain restrictive conditions, obtaining a PDE with variable coefficients which was reduced to two ODEs once made suitable transformations; one of which was solved by Fourier-Bessel technique, and the another one was reduced to a Sturm-Liouville problem . General solution was represented by a series given by a linear (infinite) combination of the eigenfunctions and its correspondent eigenvalues.

>Also, it might be good to point out that you have divided by density so the forcing function is dependent on this variable density.

-Well, after of some working one arrives to the PDE

\rho(x).\partial^2{u}/\partial{x^2} - T_0.\partial^2{u}/\partial{t^2} = \rho(x).f(x,t). (1)

So, what you call "forcing function" is simply dividing (1) by \rho(x), and renaming T_0/\rho(x) = c^2(x).

perucho

Re: Equations

Thanks for this. My area is water waves and we obtain a similar equation for shallow water waves in variable depth, except that the depth term appears exactly in the way I suggested the density should for a string. This is what made me think your equations were wrong.