What determines the point of energy spillover to higher modes of a standing wave resonator? One of the better known physics demonstrations for standing wave resonance is the singing rod . By holding the rod exactly in the middle the demonstrator constrains the first mode of excitation - the mode with wavelength the size of the length of the rod.
I've done this demonstration many times and found that if I persist pumping energy into the rod that the first mode of excitation appears to saturate, and at that point it further appears that the additional energy spills over into a higher frequency mode - a mode in which the wavelength is equal to half the length of the rod. This is heard as an overtone - a frequency twice the frequency of the first mode.
So my question is what physical parameter(s) in the rod determine the point at which the first mode saturates (where energy 'fills' that mode) and begins to spill over into the other mode?
Also can the answer be generalized to other systems that exhibit standing wave resonance, such as the Rijke Tube, for example?
My experience is that saturation comes about from something nonlinear in the physics. But specifically what is this nonlinearity
 A: Generally, in linear systems modes are independent. Energy does not flow from one mode to another. What causes the coupling is a nonlinearity. The nonlinearity reveals itself at higher amplitudes (nonlinear terms are small at small amplitudes). Thus, when you drive the rod just a little bit the energy DOES go to the higher harmonics, but the coupling is weak and only a small fraction of energy flows. As you drive the first mode stronger the nonlinearity becomes more and more pronounced and you can detect the energy exchange between the modes.
What kind of nonlinearity? First, there is material nonlinearity. Hooke's law is just a linear approximation, in reality elastic forces are not linear in displacement. Second, there is geometric nonlinearity which takes into account change in geometry of a structure as it deforms. There may be more system-specific causes of nonlinearities as well. 
EDIT: I am not sure that in this demonstration you drive only the first mode. So in addition to the mode coupling there may be just direct excitation of higher harmonics by driving.
A: A possible answer for that might be that if you have a rope with the length $L$, you have a frequency $f$ as the first harmonic frequency with $T=\frac{1}{f}$ as the time between two amplitude maxima. This time is determined by the frequency how fast the wave can propagate in the rope, and is therefore bound to the speed of sound in the rope $$\nu = \left(\frac{E}{\rho}\right)^\frac{1}{2}$$
The energy of a wave with a wavelength $\lambda$ over one wavelength his equal to 
$$E_\lambda=\frac{1}{2}\mu\omega^2A^2\nu$$ 
with $\mu$ equals the mass per unit length of the rope, $A$ the wave amplitude and $nu$ the velocity of the waves moving in the rope. 
So set together: If we have a standing wave in the basic frequency, the energy is $E_0$ with 
$$E_0 = \frac{1}{4}\mu\omega_0^2A^2\nu$$
Here we have only $\frac{1}{4}$ because we only have half the wavelength. When increasing the frequency so that we get the next harmonic with $\lambda_1 = \frac{1}{2}\cdot\lambda_0$, we get an energy of 
$$E_1 = \frac{1}{2}\mu\omega_1^2A^2\nu = \frac{1}{2}\mu\omega_0^2\cdot4\cdot A^2\cdot\nu = 4\,E_0$$
as additional energy, and the total energy in the system is 
$$E = E_1 + E_0$$
This can be continued for higher frequencies, too, so the complete energy in the system is 
$$E = \sum_i E_i$$
Thus, I would say that if you put energy in the system, all modes are stimulated, but you can not see the higher modes because of the bigger amount of energy which is needed. The material constants are completely linear in this calculation.
Disclaimer: I might not completely be correct, and there could be some calculation mistakes in it.
