How to solve the monstrous wave equation of an inextensible string?

Question

Let $x(s,t)$ and $y(s,t)$ be two unknown functions. How to solve the following system of PDE (with known initial conditions)?

$$x_{tt}=\frac{\partial}{\partial s}\frac{y_{tt}-(y_s/x_s)x_{tt}}{\frac{\partial}{\partial s}(y_s/x_s)}$$ $$x_s^2+y_s^2=1$$

Background

In the 1-D wave equation for a string $\frac{\partial^2 y}{\partial x^2}=\frac1{c^2} \frac{\partial^2 y}{\partial t^2} $ it is assumed that the string can be slightly extended.

Now consider the case of an completely inextensible unifrom string: a string originally lies straight on the x-axis, and at time $t$ the particle originally at $x=s$ moves to $(x(s,t),y(s,t))$. The fundamental equations are

1. Inextensibility: $$x_s^2+y_s^2=1$$
2. Newton's second law in x-direction: $$\frac{\partial T_x}{\partial s}=\lambda \frac{\partial^2 x}{\partial t^2}$$ where $T_x(s,t)$ is the x-direction tension on the particle originally at $x=s$, at time $t$; $\lambda$ is the linear density of the string.
3. Newton's second law in y-direction: $$\frac{\partial T_y}{\partial s}=\lambda \frac{\partial^2 y}{\partial t^2}$$
4. Tension force vector is tangential to the string: $$\frac{T_y}{T_x}=\frac{y_s}{x_s}$$

Eliminating $T_x$ and $T_y$ one arrives at the above system of PDE.

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With very limited knowledge in the theory of PDE, I am not able to make any progress. I guess an analytical solution rarely exists, so by 'solve' I mean 'simplify'/'study', possibly in the following directions:

1. Converting this system into two new PDEs, each depending on $x(s,t)$ or $y(s,t)$ only.
2. Making physically reasonable approximations that preserves the property of inextensibility.
3. Rewriting the system in a form that suits numerical solutions.
4. Solving exactly/approximately for particular (non-trivial) boundary conditions and/or initial conditions, e.g. $x(0,t)=0,y(0,t)=A\sin(\omega t)$.
5. Interesting properties of the PDEs.

etc...

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The problem arises from a physical framework but boils down to a highly mathematical question, with (possibly) only slight physical interpretation. Therefore I would regard it as suitable for Math SE rather than Physics SE.

Thanks in advance.