1. Research Methods in Acoustics Lecture 3: Damped and Forced Oscillators
Jonas Braasch

2. Helmholtz Resonator harmonic oscillator mass spring mechanical
acoustical
oscillator

4. Quantitative Helmholtz Resonator
mass: mairL·S·r with L: length of bottle neck
S: Cross area of bottle neck
r: density of air

5. Quantitative Helmholtz Resonator
g, the ratio of specific heats

6. Gas γ H2
1.41 He
1.66
H2O
1.33 Ar
1.67
Dry Air
1.40
CO2
1.30 CO O2 NO
N2O
1.31
Cl2
1.34
CH4
1.32
The heat capacity ratio γ is simply the ratio of the heat capacity at constant pressure to that at constant volume
Heat capacity is a measurable physical quantity that characterizes the ability of a body to store heat as it changes in temperature. In the International System of Units, heat capacity is expressed in units of joules per kelvin.

7. Quantitative Helmholtz Resonator

8. Quantitative Helmholtz Resonator
Solution:

9. Quantitative Helmholtz Resonator

10. Quantitative Helmholtz Resonator
Numbers:
V=1l S = 3 cm2, L = 5 cm
f= Hz, the C below middle C

13. Helmholtz Resonator
University of
Toronto (1876)

14. Helmholtz Resonator w d r
f = resonance frequency in Hertz [Hz] r = slot width [mm]
w = slat width [mm] d = effective depth of slot [mm]
(1.2 x the actual thickness of the slat) D = depth of box [mm]. D

15. Helmholtz Absorber

16. Helmholtz Resonator Example: r=6 mm w=90mm d=30mm D=450mm
f = resonance frequency in Hertz [Hz] r = slot width [mm]
w = slat width [mm] d = effective depth of slot [mm]
(1.2 x the actual thickness of the slat) D = depth of box [mm].

17. The damped oscillator Before we start to deal with the damped
Oscillator. Let us derive the ideal oscillator using the complex e-function ejwt
The derivation of the e-function is an e-function. Therefore:

18. The exponential solution
We can insert it into our differential equation

19. The damped oscillator

20. General solution

21. General solution Case 1: overdamping Case 2: critical damping Case 3:
underdamping

22. Case 1: Overdamping with
Note that the term under the square root is positive. Our general
solution is:

23. Case 1: Overdamping
Note how we separated the ‘±’-sign into two separate additive
solutions (superposition) with each its own amplitude x1,2. It is
also important that the exponential term is always negative since:

24. Case 1: Overdamping

25. Case 2: Critical Damping
with
Note that the term under the square root is zero. Our general
solution is:

26. Case 2: Critical Damping

27. Case 2: Critical Damping
Since the exponent is always smaller than one of the two
solutions in the overdamping case:
Exp critical damping
Exp. overdamping
Condition for overdamping
The critical damping case is the case in which the oscillator comes
soonest to a rest!

28. Case 3: Underdamping with
Note that the term under the square root is negative. Our general
solution is:

29. Case 3: Underdamping
Again, we separated the ± into two separate additive solutions
(superposition) with each its own amplitude x1,2. We pulled out
exponential decaying real part from the oscillating imaginary part.

30. Case 3: Underdamping

31. The forced oscillator

34. Phase

35. Amplitude

36. Resonance-Amplitude
g=10
w=200 Hz
g=1
w=200 Hz

37. Resonance-Amplitude
g=10
w=150 Hz
g=10
w=300 Hz

38. Amplitude

39. References
T.D. Rossing: The Science of Sound, Addison Wesley; 1st edition (1982) ISBN:
Jens Blauert, Script Communication Acoustics I (wave equation derivation), The script is currently translated by Ning into English.