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

Helmholtz Resonator harmonic oscillator mass spring mechanical acoustical oscillator

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

Quantitative Helmholtz Resonator g, the ratio of specific heats

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. http://en.wikipedia.org/wiki/Heat_capacity_ratio

Quantitative Helmholtz Resonator http://en.wikipedia.org/wiki/Density_of_air

Quantitative Helmholtz Resonator Solution:

Quantitative Helmholtz Resonator

Quantitative Helmholtz Resonator Numbers: V=1l S = 3 cm2, L = 5 cm f= 129.1275 Hz, the C below middle C

Helmholtz Resonator University of Toronto (1876)

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

Helmholtz Absorber

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].

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:

The exponential solution We can insert it into our differential equation

The damped oscillator

General solution

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

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

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:

Case 1: Overdamping

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

Case 2: Critical Damping

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!

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

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.

Case 3: Underdamping

The forced oscillator

Phase

Amplitude

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

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

Amplitude

References T.D. Rossing: The Science of Sound, Addison Wesley; 1st edition (1982) ISBN: 0805385657 Jens Blauert, Script Communication Acoustics I (wave equation derivation), The script is currently translated by Ning into English. http://www.phys.unsw.edu.au/~jw/Helmholtz.html