Vibrationel energy of vibration drop?

[M_1]

I want to calculate the vibrational energy of vibrating water drops. To start with I assume water is inviscid.

The frequency spectrum can be found in for example Landau and Lifshitz. But I would like to know the vibrational energy of each mode.

How should I proceed? Would it be correct to say that the vibrational energy is identical to the surface tension multiplied by (largest area - smallest area)? How can this be calculated for a given frequency and amplitude? I assume quadrupole vibrations are most important.

Thanks!

 

[eljose79]

Calculating the vibrational energy of vibrating water drops is an interesting problem. To start with, assuming water is inviscid is a good approximation for many cases, but it's important to keep in mind that real water does have some viscosity, which may affect the results. With that said, let's discuss how you can approach this problem.

First, let's define the vibrational energy as the total energy associated with the motion of the water molecules in the drop. This includes both the kinetic energy and the potential energy due to the intermolecular forces within the drop. The frequency spectrum of the drop can give you information about the different modes of vibration that are possible, but to calculate the vibrational energy of each mode, you will need to consider the amplitude of the vibration as well. This is because the energy of a vibrating system is directly proportional to the square of its amplitude.

To calculate the vibrational energy for a given mode, you can use the formula E = 1/2 * k * A^2, where E is the energy, k is the spring constant (which can be related to the frequency of the mode), and A is the amplitude of the vibration. The spring constant can be found in the literature or can be calculated using the surface tension and the size of the drop. However, this formula assumes a simple harmonic motion, so it may not be accurate for all modes of vibration. For more complex modes, you may need to use numerical methods to calculate the energy.

Regarding your question about the relationship between vibrational energy and surface tension, it is not accurate to say that the vibrational energy is equal to the surface tension multiplied by the difference in the largest and smallest areas. The surface tension only affects the potential energy of the drop, while the vibrational energy also includes the kinetic energy. Additionally, the surface tension may vary depending on the shape and size of the drop, so it cannot be used as a constant in this calculation.

In conclusion, to calculate the vibrational energy of water drops, you will need to consider the frequency spectrum, amplitude of vibration, and intermolecular forces within the drop. You may also need to use numerical methods for more complex modes of vibration. I hope this helps to guide your research and calculations. Best of luck!