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I'm trying to derive the equation for work done by a vibrating string, but I'm running into problems. The easiest way - the method used by the other question by this name - makes the approximation $\sin\theta\approx\tan\theta$, that is, the small angle approximation.

I'm fairly sure this doesn't reflect some underlying physical concept that changes the expression for high-amplitude high-frequency waves - for starters, I do have another derivation, but it makes the assumption without justification that $\frac{dK}{dx}=\frac{dU}{dx}$, $K$ kinetic energy and $U$ potential energy. So can anyone explain an alternate derivation, or else justify that assumption?

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1 Answer 1

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You might need to say some more about what you want to do with this equation, because you can descend into as much complexity as you like. Do you, for example, want to think about variable length strings, i.e. those where the tension lengthens the string and the tension itself is a function of position along the string? Do you want to think about a general, nontangential force? You could pull out Landau and Lifshitz "Theory of Elasticity" or Stephen Timoshenko "Strength of Materials Volume 2" or "Theory of Elasticity" and build something pretty complicated, but each new effect modeled is going to yield diminishing returns.

Assuming a constant tension $T$ in the string of linear density $\mu$ along its length $z$ and assuming still predominantly transverse motion $y(z,\,t)$ in one plane, I get:

$$T\, \cos\theta(z,t)\, \kappa(z,t) = \mu\, \partial_t^2 y\quad\quad\quad(1)$$

where $\theta$ is the string's angle made with the horizontal and $\kappa$ its curvature. Substituting for $\cos\theta$ and $\kappa$ yields:

$$T\,\partial_z^2 y = \mu\,(1+(\partial_z y)^2)^2\, \partial_t^2 y\quad\quad\quad(2)$$

which will give you a nice nonlinearity to chew on.

Next, you might consider a constant tension, constant length string with vibration but with motion in both transverse directions. So you're going to get two coupled nonlinear differential equations. Let our two transverse displacement components be $x(z,t)$ and $y(z,t)$, then the local tangent to the string be defined by the unit vector components $X = \partial_z x/\sqrt{1 + (\partial_z x)^2+(\partial_z y)^2}$ and $Y = \partial_z y/\sqrt{1 + (\partial_z x)^2+(\partial_z y)^2}$ so that (here $s$ is the arclength):

$$T\, \mathrm{d}_s X = T\,\partial_z\left(\frac{\partial_z x}{\sqrt{1+(\partial_z x)^2+(\partial_z y)^2}}\right) \mathrm{d}_s z = \mu\, \partial_t^2 x\quad\quad\quad(3)$$ $$T\, \mathrm{d}_s Y = T\,\partial_z\left(\frac{\partial_z y}{\sqrt{1+(\partial_z x)^2+(\partial_z y)^2}}\right) \mathrm{d}_s z = \mu\, \partial_t^2 y\quad\quad\quad(4)$$

whence (since $\mathrm{d}_z s = \sqrt{1+(\partial_z x)^2 + (\partial_z y)^2}$):

$$\left(1+ (\partial_z y)^2\right)\,\partial_z^2 x- \partial_z x\,\partial_z y\,\partial_z^2 y = \frac{\mu}{T}\,\left(1+(\partial_z x)^2+(\partial_z y)^2\right)^2\, \partial_t^2 x\quad\quad\quad(5)$$ $$\left(1+(\partial_z x)^2\right)\,\partial_z^2 y- \partial_z x\,\partial_z y\,\partial_z^2 x = \frac{\mu}{T}\,\left(1+(\partial_z x)^2+(\partial_z y)^2\right)^2\, \partial_t^2 y\quad\quad\quad(6)$$

which reduce to Eq. (2) when there is vibration in one plane only. You'll get some really interesting effects from these coupled equations: whirling, coupling of energy from $x$ to $y$ and back again and so forth.

The next step would be to think of the axial motion of the string and the attendant variable tension along the string's length. This would only be apparent well into the nonlinear régime and likely (5) and (6) should model most of the nonlinear effects you will need.


Energies in the String

If you are seeking to find out the work done by the end of the string, then you would need a model of what it's linked to and therefore a tension to displacement expression - likely a differential equation, which will be a differential equation. Now the tension $T$ is a function of time, so you're beginning to get seriously interesting! You might also be interested in looking at a tension varying with length at this point, with the local tension defined by $E\,A\,\epsilon(z,t) = k_T\, \epsilon(z,t)$, where $E$ is the string's Young's modulus, $A$ its cross-sectional area and $\epsilon(z,t)$ the strain. It makes more sense to use $k_T$ and measure experimentally: it's not going to be easy to work out $k_T$ from first principles from the material elastic constants for a braided or stranded string! The sting's curvature begets the strain: $\mathrm{d} s = \sqrt{1+(\partial_z x)^2 + (\partial_z y)^2} \mathrm{d} z$ so that

$$\epsilon(z, t) =\sqrt{1+(\partial_z x)^2 + (\partial_z y)^2} - 1 \approx \frac{1}{2}\left((\partial_z x)^2 + (\partial_z y)^2\right)\quad\quad\quad(7)$$

If you looking for loss in the string, a good model of air drag force is $−\lambda\,\partial_t x$ , $−\lambda\,\partial_t y$ (i.e. proportional to transverse velocity), which terms you'll need to include in the dynamical equation, then work out loss from the power dissipated by these terms. Internal material bending losses are complicated to model: often you can do this kind of thing by replacing material elastic constants with lossy elastic operators - so you would replace the Young's modulus $E$ for example by something of the form $E+E_t \partial_t$, for some loss constant $E_t$; equivalently, you would work with $k_T + k_1 \partial_t$ for the string's effective srping constant.

But, at last, if you, as I now understand from your questions, are looking simply to find out the energy needed to set the vibration up (the energy stored in the string) in a lossless string, then you can work as follows. The kinetic energy per unit length is obvious: it is simply:

$$K(z,t) = \frac{1}{2}\,\mu\,\left((\partial_t x)^2 + (\partial_t y)^2\right)\quad\quad\quad(8)$$

Now, if we assume that the displacement is small, such that the at first high tension $T$ does not change much as the string vibrates, then the work done by $T$ in straining a length $\mathrm{d}z$ of string is $T\,\epsilon\,\mathrm{d}z$, so that the potential energy stored per unit length is, from Eq. (7):

$$U(z,t) = T\,\epsilon\ = \left(\sqrt{1+(\partial_z x)^2 + (\partial_z y)^2} - 1\right)\,T\approx\frac{1}{2}\,T\, \left((\partial_z x)^2 + (\partial_z y)^2\right)\quad\quad\quad(9)$$

the approximation holding when $|\partial_z x|,\,|\partial_z y|\ll 1$. These are the general equations. To find the dispersion relationship for the uncoupled linear vibration equations $T\,\partial_z^2 y = \mu\, \partial_t^2 y$, $T\,\partial_z^2 x = \mu\, \partial_t^2 x$ we study solutions of the form $\exp(i\,(k\,z\pm\omega\,t))$ where $k$ is the wavenumber and $\omega$ the angular frequency; on substitution into the linear equations, we get $T\,k^2 = \mu\,\omega^2$ or:

$$c = \left|\frac{\omega}{k}\right| = \sqrt{\frac{T}{\mu}}\quad\quad\quad(10)$$

so for such a wave, Eq. (8) and Eq. (9) (the latter in the small vibration $|\partial_z x|,\,|\partial_z y|\ll 1$ approximation) can be combined to show that $U(z,t) = K(z,t)$, as you state. Likewise, by using this relationship as well as Parseval's theorem for Fourier series for any superposition of frequencies such that the waveshape is periodic, you can prove that the total kinetic and potential energies integrated over a wavelength are equal. But this is for the linear régime only. More generally, you must use Eq. (5) and Eq. (6) together with Eq. (8) and Eq. (9) separately. Even with these equations, it would be altogether reasonable to assume the small vibration approximation with Eq. (9), because none of the above considers $z$-directed components of the force, which will become significant with angles that are big enough to make the small vibration approximation of Eq. (9) invalid. Therefore, your final set (approximating the RHS of (5) and (6) in the same way as (9))might be:

$$\begin{array}{rcl} \left(1+ (\partial_z y)^2\right)\,\partial_z^2 x- \partial_z x\,\partial_z y\,\partial_z^2 y &=& c^2\,\left(1+2\,(\partial_z x)^2+2\,(\partial_z y)^2\right)\, \partial_t^2 x\\ \left(1+ (\partial_z x)^2\right)\,\partial_z^2 y- \partial_z x\,\partial_z y\,\partial_z^2 x &=& c^2\,\left(1+2\,(\partial_z x)^2+2\,(\partial_z y)^2\right)\, \partial_t^2 y\\ K(z,t) &=& \frac{1}{2}\,\mu\,\left((\partial_t x)^2 + (\partial_t y)^2\right)\\ U(z,t) &=& \frac{1}{2}\,\mu\,c^2\, \left((\partial_z x)^2 + (\partial_z y)^2\right)\end{array}\quad\quad\quad(11)$$

with $c$ defined by Eq.(10).

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