Question

explain using a non homogeneous spring de, why do wine glasses shatter at certain sonic pitches. give a spring de that could fit this property and show why

Answer 1

To model the wine glass problem, we can look at the anaogy from the ode for a forced damped harmonic oscilator which is,

Where gama is the daming factor, is the natural or resonance frequeency of the system, is the time varying force acting on the system with a frequency of omega .

This time varying force is due to pressure variance(sound) in the case of our wine glass problem. Here we can assume that the pressure is of the form,

Then our ode will become:

ie,

To solve the equation above, we can substitute,

Where B and C are arbitrary constants.

On solving the ode, we can obtain the amplitude as a function of applied frequency,

We can see that the amplitude shoots up when the applied frequency is at the resonance frequency of the material.

This amplitude will be too large for the wine glass to withstand and it will break, this phenomena will usualyy occur at the ultrasonic level since the natural frequency of glass is high!

0 cos wt ?7m