envelope

The first is the sound's attack, or the initial peak of the sound wave. The second is the sound's sustain, or the plateau of the sound wave after the initial peak, but before losing energy. The third is the sound's decay, starting at the point the wave begins to decrease in amplitude until the forth stage, release, the part of the sound that trails off into the noise floor.

This is the huge colorful fabric "bag part" of a hot air balloon. These are made with a special kind of coated rip-stop nylon. The coating serves to add heat and UV resistance.

An envelope has vertical nylon webbing or steel cables (now out of favor, since they don't agree with power lines) running the entire height of the envelope, converging at the bottom into four cables or ropes from which the gondola is suspended.

At the bottom of an envelope, there is a detachable skirt which is made of a thicker, more heat resistant material. It serves to shield the burner flame from any wind gusts, as well as to sort of funnel the hot air into the envelope.

At the tip-top of the envelope is a parachute top. This is a detachable circular section of fabric which is fastened loosely to the envelope with velcro. Running down from the parachute top is the vent line which pulls the parachute top down a bit to vent hot air and cause descent. Pulling HARD on this rope separates the velcro and deflates the balloon.

Overheating, burns and rips are the greatest enemies of an envelope. Overheating reduces the life of the envelope by weakening the nylon. Burns may happen upon landing, when the gondola tips over and the hot burner element touches the skirt. Rips are not uncommon after a rough landing.

Fortunately, in the US, the FAA requires an annual or 100 hour inspection of envelopes to ensure that they are flightworthy

A typical envelope is around 70,000 - 90,000 cu ft in volume, 50 - 60 ft wide, and may stand as tall as a 7 story building when fully inflated.

There are four common types of patterns of envelopes:

• Chevron - Vertical longitudinal strips, each comprised of diagonally biased strips. Adjacent vertical strips have opposite diagonal bias, so that it forms chevrons.
• Sawtooth - Fabric is divided into "square" sections, just like longitude and latitude on the globe. However, the color varies such that the same color is used in a given panel, and also in the panel which is one unit North and one unit East and so forth. Thus, the pattern of the fabric is squares, but the color goes diagonally around the envelope.
• Vertical - Fabric sections are longitudinal, with one color being used for the entire strip.
• "Special Shapes" - These are the comically shaped balloons you sometimes see, which are shaped like sneakers, birthday cakes, cartoon characters, etc.

• http://balloonzone.com/balloonparts.html
• http://www.howstuffworks.com/hot-air-balloon2.htm
• Personal experience crewing with members of the Plano Ballooning Association

A very basic definition of a mathematical envelope is a bunch of lines put together to form a curve (see Webster 1913's definition). However, it doesn’t have to be just lines. Envelopes can use circles to form cardioids, planes to form 3-D figures, or any number of things. The simplest envelopes aren’t very hard to draw/create. In fact, I made them in grade school as a young child without even realizing it. In first grade, my teacher had us make string art. She gave us a piece of wood with nails hammered into it in a square formation. Everyone in class was given one piece of string and told to loop the string around the nails in straight lines to see if we could form curves. Ahhh, the joys of discovery. I stumbled onto the creation of my first envelope. Now that I am a little more mature, I tend to use graph paper, a pencil, and a ruler instead of wood, nails, and string.

Here is a simple envelope and the method used to find its equation:

On a Cartesian coordinate system, connect the points (0,8) and (1,0), (0,6) and (2,0), and (0,4) and (3,0). Continue the pattern until the envelope can be seen.

For a line to be a part of this curve, it must be tangent to the curve. In other words, it must touch the curve and have the same slope as the curve at that point. This takes two equations; one for the point on the curve (the purpose of the point-slope equation below) and one to be sure it has the same slope (the derivative of the point-slope equation).

The first thing that must be done in the process of finding the equation to any envelope is to choose a parameter that will result in only one line of the envelope. For this particular problem, I chose the y-intercept, using the generic point (0,β) to write an equation for all the lines. The corresponding x-intercept for the point (0, β) is (5-β/2,0). Using the point-slope equation for a line, y₁-y₂=m(x₁-x₂), I plugged in a point and my slope to get this equation.

y – β = ((-β)/(5 – β/2)) x *or* y = ((-β)/(5 – β/2)) x + β

From here I simplified to get the equation 10y – yβ = -2βx + 10β – β²

I then took the derivative with β as my variable and obtained –y = -2x + 10 – 2β. Set these two equations up as a system of equations. Solve the second equation for β, and plug into the first equation of the system. After quite a bit of simplifying, I arrived at the equation

0 = ¼ y² - xy + x² -5y –10x + 25

This fits the standard formula for a conic 0 = Ax² + Bxy + Cy² + Dx + Ey + F. This particular envelope happens to be a parabola.

All it takes to solve envelopes is the right parameter and the willingness to plow through some nasty, and occasionally impossible, algebra.

Envelopes, also known as contours, are waveforms that don't repeat. Like the waveforms generated by LFOs, they move back and forth far too slowly to be audible, and are never heard directly; instead, they are used to control other parameters, such as a filter's cutoff point or an attenuator's level. This is what enables synthesisers to fade in a note when you press a key, then fade it out again when you let go of that key.

When you press down a key on a monophonic, analogue synthesiser, it does two things in addition to changing the oscillator's frequency: it activates a trigger, which is a quick electronic pulse, and it opens a gate, which is a sustained electronic signal. Playing another note while still holding down the first one causes the trigger to be activated again. Letting go of all the keys closes the gate. The signals for the trigger and gate are usually fed through to an envelope generator, which uses this information to produce envelope waves in time with the individual notes.

There are several different kinds of envelope, but the most popular are the decay, trapezoid and ADSR types.

Decay envelope

The simplest kind of envelope is a decaying line or curve. It's useful for percussive sounds. For example, white noise that's being attenuated with a decay envelope makes a simple but effective snare drum substitute. As the decay envelope doesn't sustain a sound, it only makes use of the trigger, not the gate.

Trapezoid envelope

A trapezoid envelope rises to its highest point, then stays at that constant amplitude until the gate controlling it is turned off. It then falls back down to zero. The trapezoid envelope is good for attenuating organ and pad sounds, especially long notes that slowly fade in and out.

ADSR envelope

By far the most common envelope is the surprisingly versatile ADSR envelope, named after its four parameters: attack, decay, sustain and release. It is similar to the trapezoid envelope, except that after it rises to its peak at the start (the attack time), it then falls back down to a more comfortable level for the sustained part (the decay time and sustain level). As with the trapezoid envelope, once the gate is closed, it falls back down to silence (the release time). When used the way it's intended, it is suitable for piano-style sounds, although its popularity is probably mostly to its versatility: by turning the attack time, sustain level and release time down to zero and setting a decay, it emulates a decay envelope; by turning the decay time down to zero and the sustain level up to the maximum setting, it emulates a trapezoid envelope. Because of this, the ADSR envelope generator has essentially made dedicated decay and trapezoid envelope generators obsolete.

More complex envelopes

There's no reason to stop at the ADSR envelope: some synthesisers let you go as far as to specify many different points in time and amplitude, connecting the dots with either straight lines or curves. These can be much more versatile than ADSR envelopes, but are curiously rare. Most people still seem to be content with ADSR envelopes, and they have become the standard.

En"vel*ope [F. enveloppe.]

1.

That which envelops, wraps up, encases, or surrounds; a wrapper; an inclosing cover; esp., the cover or wrapper of a document, as of a letter.

2. Astron.

The nebulous covering of the head or nucleus of a comet; -- called also coma.

3. Fort.

A work of earth, in the form of a single parapet or of a small rampart. It is sometimes raised in the ditch and sometimes beyond it.

Wilhelm.

4. Geom.

A curve or surface which is tangent to each member of a system of curves or surfaces, the form and position of the members of the system being allowed to vary according to some continuous law. Thus, any curve is the envelope of its tangents.

<-- 4. A set of limits for the performance capabilities of some type of machine, originally used to refer to aircraft. Now also used metaphorically to refer to capabilities of any system in general, including human organizations, esp. in the phrase push the envelope. It is used to refer to the maximum performance available at the current state of the technology, and therefore refers to a class of machines in general, not a specific machine.

push the envelope Increase the capability of some type of machine or system; -- usu. by technological development.

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© Webster 1913.