Why Soft Sounds Are Pleasurable

At the other extreme of physical pain, are soft, tactile impressions that are just above the threshold of perception. Imagine a gentle wind finding its way through body hair or the lightest of drizzles falling on one’s bare shoulders. The acoustic equivalent of such pleasures is the distant sound of rustling leaves. At about 10 decibels it has less than 10 times the intensity of the faintest audible sound. And it has only one of ten trillionths of the power per area of a sound wave that could cause pain. It seems that we have evolved not only to react to extreme stimuli that endanger us but also to be rewarded when the environment toys with the lower limits of our senses.

Although the above numbers are mathematically accurate—-they are based on the formula which equates decibels with 10 times the logarithmic ratio of a sound’s intensity to our threshold intensity of 10-12 watts/meter2—they exaggerate what our eardrums actually experience.

pressurewave
from hyperphysics.phy-astr.gsu.edu  Check link for neat demo idea using methane in pipe to visualise pressure variations.

Before delving further, just what is sound? When a leaf moves back and forth, or some other body vibrates, the oscillation causes a periodic disturbance of the surrounding air molecules. The movement in one direction causes molecules of the air (or other transmitting medium) to bunch up, increasing pressure in one region. As the object reverts to its original position, a region of less crowding among molecules occurs as well. Meanwhile, the crowded region transmits its kinetic energy to other particles of the medium, repeating the pattern of high and low densities as the pressure wave propagates away from its source.

As a result, what’s more appropriate in comparing sound-strength is the use of pressure amplitude, which is proportional to the square root of a sound’s intensity. For example if our slightly rustling leaves come in at 10 dB and a pneumatic sidewalk-breaker next to us is painfully wreaking havoc at 120 dB, first we figure out the intensity ratio from the exponential version of the decibel-log formula:

I2/I1 = 10 0.1(dB2 – dB1) =10 0.1(120 – 10) = 1011.

Then by taking the square root  of 1011, we obtain the pressure ratio— about 316 000. That’s still a huge factor for how much louder the pneumatic drill is in comparison to the distant sound of gently moving leaves.

A little more number-crunching can make us appreciate how wonderfully sensitive our ears are, at least when they are working well. The exact pressure amplitude(ΔPm) of a rustling sound can be calculated from:

ΔPm = √(2Iρv) ,

where I is the intensity, and ρ and v are the density and velocity of the medium. For air at room temperature the latter numbers are 1.2 kg per meter cubed and 343 m/s, and for a 10 dB-sound, I = 10-11 W/m². Thus ΔPm is  9.07 X 10-5 pascals. Comparing it to the atmospheric pressure of 101.3 kPa, we can actually detect a difference of 1 part in a billion, and it’s enough to give us pleasure! When you consider that it takes a few parts per million of psychotropic substances to buzz the brain, something that many marvel at, to me our acute sensitivity to sound seems more remarkable, while no drugs are necessary.

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Where Have All the Clotheslines Gone?

It’s always tempting to use a simplistic view of our surroundings because, in the short term, it seems to conserve energy and appear practical. But in many cases, and certainly in the case of clotheslines, the opposite is true.

Let’s start with an apparently trivial fact about a clothesline. No matter how taut you pull it, the line sags a little under its own weight. The curve is accentuated if the line’s material is heavier or if you start hanging clothes on it. It may superficially resemble the familiar parabola from high school texts, but it’s actually closer to being a catenary described by a hyperbolic cosine function.

Parabolic shapes in both artificial situations and in nature are actually less common than people imagine. To get a parabolic path from, say, a baseball hit upwards at an angle, there would have to be only gravity acting on it. Then the angle in flight would only be the result of the diminishing and then increasing vertical component combined with a constant horizontal component. But in reality air friction and wind change the flight path into a more complicated exponential function.

The tension of a clothesline has a vertical and horizontal component at every point along its curve. Being at equilibrium the chain-tension’s vertical components balances gravity, while its horizontal components are also countered by forces in the opposite direction. The fact that the tension’s two components constantly change with the rope’s varying angle over every little length of the clothesline is what gives rise to the catenary.

Clotheslines have disappeared from many neighborhoods not only because many people do not appreciate the mathematics of pedestrian objects. If they are a rarity it’s partly because of city bylaws inspired by their perceived unsightliness and the way they hinder the view of more pleasant things like trees and sky. What gives people the luxury of giving aesthetics priority is the existence of the clothes dryer. But clothes dryers, as essential as they may be to those with small apartments and living in temperate climates, suck up a great deal of energy, an estimated 6% of all power generated in the province of Ontario, Canada, for example. Compared to the wind-and-sun-option of clothes-drying, the combination of the mechanical dryer’s tumbling action and high heat removes more lint from clothes, wearing them out faster.

While giving convenience priority over environmental matters and household budgets, people also imagine an unnecessary dichotomy between dryers and outdoor clothes lines. But clothes can also be laid out to dry naturally and discreetly in garages and on racks made for decks and balconies. Without intruding on anyone’s views,  CO2 emissions or radioactive wastes will be reduced, assuming that certain homes rely on an electrical grid still dependent on methane combustion, coal or nuclear power. If not, there are still more benefits of natural drying:

  • as we mentioned, the lengthened life cycle of clothes and the associated savings;
  • the energy that a greener grid saves can be distributed to areas with a bigger ecological footprint;
  • a smaller contribution to the heat island effect of urban areas;
  • clothes racks do not need much maintenance;
  • With less use, dryers last a lot longer. We gave a dryer to my parents and since it shares their burden of laundry with outdoor and indoor laundry lines, twenty-eight years later, it is still working.

Keep in mind that real progress lies not in developing technologies in order to be enslaved to them. We progress when we constantly evaluate their interaction with human nature and assess their health/ecological impact and adjust accordingly.

Science from An Old Bucket of Water

wallpaper-mountains-1366x768
Trees’s growth  ignore the slope. Picture Source: https://previews.123rf.com/images/waldru/waldru1303/waldru130300029/18594282-Trees-on-the-mountain-slope-Winter-without-snow-Stock-Photo.jpg

If you climb a hill, you will notice that the trees are not perpendicular to the slope. With some variations due to wind, tree trunks generally meet the horizon at a right angle. Similarly, if you are collecting rain water with an old bucket on a slanted driveway, you will notice that the water level is not parallel to the asphalt. More water leans against the walls of the bucket facing downslope.

bucket

If you wait for more rain to fall into the bucket it will look like the second illustration. Wait longer and liquid water will not reach point C.  Instead it will overflow at point A. The bucket never fills with liquid.

All this is a reminder that gravity neither acts from the surface just below trees, nor from the surface below the driveway. It acts from the center of the Earth. We know from geometry that any radius is perpendicular to a line tangent to the circle. The horizon is equivalent to a tangent line, hence the reason for the alignment of trees and for the fact that points A and B in the bucket are at the same height above the horizon. (see green arrows in the diagram) It’s at those positions that they have the same potential energy, the product of weight and distance from the plane’s center.

If you leave the bucket out in late autumn, you will be in for a more pleasant surprise. After I knocked the bucket down, here’s what slid out along with the water that had not yet frozen.

icebucket
Two views of the same hollow cylinder of ice from a bucket. My keys are at the base for a sense of scale.

The ice is in the form of a hollow cylinder, one that eventually would have filled the whole bucket. The sight is a little deceiving. Unfrozen water is not the only thing missing from the picture. On its way out, the unfrozen water broke through a thin layer of surface-ice, which formed first because it’s the only part of the water that was directly in contact with the cold air. But why was the core of the liquid left unfrozen? The inner plastic walls not only cooled off faster than the subsurface water due to the lower specific heat of plastic, but the impurities and imperfections on its surface also provided nucleation sites for ice crystals to form. Even at the very beginning, one observes a crescent of thin ice on the colder surface, not coincidentally resting on the side of the bucket with more plastic exposed.

If we had waited long enough, why would the entire bucket have been filled with ice, something liquid water is incapable of doing when the bucket is sitting on a slope? On average, in liquid water, each molecule is hydrogen-bonded to about 3.4 neighboring molecules that constantly break and reform. But in ice each each molecule is hydrogen-bonded to 4 other molecules in a more stable fashion. This spreads out the H2O molecules in the ice structure, lowering its density. It’s the reason ice expands as it freezes into a hexagonal network, one that’s 3 kJ/mole stronger than that of a non-supercooled liquid network. It’s also the reason ice can’t flow like liquid water.

ice2
Text and image modified from a combo of AP Biology and chemguide.co.uk diagrams.

So after the ice starts to form on top and then along the internal walls of the bucket, the frozen base and the circular rim begin to thicken. The rest of the forming crystals grow until they reach the middle of the bucket. With molecules that are more tightly bound, the ice at point A cannot flow out of the bucket as it did when it was in a liquid state. But the air space in the bucket above the slant will be occupied as the ice from the freezing core expands and pushes upwards. The bucket, as a result, even though on a slant, gets completely filled with ice, which also expands against the bottom, deforming the plastic base.

If you slide the ice back into the bucket and wait for a warm day to melt it, water will reveal its intrinsic color. Most glass containers aren’t large enough for water to absorb enough red light to reveal a perceptible hue of greenish blue. Too often we get the false impression that water is transparent. But the white walls of the bucket will provide enough internal reflection to increase the path length, and water’s color becomes noticeable. I’ve ventured a little deeper into that idea in this blog entry. If you’re not interested, hopefully I’ve nevertheless shown that there are some side-benefits to saving water and energy while collecting rain.

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