Collapsing Bridges   --An Investigation of Resonance
This document supplements chapter 11 of Partial Differential Equations and Fourier Analysis by K.K. Tung.
The equation numbers cited here refer to that document.

Consider the forced wave equation         (11.1-11.4)

            

In section 11.5 this system is  solved for         (11.16)

            

The solution is given by        (11.19-11.20)
 
            
 

In the simulations below,

            

First, consider a forcing function where
              
Since the second mode is absent, this forcing function is ``far'' from the resonant a frequency.  The simulation is on the   mode2    page.  Note that the maximum deflection of the bridge is less than 0.3% of the bridge length.

Next, consider a forcing function where
            
This forcing function is 1% above the frequency of the first harmonic mode.  The simulation is on the   mode1    page.  After about ten seconds the maximum deflection of the bridge is around 12% of the bridge length.  The deflections are already forty times greater than the maximum deflections that are ever seen in the last example!

Finally, consider a forcing function where
            
This forcing function is 1% above the frequency of the third fundamental mode.  The simulation is on the   mode3   page. As predicted in the text, the middle third of the bridge is out of phase with the ends.