1. Passive Acoustic Radiators Justin Yates, Wittenberg University Spring 2014

2. Outline Review of Passive Acoustic Radiator (PAR or PR) Model of PAR system as a damped, driven oscillator Solutions to the system and their physical meanings Designing a PAR Questions

3. Review We can model the passive cone of a PAR as a mass on a spring The oscillator then, is both driven and damped β x k F(t)

4. Review PR’s need to be tuned Tuning can be most easily achieved by changing the mass Masses (m) Restoring spring (k)

5. Damped Harmonic Oscillator Modeling the PR as a damped, driven oscillator gives us the 2 nd order linear differential equation for the position of the cone; This equation comes from the equation of motion for the driving force of the oscillator Where I have simplified by dividing by m and substituting equivalent coefficients. The driving force in a PAR is actually a sinusoidal function of time so Eq. (1)

6. Solutions The homogeneous equation corresponding to Eq. (1) has the characteristic equation whose solution takes the form After finding the roots and simplifying the corresponding solution is

7. Solutions The homogeneous equation corresponding to Eq. (1) has the particular solution of the form The complex part can be drawn out into the coefficients, let and multiply C by its conjugate, C* so when solved for A 2 gives and thus the particular solution is

8. Discussion The general solution is It is important to note that the solution consists of two distinct parts. Of the two terms, one is transient and the other is harmonic, or steady. TransientHarmonic

9. Physical Meaning The full solution and its parts

10. Designing a PAR We can take what we have learned about the physical meanings of the parts of the general solution to examine the ideal conditions for a PAR system. – General solution – Damping coeff. and the quality factor – The “notch”

11. Designing a PAR A PAR is designed to exploit the harmonic term of the general solution. To physically reason this, lets recall the period of a sinusoidal function Where f is the frequency. When the frequency decreases, the period of the wave increases. The typical frequency range for bass notes is around 120-20Hz. This means that bass notes tend to hang around longer compared to higher frequencies. This make them correspond to the harmonic term.

12. Designing a PAR PAR’s work best with a low Q-factor. Increasing β is going to decrease the Q-factor. Another way to think about desired Q and β, is with the transient term. Oscillations die out in the absence of a driving force according to the exponential in the term When designing a PAR you do not want the system to continue to “ring” long after a note is over. We want this term to die out quickly. This means that a properly designed PAR should not have β << ω 0 to keep Q low.

13. Frequency vs. Output Red—ported cabinet Green—Sealed cabinet Blue—PAR (ω 0 = about 15Hz) The PAR is being driven above resonance here and the notch occurs below the range of human hearing 20

14. Further Exploration Effects of: – box size on output – Surface area of cone – Resonant frequency on the notch (superposition) Why the “notch” occurs

15. Questions?

16. References Passive Radiator Systems. (2010, August 16). Retrieved Febuary 1, 2014, from DIY Subwoofers: http://www.diysubwoofers.org/prd/ Transient (acoustics). (2014, January). Retrieved January 30, 2014, from Wikipedia: http://en.wikipedia.org/wiki/Transient_%28acousti cs%29 http://en.wikipedia.org/wiki/Transient_%28acousti Taylor, J. R. (2005). Classical Mechanics. University Science Books. William E. Boyce, R. C. (2009). Elementary Differential Equations and Boundary Value Problems. John Wiley and Sons Inc.