Acoustics from A to Z: K&L

KINETIC is the energy

That always works in synergy  

With energy that is potential. 

Both of them are quintessential 

For unforced periodic motion. 

We sometimes fail to grasp this notion. 

I have often expressed amazement about how much one can understand about vibrations from studying the simplest of all the conceptual vibrating systems – namely, a mass connected to a spring, possibly with an added damping element. Of course, some of the utility of the mass-spring model stems from the fact that the behavior of any mode of a dynamic system corresponds to that of an equivalent mass-spring system. But why is a simple mass-spring assemblage an appropriate representation of anything that can vibrate? 

The answer is that such an assemblage incorporates an element that can store kinetic energy and one that can store potential energy – and an interchange between potential and kinetic energy is at the core of any vibration. Let’s consider the simplest situation of free (unforced) vibration of a spring-mass system. If the mass deflected from equilibrium and released with zero velocity, then it initially has no kinetic energy, but there is potential energy stored in the spring. The spring accelerates the mass, giving it kinetic energy, but losing some of its potential energy in the process. This goes on until the mass reaches the equilibrium position, where the spring’s potential energy storage is zero and all of the energy of the system is kinetic. As the mass moves further, it deflects the spring, causing energy to be stored in it, but giving up a corresponding amount of kinetic energy. And so on. 

A few years ago Sound and Vibration magazine and I offered a prize to that reader who would give me the best physical explanation (that is, without the use of mathematics) of why a simple undamped mass-spring system has a definite natural frequency. Although S&V gave away the prize, I was not entirely satisfied with the explanation. So, here is mine. Can you challenge or improve upon it? 

A system has a natural frequency if, as it vibrates in absence of external forces, it takes the same time to complete each cycle. Because the total energy (the sum of the kinetic and the potential energy) in an undamped freely vibrating system is constant, the instantaneous magnitude of the potential energy determines the corresponding instantaneous magnitude of the kinetic energy. This now implies that whenever the mass passes a given location (measured in terms of displacement from equilibrium) it does so with the same velocity. This velocity establishes the time interval it takes for the mass to move from one point to the neighboring one. Therefore, the mass always takes the same time to move from one point to any other point and it always takes the time for the entire round-trip it makes in a cycle. In other words, all cycles have the same period and thus the same frequency. 

The foregoing argument, incidentally, is not limited to linear springs, but also applies to springs with nonlinearites. For nonlinear springs, the natural frequency depends on the amplitude, as determined by the initial displacement and velocity. Why is the natural frequency of an undamped mass-spring system with a linear spring (a spring whose deflection is proportional to the applied force) independent of the amplitude? Borrowing from the language of textbooks: this is left as an exercise for the reader. 

The LOUDNESS of a sound we hear 

Tells how intense it may appear. 

For simple or for complex tones,  

One measures it in phons or sones.  

But how we do perceive a sound?  

Depends on what else is around. 

  Loudness refers to the subjective evaluation of a sound. A 3 dB change in sound pressure level (which corresponds to a doubling or halving of the sound power) results in a barely perceptible change in the perceived loudness. A 10 dB increase in the sound pressure level (which corresponds to a tenfold increase in the sound power) is judged as doubling the loudness. According to Bies and Hansen [1] if one started with 100 trombone players behind a screen, all doing their best, and if 99 of them leave, the audience would perceive a loudness reduction by a factor of four. Advertisements that claim a 99% noise reduction for similar scenarios “are written by the uninformed for the ignorant.” 

One ‘sone’ is defined as the loudness of a 1-kHz tone at a sound pressure level of 40 dB. A 1-kHz tone at N sones is N times as loud as this 40-dB tone. A 10 dB increase in the sound pressure level results in doubling of the loudness in sones. Plots of the frequency variations of the sound pressure levels that correspond to a given loudness are called “equal-loudness contours,” which are labeled by ‘phon’ numbers. All points on such a contour correspond to the same perceived loudness; thus, a 40 phon tone at 60 Hz sounds just as loud as a 40 phon tone at 8000 Hz, even though the related sound pressures may be quite different. For pure tones, the sone and phon measures are simply related, but for more complex sounds the situation becomes more phoney [2] Methods for estimating the loudness of sounds that are not pure tones are discussed by Small and Gales [3] for example. 

‘Masking’ – interference in the perception of one sound by the presence of another sound – may make communication difficult. And, it may constitute a critical safety issue, for example, where construction noise may mask an alarm signal or where a pedestrian’s earphones may mask the sound of an oncoming car. Sound masking may also have beneficial effects, some of which are realized by the installation of sound masking systems in open-plan offices in order to eliminate the distractions caused by neighboring conversations. Unfortunately, the masking needed to cover up the rock music from a neighboring apartment would have to be so loud that it would lead to more insanity than the music itself. 

1. Engineering Noise Control, D. Bies and C. H. Hansen, Unwin Hyman Ltd., 1988. 

2. David Towers of HMMH, an experienced punster, felt the need to top my “phon number” and “phoney” wordplay. His comment: “To each his sone!” 

3. “Hearing Characteristics,” A. M. Small, Jr., and R. S. Gales; Chapter 17 of Handbook of Acoustical Measurements and Noise Control, C. M. Harris, Ed.; Mc-Graw-Hill, Inc., 3rd Edition, 1991.