Invisible and relentless, sound is seemingly just there, traveling through our surroundings to carry beautiful music or annoying noises. In this article I’ll explain what sound is, how it’s created and propagated.
Throughout this presentation you will be hearing different sounds, which you will often play yourself on little keyboards like the one below. You can either click its keys with your mouse or use WER keys on your computer keyboard, but before you do so make sure your system volume is at a reasonable level:You can press its keys with your fingers, but before you do so make sure your system volume is at a reasonable level. You may also need to unmute your device:
We’ll eventually understand how these sounds are created and how they get to your ears, but we have to start by talking about the medium that is most commonly associated with sound – air.
You’re probably aware that at microscopic level air consists of individual, tiny particles. Let’s look at the behavior of those molecules up close. In the demonstration below, I parceled out a tiny cube of air – notice how the molecules freely enter and leave this space. You can drag the cube around to change the viewing angle and you can also control the speed of the flow of time with the slider:
I need to note that this visualization simplifies things a little. The air consists primarily of nitrogen and oxygen – their particles contain two atoms bound together, but for simplicity I’m drawing all molecules as simple spheres.
Each particle of air travels through space with some velocity in a more or less random direction. While some molecules are faster, and some are slower, at room temperature the average velocity of a particle is a staggering 1510 ft/s460 m/s so to make the particles visible I’m showing their motion significantly slowed down. As a side note, I’m using imperialmetric units here, but you can switch to the metricthe imperial system if you prefer.
With all the commotion in the previous simulation it may have been hard to see that the air particles collide with each other. In the demonstration below, I’m flashing the particles red and making them a little bigger right after they collide. To make these events easier to see you can control the speed of time with the slider:
Naturally, those particles can also collide with other things in the environment. In the simulation below I put up walls around that parcel of air. While at this scale we could also see the individual atoms constituting the walls of the box, I’m not showing them here to make things clearer. The particles of air are bouncing off these sides and every time that happens I’m drawing a little blue glow on the wall:
When an air particle bounces off, it imparts a small force on the wall. Although a single collision doesn’t do that much, with enough particles present the walls experience a constant barrage of collisions which exert expanding pressure on the container. The magnitude of that pressure is the sum of all these forces of impacts divided by the area of the box. Intuitively, for a box of a fixed size, the more particles there are, the heavier they are, and the faster they move, the higher the pressure.
Let’s see what happens if we resize this box a little while keeping the number of particles constant. In the demonstration below, you can drag the slider to change the length of the box, the plot in the bottom tracks the number of collisions per area of the box in a unit of time:
The number of collisions per area, and thus the pressure, increases as we squeeze particles closer together, and it decreases as we let them loose. Remember that we’re observing a significantly slowed down simulation. Because of the speeds involved here, I’m only letting you resize the box at a limited pace. That limitation prevents the walls from moving at enormous speeds, which would create some truly extreme pressure conditions.
It may not be immediately clear, but the experience of pressure is not limited to the walls of the box we built. Molecules in any open region of gas also experience compressing forces exerted by collisions with molecules from the neighboring regions. Those collisions aren’t as neatly aligned in space, but in aggregate it makes no difference if particles are bouncing off a wall or off particles from the neighboring area – the pressure is still there.
All the collisions between particles also have other consequences, but to look at them I need to adjust the scale at which we’ll be looking at – you can do this with a slider below:
The small cube of air from the previous demonstration is now just part of a larger volume. While this large cube is 20 times larger in each dimension, its sides measure only 1 micrometer – a few times smaller than a width of spider silk. Unfortunately, it would be both chaotic to display and taxing to correctly simulate the over 26 million air particles that one could find in this area. Instead, I’ll just visualize the motion of a single molecule as it travels through this space. In the demonstration below, I enlarged that particle to make it visible, you can also control the flow of time with the slider:
Due to collisions with other molecules, it may take a while for this particle to travel a significant distance and get out of this larger volume. You probably experienced this yourself – when someone sprays perfume in a distant part of a room the scent doesn’t instantly get to you. It’s only after some lucky aromatic particles manage to bounce into the vicinity of your nose that you get to experience the smell. If it wasn’t for the collisions with other molecules, those fragrant particles could travel the distance to your nose in a fraction of a second.
The important point is that, while the air particles move very fast, the collisions with other particles keep them mostly contained inside their local area. Over a longer period of time they will eventually all mix up, but at the short timeframes we’ll be interested in here, each reasonably sized parcel of air contains mostly the same particles.
All the microscopic behavior of individual particles we’ve observed ends up contributing to properties of air at larger scales. From this point on, we’ll move past looking at individual molecules and instead we’ll focus on regions of air that more closely match the sizes we’re used to. Those particles are still there, and they’re what actually end up distributing changes in gases, but we need to aggregate a huge collection of the molecules to end up with more discernible motions.
Let’s witness some of this large scale behavior when a section of air in a tube gets pushed on by a moving plate. In the demonstration below, I divided the air next to that circular plate into individual slices, which I’m drawing with thin lines. Although these boundaries are imaginary, they let us see what happens to different sections of air as you move the plate around with the slider:
Notice that the individual parcels of air move only a little from their initial position, but the disturbance caused by the motion of the plate seems to propagate through them with some speed. Moreover, that disturbance moves away from the plate regardless of the direction of the plate’s motion.
If you recall our experiment with a resized cube of air having more or less available space, you’ll notice that a very similar situation happens here inside each slice – this tells us that we’re witnessing some changes in pressure.
Let’s look at this situation from the side, which will make it easier to see how these pressure changes propagate. In the following demonstration I’m coloring each slice of air red when it’s compressed and the pressure increases, and I’m coloring it blue when it’s expanded and the pressure decreases:
When the plate moves to the right, it compresses the air particles in its vicinity, which increases the pressure in this region. Those particles then push and expand into particles in the neighboring slice of air, which then push onto the next slice and so on. All of this results in an area of higher pressure propagating away from the plate.
On the flip side, when the plate moves to the left, it creates an area of depletion, or rarefaction, which the neighboring slice of air rushes to fill in. This then reduces the pressure in the vicinity of the next slice and so on. While the slices move towards the receding plate, the pressure disturbance still moves away from the plate.
What we’ve just seen more clearly were pressure waves propagating through the air. When those pressure waves reach our ears we perceive them as sound. By moving the plate we’ve effectively created a very crude speaker that emits sounds as its moving element changes position.
While this simple demonstration helps to build an intuitive understanding of the propagation of pressure waves, the relation between the vibration of the object and the pressure disturbances it generates is much more complicated. We’ll soon explore some of those details, but for now we’ll focus on the relation between the movement of that vibrating element and the sound it produces.
Let’s begin by introducing a simple musical instrument. By pressing a key down, you can request the plate to rapidly change its position. When you let that key go, the plate will move back by the same amount. Notice that each of the three keys moves the plate by a different amount, with red marker showing the total offset:
Despite its simplicity, this device lets us observe a few things. Firstly, a sudden jump of a plate creates a popping sound. It also doesn’t matter if the object has jumped forwards or backwards – the effect sounds the same.
Secondly, the loudness of that pop depends on the magnitude of the jump – the larger the jump the louder the pop.
Finally, this plate can jump forwards or backwards from any position – you can verify this by holding one of the keys and then pressing some other key. That pop sounds exactly the same as the sound created by a standalone key press.
Three pops of different loudness is not much to create music with as this only lets us play some simple beats. You may have tried pressing the buttons as fast as possible to create more upbeat tunes, but there are limits to how fast human fingers can press buttons.
Thankfully, we can ask the device you’re using to repeat the pops at a certain rate. In the demonstration below, you can drag the first slider to change how many times per second the plate should jump forwards and backwards. You can also control how far the plate moves with each jump using the second slider:
To describe the frequency of vibration, I’m using the unit of hertz, often abbreviated as Hz – it describes how many full cycles of motion happen within a single second. For example, a frequency of 2 Hz means that two full cycles happened in a second which means that the plate moved back and forth and back and forth again.
While at lower frequencies these patterns sound like a rapid series of pops, at higher frequencies these sounds blend together into a discernible note. The most important observation we can make here is that the higher the frequency with which the plate vibrates, the higher the perceived pitch of the resulting sound is.
You’ve probably observed that at lower rates you could follow the motion of the vibrating plate, but as the frequency increases, it becomes impossible to reliably track its movement. Unfortunately, we’re hitting the limits of frequencies we can easily display on a screen and perceive with our eyes, so from this point on I’ll significantly slow down the motion of the plate. The patterns of motion will still be correct, they’ll just happen at much slower pace so that we can actually see what’s going on.
With that in mind, let’s make our music synthesizer create those very frequent jumps of the plate. We’ll assign a different frequency to each key – you can see them displayed above the keys. I’m also drawing three plots corresponding to each key, and an additional plot showing the cumulative effect of the individual motions.
These plots, or waveforms, represent the motion of the plate as requested by the keys. As the plots pass through the center line they reflect the current position of the plate, but remember that the motions we see here are a few hundred times slower than the actual ones:
Notice that when you press two keys at the same time you can still hear both sounds playing through – we’ve already seen that the final plate movement is just the sum of individual motions that each key requests.
You may also have realized that I’ve shifted the center line of the plate. What we consider a base position of that object is ultimately arbitrary. The keys simply request the plate to move forward or backward and they don’t care from where it’s starting, but having the object oscillate around a well defined origin will be more convenient.
If you look closely at the plots when two keys are pressed, you may have also noticed that there are some areas where one of the keys wants the plate to move forward, while the other wants it to move back. As a result, the plate stays at the same position.
What may be a little surprising is that there is nothing magical about the sound emitted by the plate when two keys are pressed – the emitted sound depends purely on the final movement of the plate. In the demonstration below, I made the third key emit the same pattern of motion as the combination of the first two:
The sound emitted when the third key is pressed is exactly the same as the sound emitted when the first two keys are pressed together. This is a very important observation, as it lets us explore some other patterns of motion that the keys could induce.
We’ll expand on this idea by changing the way in which the plate vibrates. Instead of having it rapidly jump back and forth, we can request the plate to move with constant velocity, following a triangular pattern. Let’s look at that pattern up close. Like other periodic shapes, a triangular waveform can be characterized by its amplitude, frequency, and phase:
In these visual examples the amplitude controls the height of the waveform. The frequency controls the width of the repeating pattern, which affects how many repetitions of the up-and-down triangles can fit inside a single second. The phase just shifts the shape around to change what we consider the beginning of that pattern.
We’ve already seen how the amplitude of the plate’s motion affects the perceived loudness, and this triangular motion is no different. In the demonstration below, all three keys move the plate in the same pattern, only their amplitude is different:
Similarly to the original rectangular pattern, we perceive different frequencies of this triangular motion as different pitches. In the following demonstration, each key has a different frequency as marked by the display in the upper part of the keyboard. The higher the frequency the higher the perceived pitch:
With these more complicated triangular shapes in place it may be a bit harder to understand how all these waveforms add up to form the final plot. Let’s visualize this process step-by-step using a few examples that you can select in the demonstration below. The slider tracks the process of the addition of four different triangular shapes of different amplitude:
Notice that each waveform expresses some shift from the current position. This offset can be positive or negative – I’m visualizing it using small arrows. To add the waveforms we just need to apply these arrows to the existing shape, which we can do by placing the arrows on that shape and then deforming the shape as the arrows demand.
The yellow bars in the bottom part symbolize the amplitude of the specific triangular waveforms at different frequencies. Notice that it’s completely fine for an amplitude to be zero – that element just won’t participate in the creation of the final cumulative waveform.
The last parameter of any periodic function is phase. On its own, phase doesn’t do much for sound, but it may give us some trouble when a phase-shifted shape is combined with another shape of the same frequency. In the demonstration below, all three keys of our synthesizer vibrate at the same frequency and amplitude, but each one is shifted in phase by a quarter of a cycle:
Individually, each of the three keys sounds exactly the same, but once we combine the keys the trouble begins. When the middle key is combined with the left or the right key, the resulting sound is slightly louder. However, when the first and last keys combine, their waveforms perfectly cancel each other out causing no plate movement – no sounds gets emitted! We’re experiencing a destructive interference where the peaks of one waveform end up getting cancelled out by the valleys of the other waveform.
Triangular functions were a little more intricate than simple jumps, but nothing prevents us from using completely arbitrary motions. In the top part of the demonstration below, you can draw your own shape of the waveform that defines the vibration of the plate. I then repeat this pattern at frequency prescribed by each keyboard key, letting you hear the result through your speakers:
After playing with different shapes you may have noticed that while they have a different character, they all share a very similar pitch. Different shapes end up creating different timbre which is a property of sound that lets us distinguish different instruments playing the same note.
So far we’ve been building waveforms of increasing complexity, but it’s time we reversed this direction by discussing sinusoidal waves, which form the most fundamental periodic functions.
Let’s try to create a sine wave from scratch. In the demonstration below, you can witness a rotating circle with a small red point marked on it. If we track the vertical position of that point over time we’ll draw a perfect sine wave. While we can assign amplitude, frequency, and phase to any other periodic function, those properties beautifully correspond to radius, speed, and initial angle of the rotating circle that generates this wave:
Let’s see how a sinusoidal vibration of a plate sounds. In the demonstration below, you can experience the whole range of sine waves with different frequencies and amplitudes playing through your speakers. Some of the tones are expected to be imperceptible, so do not adjust your sound volume until you reach the middle frequencies:
You’ve probably noticed that low frequencies start inaudible, and they increase in loudness as the frequency increases. However, as the frequencies go toward the higher end the sound fades away into silence. Although your speaker and audio settings may to some extent contribute to this effect, it’s otherwise perfectly natural – humans can hear in a limited range of frequencies and the perceived loudness changes with frequency.
In the physical world, these pure sine tones are easily generated by striking different tuning forks, but on this website we have to resolve to making our synthesizer play some of these notes at different frequencies:
Once again, as each key requests the plate to vibrate in a certain way, its final motion is the sum of those individual requests. That addition can sometimes create unexpected effects when the frequencies of played sounds are close to each other, which you can experience in the demonstration below. Notice the frequencies of sounds as shown above the keyboard keys:
When you press two keys at the same time you’ll hear the volume of sound increasing and decreasing in a phenomena known as beats. It’s not your ears misleading you – it’s a real effect caused by the two original frequencies interfering with each other, which you can observe on the plots. This effect can happen with all periodic signals and you can sometimes hear it when sitting in an airplane – the rotation speeds of two engines may not be perfectly synchronized and you can hear the variation in slowly changing volume of the resulting sound.
Similarly to triangular waves, we can fairly easily visualize how the addition of multiple sine waves with different amplitudes, frequencies, and phases creates more complex shapes. In the demonstration below, I’m adding a bunch of these sine waves. You can use the control below to choose the shape we’re creating:
Observe that I’m also using a “special” sine wave with frequency of 0 Hz – it reflects a constant value that is useful to represent waveforms that are, on average, not centered around the baseline. Looking at the construction of these waveforms, we can easily reason that the final shape is actually created by 6 different sine waves with varied amplitude, frequency, and phase.
Instead of adding these sinusoidal basic blocks to create a complicated function, we can, perhaps surprisingly, invert this process and decompose any repeating waveform into its constituent sine waves. Let’s see this in action. In the demonstration below, you can draw the desired shape in the top part and I’ll do my best to express it using the sum of 64 sine waves with increasing frequencies. For clarity, I’ll skip the animations of green and red arrows being added:
In the bottom part I’m once again showing the amplitudes of individual sine waves creating the final shape, but since we have so many of them I just pack their amplitude bars next to each other to create a cumulative plot of amplitudes.
Depending on how the original signal is expressed, this method of decomposing periodic functions into sines is known as Fourier series or Fourier transform. It lets us deconstruct a waveform expressed as a function of time into a waveform expressed as a collection of individual frequencies that are contained in that waveform.
We can look at any repeating pattern either as a collection of individual values over time or as a collection of distinct sine waves that add up to the same pattern. Both views represent the same shape, they just present it in a different way.
In general, Fourier transform provides us with both amplitude and phase of each contributing sine wave. As we’ve seen, however, human hearing is almost completely agnostic to phase shifts so amplitudes typically provide enough information to express the original shape.
Let me bring in the little waveform decomposer once more, this time I’ll fill it with a square function corresponding to the simple back-and-forth jumps that our plate did initially:
As you drag the slider to the end you’ll notice that the decomposition of a waveform that rapidly changes values ends up creating ripples in the recreated function in an effect known as Gibbs phenomenon. Notice, however, that as we add more sines of high frequency the width of that ripply region decreases.
More importantly, this square pattern that repeats once per second contains in it many sine waves of much higher frequencies. This explains why we were able to hear a plate jumping back and forth once per second – while we can’t hear many of those lower frequencies, we can hear the higher pure tones. A sine wave at 1Hz frequency contains only a 1Hz signal that we can’t hear.
Let’s look at the frequency spectrum of the things in your environment. In the demonstration below, you can press the button to start using your microphone. I’ll then draw a plot of different frequencies and their amplitudes as heard by the device:
I recommend whistling, or singing “aaaaaa” using lower and higher voice to see different frequencies appear. If you have any musical instruments handy or applications that can recreate their sounds, the resulting diagrams will in many cases also show interesting distributions. More complicated sounds like speech or music will rapidly change their shapes, which reflects their dynamic nature.
When singing or playing your simple notes into the microphone you may have noticed repeating patterns of frequencies – a certain base frequency gets a strong sibling at double, triple, and higher multiples of that initial frequency. These repeating patterns can happen for different reasons, and in the next section I’ll explain how they’re created by some of the simplest vibrating objects.
To explore these vibrations, we’ll start by building a simple system consisting of a small mass attached to two springs. In the demonstration you can move that mass up and down with the slider:
At the small amplitudes of motion we’re considering here, the system oscillates sinusoidally with a certain natural frequency that depends on the mass of the object and the stiffness of the springs. The friction between moving parts and the air resistance causes the oscillations to die out after a while.
You may have realized that in this simulation the mass moves only up and down, but in principle it could also move left and right. However, these horizontal motions won’t be of much use to us, so we’ll focus purely on the vertical movements.
Let’s add another mass and spring to this system:
Even with this simple setup the behavior of this system can get fairly complicated, including some unusual patterns of motion, but all of these scenarios can actually be decomposed into the sum of two much simpler motions.
In the first motion the masses move together in the same direction, and in the other motion the masses move in the opposite directions. In the demonstration below, you no longer control the position of the masses, but instead the amplitudes of those two separate modes:
With careful arrangements, you can recreate the interesting motion patterns that this system can create. Notice that the second mode oscillates faster than the first mode. This should make intuitive sense since the central spring is getting stretched, which adds an additional pulling force to the system.
Any pattern of vertical motion of those two identical masses and three springs can be decomposed into the two normal modes that we’ve just looked at. That idea extends to even more complicated systems. In the demonstration below, you can see the five normal modes of a system consisting of five masses and six springs:
As we increase the number of masses and springs to the extreme, each segment starts to resemble a tiny section of a string that has some mass and elasticity. Conceptually, a simple string can be thought of as a system of a huge amount of very small spring-and-mass segments – you can see one in the following demonstration. Let’s look at the first few normal modes of this system:
Since the endpoints of the string can’t move, only certain wave patterns are allowed – no wave shapes that would cause the ends to wiggle up and down can be expressed with this setup. The frequency of the first mode is known as the fundamental frequency. Each higher mode has a frequency that is an integer multiple of that base frequency – for instance the third mode has three times as high frequency as the fundamental mode. These modes of vibration are known as harmonics.
Similarly to the more primitive systems, any motion of a string tied to two ends can be decomposed into the sum of these individual motions, each with a different amplitude. These concepts are very similar to the Fourier transform operations we did for the waveform shapes, but this time they’re applied to the shape of a string.
What this all ultimately means is that the motion of a vibrating string is the sum of the individual harmonic modes, each with an arbitrary amplitude, and a prescribed frequency that is a multiple of the fundamental frequency. Let’s see and hear how this manifests in practice. In the demonstration below, you can pluck the string by clicking and dragging over with your mousewith your finger:
You may have noticed that the place at which you pluck the string changes the sound it produces – it’s just a result of different ratios of harmonics created by the shape of the plucked string right before it’s released. The extent to which this string can be pulled is quite exaggerated here making it look a little unrealistic, but a slow-motion capture of a guitar string reveals a very similar behavior.
The fundamental frequency of vibration of a taut string depends on its length, weight, and tension. The shorter the length, the higher the frequency – this is the main reason why a violin has a much higher pitch than a double bass.
More importantly, by placing a finger on a string and thus changing its effective length the performer can play different notes using a limited set of strings. In the demonstration below, you can hear the variation in pitch as you drag the slider to change the location of the finger holding down the string:
The thicker the string the more inertia it has and the more difficult it is to move it around, so the frequency of the sound decreases as string gets heavier, or, more specifically, as its linear density increases. By using strings of different thickness or material, musical instruments can achieve different pitches on strings of roughly the same length – you can play with that parameter in the demonstration below:
Finally, the more taut the string is, the faster it snaps back, so a higher tension increases the frequency – by turning the tuning pegs of a string instrument one can tweak the fundamental frequency of each string.
On its own, a vibrating string doesn’t move much air and it’s not very loud. To make the sound more pronounced, the motion of the string is transferred to a larger surface to increase the vibrating area and the volume of emitted sound. The shape and materials used in the construction of the body of an instrument strongly contribute to different sound frequencies getting amplified or absorbed.
Our models of springs and masses also extends to other vibrating elements. Membranes of drums form a two dimensional grid of vibrating elements. Tightly packed particles of solids form a three dimensional lattice – its elements also elastically interact with each other, which at larger scales causes cymbals, gongs, and bells to vibrate with various natural frequencies that depend on the shape and construction of these instruments.
Some other instruments like flutes or organs rely on a column of air rapidly traveling back and forth through their pipe-like bodies, forming yet another way to make the surrounding air vibrate at certain rates. When we sing, the vocal cords rhythmically open and close creating puffs of air – those also induce pressure disturbances that then travel around.
In the final part of this article we’ll explore how pressure waves get emitted from vibrating objects and how they end up spreading through the air and interacting with the environment.
Let me bring back the vibrating plate inside a tube that we could control directly with a slider:
While easy to understand, the contained nature of this model makes it a little simplified as it doesn’t match the behavior of typical instruments or speakers. To explore more realistic scenarios of sound propagation, we’ll firstly use a simple sphere that can inflate and deflate to quickly change its radius – you can think of it as some sort of water-filled balloon that we can rapidly add or remove liquid from. In the demonstration below, you can see an example of that simple pulsing sphere with the slider controlling the frequency of its vibration:
Let’s look at the pressure near the surface of this sphere up close using our method of slicing the air, this time into thin shells surrounding this sphere. At high frequencies of vibration the behavior of the air surrounding the sphere resembles that of the simple plate in a tube, which you can experience in the following demonstration. The slider controls the speed of the animation, which lets you see the details, but the sphere vibrates with the same frequency at all times:
As this sphere finishes expanding, it creates an area of higher pressure near its vicinity. However, at lower frequencies of vibration the situation is a bit different – you can see this in the simulation below. Notice that when the sphere finishes expanding, the pressure decreases. I’m exaggerating the size changes of the sphere to make it more visible:
What we’re witnessing here is the air’s inertia. The initial expansion of the sphere accelerates the neighboring air. When the sphere stops growing, that air still has some velocity so it moves away from the sphere, creating the local rarefaction. Similarly, as the sphere shrinks, the surrounding air starts getting pulled towards it, and it continues to bunch up even after the sphere reaches its smallest size.
The analytical derivation of this behavior is quite complicated, but ultimately the range of frequencies at which these different behaviors occur depends on the size and shape of the vibrating object.
A sphere forms only a simple model of a speaker – a more realistic example would be a vibrating circular plate surrounded by a very large wall:
Similarly to a sphere, the relation between the position of this plate and the resulting pressure also depends on frequency of the sound. However, the sphere was perfectly symmetrical, which made the intensity of the created pressure waves independent of direction. For a vibrating plate some of those symmetries are lost and the intensity of the pressure disturbances also depends on the direction relative to the source:
As you increase the frequency of the sound, the pressure waves focus in the frontal direction. Most commonly, lower frequencies emitted by speakers are much more widespread than higher frequencies. Notice that I also stopped drawing the thin lines separating individual sections of air – we’ll now more realistically assume that the pressure changes without any discrete steps.
As we’ve seen, the distribution of pressure that a vibrating object creates can get a little complicated, but, thankfully, at a reasonable distance, a steady sinusoidal oscillation of the source creates a steady sinusoidal variation in pressure that travels through the air. You can see the “side” view of that pressure variation on the dotted line in the bottom part of the following simulation:
This means that everything related to sinusoidal vibrations of plates like the amplitude, frequency, and phase of their motion also applies to the pressure waves themselves.
While these colorful diagrams made it easier to see how pressure changes propagate through air, they’re unfortunately a little deceptive. Firstly, I’m slowing things down significantly to make the propagation of waves visible, otherwise they’d just zip across your screen in a fraction of a second. At room temperatures the speed of sound in air is around 1125 ft/s or 767 mph343 m/s or 1,235 km/h.
Moreover, the actual pressure variation in sound waves is very small. When a person speaks at a normal volume the pressure increases and decreases by around 0.02 pascals or around 0.00002% of the atmospheric pressure. The red and blue colors make it look like pressure is changing a lot, but in practice these variations are minuscule – it’s quite astonishing that our ears pick up differences that small.
I also need to note that the pressure simulations we’ve just seen were fairly simplified. The behavior of a typical speaker is more complicated and the way it impacts the surrounding air depends on the speaker’s size, shape, and overall construction. All of these intricate behaviors may obscure some other phenomena that we’ll discuss, so for the following examples I’ll simplify things a little by using a basic, omnidirectional source of sound, similar to the expanding sphere we’ve used.
In the demonstration below, the blue speaker icon symbolizes that source and the yellow ear icon shows the location at which you can hear things – you can drag the latter around to change where you are:
You may have noticed a few interesting phenomena. Firstly, as you move away from the source it may take some time for the sound to get to you. This effect is just a consequence of the finite speed of sound. While at small distances we don’t notice that delay, it’s easily observed when a lightning bolt strikes and the sound gets to us with a perceptible lag. In this demonstration the distances are also quite large which you can see on the scale in the bottom left corner.
Secondly, the loudness of the sound we hear depends on the distance from the source. With a simple source like this the pressure waves travel away radially. While the waves look like circles in this two dimensional demonstration, in the physical world they form expanding spheres. The amplitude of pressure p of these waves is inversely proportional to the distance r from the source:
This effectively means that when we double the distance from the source the amplitude of the sound pressure wave gets halved. As we’ve seen a few times with vibrating plates, a change in amplitude of their motion, and thus the change of amplitude of the generated pressure waves, affects how loud a sound sounds.
The difference in magnitude of those pressure variations between loud and quiet sounds can get very large – the amplitude of sound pressure waves from a roaring jet engine is around 10000 times larger than that of a faint whisper. Comparing values using numbers that large would be inconvenient and vague terms like “faint whisper” aren’t particularly well-defined.
To solve these problems the concept of sound pressure level or SPL gets introduced:
Let’s unpack this equation. For reference pressure p is the average measured pressure for which we’re calculating the SPL.the commonly used value is 20 µPa, or 20 micropascals, since it is considered to be the threshold of human hearing, while
The logarithm compacts large numbers into much more manageable values. Speaking very roughly, this logarithm is related to the number of digits in the decimal representation of a number – making a number 10 times as large, or, equivalently, adding one more 0 digit to it, increases the value of a base-10 logarithm by 1.
What we’re calculating here is the logarithm of the ratio of the measured pressure p to the reference pressure . All of that is scaled by number 20, which gives us the result in units of decibels, often abbreviated as dB.
Let’s see how this equation can be used in practice. In the demonstration below, you can change the value of pressure and see how it affects the computed value of sound pressure level:
Recall that doubling distance halves the pressure, which effectively means that every doubling of distances reduces sound pressure level by 6 decibels. On the flip side, using two speakers instead of one also increases the SPL by just 6 decibels. Note that these equations can also be used to compare the SPL of any two sources – for example, if two sounds differ by 20 dB then the amplitude of one is 10 times larger than that of the other.
The perceived loudness of sound very strongly depends on the sound pressure level, however, that loudness also depends on the frequency of the sound – you’ve experienced this when tinkering with the plate moving in a sinusoidal motion. To account for this, the value of pressure amplitude at different frequencies is often weighted so that the decibel values match human perception more closely.
It’s also worth mentioning another quantity related to the “amount” of sound – the sound intensity I which measures the power of the sound per area. The power of any wave depends on the square of its amplitude, so for simple spherical waves the sound intensity is proportional to the inverse square of the distance in, already mentioned on this blog, inverse square law:
Similarly to pressure, one can also define a sound intensity level, which expresses the ratio of intensity to a reference intensity.
Another aspect affecting the propagation of sound is acoustic attenuation caused by e.g. viscosity of air and thermal conduction between its molecules. These interactions cause additional decrease in amplitude of the pressure waves as they move away from the source. Attenuation depends on temperature and humidity of air and it also varies with the frequency of the sound. These effects are fairly complicated so I won’t account for them here.
The last demonstration had one final hidden feature that you may have experienced when simultaneously pressing the keyboard keys and changing your hearing position relative to the speaker. This effect is more commonly experienced when it’s the source of the sound that’s moving relative to the receiver, like in the demonstration below, where you can control the velocity of the speaker using the slider:
Notice that when the speaker moves toward the receiver the pitch increases, and when it moves away from the receiver the pitch decreases in the phenomenon known as Doppler effect – you’ve probably experienced this in real life when an emergency vehicle with blazing sirens or a motorcycle with a loud engine was passing by.
This effect is easiest to understand by looking at the individual peaks of the sound waves. In the demonstration below, you can control both the flow of time and the velocity of the speaker. The plot at the bottom tracks when the peaks of the sound wave are emitted, and when they’re heard at the center of their respective icons:
The speaker emits the peaks at a certain fixed cadence, which depends on the sound frequency. When a first peak is emitted, it will take some time for it to get to the receiver. However, as the speaker moves towards the listener, the next peak will be emitted from a closer distance than the previous one, and it will take it less time to get to the receiver – the receiver will witness the peaks coming in at faster rate, or higher frequency, which we perceive as higher pitch.
When the speaker moves away from the receiver, the situation is reversed – each new peak has to travel a bit longer distance to get to the listener, making the peaks more spread out for the receiver causing the frequency and the pitch to drop.
You can see both of those effects in the plot in the bottom part of the demonstration – the yellow bars get bunched up or loosened up relative to the blue bars, which implies increased or decreased frequency of received sound.
The speaker is completely unaware of what’s going on – it just keeps emitting the sounds at the same frequency. However, as the speed of the sound emitter increases, the circles corresponding to the peaks of the waves get bunched up more and more creating more extreme pressure conditions. When the sound-emitting object surpasses the speed of sound it will create a sonic boom.
Although relative motion created interesting effects, even static conditions may end up forming unexpected phenomena. In the demonstration below, the pure notes played on the keyboard are emitted by two speakers at the same time. You can drag your position around to see how the interaction of pressure waves affects what you hear:
Depending on the location of the receiver the sound level increases and decreases, which we can also see in the diminished intensity of the pressure variations in those areas. This effect is caused by destructive interference of pressure waves from the two speakers – peaks from one speaker cancel out valleys from the other. As long as you have access to a two speaker system you can actually try to replicate that experiment.
Fortunately, different frequencies played by different keys end up having dead spots in different places, too. Since the vast majority of the typical sounds we hear consists of a multitude of different frequencies that constantly change, we don’t experience this loudness variation in more practical scenarios.
It’s important to note that the interference doesn’t permanently destroy the waves as the disturbances merely pass through each other. This should make intuitive sense – when a few people are simultaneously speaking in the same room their speech doesn’t become garbled or faded and we can still understand what they say.
Quite conveniently, all of our experiments so far happened in a vast open space unencumbered by any obstacles. Let’s see what happens if we put a wall in our testing environment:
When you emit a short pulse of sound while standing far enough from the wall, you’ll hear the originally emitted sound with some delay in a familiar effect known as echo. Notice the distances involved here as shown by the scale in the bottom left corner. When standing close to the wall, the delay between the original and reflected sound gets very short – when it’s smaller than around 50 ms, our auditory system doesn’t discern these two separate sounds.
You may have noticed that the reflected wave is quieter. While the increased travel distance accounts for some of that reduction, it’s also the wall itself that partially absorbs the energy of the pressure disturbances.
For the last demonstration in this article we’ll move to a completely enclosed space allowing the sound waves to bounce back and forth from the walls. You can change the size of this room using the slider and witness the reflection of sound off the walls:
Even after you stop pressing the keys it may take a while for the sound to die out, because the waves keep reflecting off the walls, losing only some portion of their energy with each bounce. You may have heard this effect known as reverb in an empty room devoid of any furniture, drapes, or clothes that absorb the pressure waves and heavily attenuate the reflection of sound.
The Science of Sound is an entry level, yet very thorough book dedicated to many topics related to sound. The three authors use a minimal amount of math and focus more on higher level descriptions of different phenomena, including perception of sound, musical scales, and audio hardware. The only drawback is the price, but thankfully one can easily find cheaper used copies online.
For further exploration of Fourier Transform and signal processing I highly recommend The Scientist and Engineer’s Guide to Digital Signal Processing by Steven W. Smith. I personally really like the author’s style of patiently explaining things step by step with multiple examples. On top of that, the entire book is available online for free on the author’s website.
Sound waves aren’t limited to gases – they spread through liquids and solids too. However, in daily life almost everything we hear are just minuscule pressure variations of air, which ultimately are just changes in the number and intensity of collisions of individual air particles.
I find it fascinating that the most irritating noises and the most inspiring music are driven by the same phenomena – it’s only the underlying shapes and magnitudes of their pressure waves that make them sound so distinctively different.
Sauti, ambayo haionekani na haitoi msamaha, inaonekana kuwapo tu, ikisafiri kupitia mazingira yetu ili kubeba muziki mzuri au kelele zenye kusumbua. Katika makala hii nitaelezea sauti ni nini, jinsi inavyoundwa na kuenezwa. Katika kipindi hiki cha maonyesho, utaweza kusikia sauti mbalimbali ambazo mara nyingi utazicheza mwenyewe kwenye keyboard ndogo kama ile iliyo hapa chini. Unaweza pia kubonyeza funguo za WER kwenye kibodi yako ya kompyuta, lakini kabla ya kufanya hivyo hakikisha kwamba kiasi cha mfumo wako ni katika kiwango cha busara. Unaweza pia kuhitaji unmute kifaa chako: Hatimaye tutaelewa jinsi sauti hizi ni kuundwa na jinsi wao kupata masikio yako, lakini tuna kuanza kwa kuzungumza juu ya kati ambayo ni kawaida kuhusishwa na sauti - hewa. Yaelekea unajua kwamba katika kiwango cha kipekee hewa ina chembe ndogo-ndogo. Hebu tuangalie tabia ya molekuli hizo kwa karibu. Katika maonyesho hapa chini, mimi parceled nje cubic ndogo ya hewa <unk> kumbuka jinsi molekuli huru kuingia na kuondoka nafasi hii. Unaweza kuhamisha kiunzi kuzunguka kubadilisha pembe ya kutazama na unaweza pia kudhibiti kasi ya mtiririko wa muda na slider. Hewa inajumuisha hasa nitrojeni na oksijeni - chembe zao zina atomu mbili zilizounganishwa pamoja, lakini kwa urahisi mimi ninachora molekuli zote kama mduara rahisi. Kila chembe ya hewa husafiri kupitia nafasi kwa mwendo fulani katika mwelekeo usio wa kawaida. Wakati baadhi ya molekuli ni haraka, na baadhi ni polepole, katika joto la chumba wastani kasi ya chembe ni staggering 1510 ft s 460 m s, hivyo kufanya chembe inayoonekana, mimi kuonyesha mwendo wao kwa kiasi kikubwa kupunguza. Kama kumbuka upande, mimi nina kutumia imperialmetric vitengo hapa, lakini unaweza kubadili kwa metric mfumo wa kifalme kama wewe kupendelea. Kwa msukosuko wote katika simulation ya awali inaweza kuwa vigumu kuona kwamba chembe za hewa kugongana na kila mmoja. Katika onyesho hapa chini, mimi nina flashing chembe nyekundu na kufanya yao kidogo kubwa haki baada ya kugongana. Ili kufanya matukio haya rahisi kuona unaweza kudhibiti kasi ya muda na slider: Kwa kawaida, chembe hizo pia inaweza kugongana na mambo mengine katika mazingira. Katika simulation hapa chini mimi kuweka juu ya kuta kuzunguka kwamba kifurushi cha hewa. Wakati katika kiwango hiki tunaweza pia kuona atomu za mtu binafsi kuunda kuta za sanduku, Mimi si kuonyesha yao hapa kufanya mambo wazi zaidi. Chembe za hewa zinarudi kutoka pande hizi na kila wakati hiyo inatokea mimi ninachora mwangaza mdogo wa bluu kwenye ukuta: Wakati chembe ya hewa inarudi, inatoa nguvu ndogo kwenye ukuta. Ingawa mgongano mmoja haufanyi mengi, na chembe za kutosha zilizopo kuta hupata msukumo wa mara kwa mara wa mgongano ambao hufanya shinikizo la kupanua kwenye chombo. Kiasi cha shinikizo ni jumla ya nguvu zote za athari zilizochanganywa na eneo la sanduku. Kwa mfano, kwa sanduku lenye ukubwa wa kawaida, kadiri chembe zinavyozidi kuwa nyingi, ndivyo zinavyokuwa nzito zaidi, na kadiri zinavyozidi kusonga, ndivyo shinikizo linavyokuwa kubwa zaidi. Hebu tuone nini kinatokea kama sisi kubadilisha ukubwa wa sanduku hili kidogo wakati kuweka idadi ya chembe mara kwa mara. Katika mfano wa chini, unaweza kuhamisha slider kubadilisha urefu wa sanduku, na plot chini kufuatilia idadi ya mgongano kwa eneo la sanduku katika kitengo cha muda: idadi ya mgongano kwa eneo, na hivyo shinikizo, huongezeka kama sisi squeeze chembe karibu pamoja, na hupungua kama sisi basi yao huru. Kumbuka kwamba sisi ni kuchunguza kwa kiasi kikubwa kupunguza kasi simulation. Kwa sababu ya kasi zinazohusika hapa, mimi tu kuruhusu ukubwa sanduku katika kasi mdogo. Kizuizi hicho huzuia kuta zisisonge kwa mwendo wa kasi sana, jambo ambalo lingetokeza hali za shinikizo kubwa sana. Huenda ikawa haijulikani mara moja, lakini uzoefu wa shinikizo hauzuii kuta za sanduku tulilojenga. Molekuli katika eneo lolote la wazi la gesi pia hupata nguvu za kukandamiza zinazotolewa na mgongano na molekuli kutoka mikoa jirani. Mgongano huo haupo sawa katika nafasi, lakini kwa ujumla haina tofauti ikiwa chembe zinaruka kutoka ukutani au kutoka kwa chembe kutoka eneo jirani - shinikizo bado lipo. Mshtuko wote kati ya chembe pia una matokeo mengine, lakini kuangalia yao mimi haja ya kurekebisha kiwango ambacho sisi <unk> itakuwa kuangalia <unk> unaweza kufanya hivyo na slider chini: Kubi ndogo ya hewa kutoka maonyesho ya awali ni sasa tu sehemu ya kiasi kikubwa. Ingawa kiunzi hiki kikubwa ni kikubwa mara 20 katika kila kipimo, pande zake zina kipimo cha micrometa moja tu - mara chache ndogo kuliko upana wa hariri ya buibui. Kwa bahati mbaya, itakuwa chaotic kuonyesha na taxing kwa usahihi kuiga zaidi ya 26 milioni chembe hewa kwamba mtu anaweza kupata katika eneo hili. Badala yake, mimi itabidi tu visualize mwendo wa molekuli moja kama ni kusafiri kwa njia ya nafasi hii. Katika maonyesho ya chini, nimeongeza chembe hiyo ili ionekane, unaweza pia kudhibiti mtiririko wa wakati na slider: Kwa sababu ya mgongano na molekuli nyingine, inaweza kuchukua muda kwa chembe hii kusafiri umbali mkubwa na kupata nje ya kiasi hiki kikubwa. Yaelekea umepata uzoefu huu mwenyewe <unk> wakati mtu anaponyoa manukato katika sehemu ya mbali ya chumba harufu haipati mara moja kwako. Ni tu baada ya baadhi ya chembe za harufu bahati kufanikiwa bounce katika ujirani wa pua yako kwamba kupata uzoefu harufu. Kama isingekuwa kwa mgongano na molekuli nyingine, chembe hizo zenye harufu nzuri zingeweza kusafiri umbali hadi pua yako katika sehemu ndogo ya sekunde. Jambo muhimu ni kwamba, wakati chembe za hewa husafiri haraka sana, mgongano na chembe nyingine huwaweka ndani ya eneo lao. Katika kipindi kirefu cha muda wao hatimaye wote kuchanganyika, lakini katika muda mfupi sisi itakuwa na nia ya hapa, kila kiasi cha ukubwa wa kifurushi cha hewa ina zaidi ya chembe sawa. Tabia zote za kipekee za chembechembe za mtu binafsi ambazo tumeziona mwishowe zinachangia mali za hewa kwa kiwango kikubwa. Kuanzia hatua hii na kuendelea, tutaendelea kuangalia molekuli za mtu binafsi na badala yake tutazingatia maeneo ya hewa ambayo yanafanana zaidi na ukubwa tunaozoea. Chembe hizo bado zipo, na ndizo ambazo kwa kweli huishia kusambaza mabadiliko katika gesi, lakini tunahitaji kukusanya mkusanyiko mkubwa wa molekuli ili kuishia na harakati zinazoonekana zaidi. Hebu kushuhudia baadhi ya tabia hii kubwa-scale wakati sehemu ya hewa katika bomba anapata kusukumwa juu na sahani kusonga. Katika maonyesho hapa chini, nimegawanya hewa karibu na sahani hiyo ya mviringo katika vipande vya mtu binafsi, ambavyo ninachora kwa mistari nyembamba. Ingawa mipaka hii ni ya kuwaziwa, inatuwezesha kuona kinachotokea kwa sehemu tofauti za hewa wakati unavyohamisha sahani na slider: Angalia kwamba vipande vya hewa vya mtu binafsi huhamia kidogo tu kutoka kwa nafasi yao ya awali, lakini usumbufu unaosababishwa na mwendo wa sahani unaonekana kuenea kupitia yao kwa kasi. Zaidi ya hayo, kwamba usumbufu anaendelea mbali na sahani bila kujali mwelekeo wa sahani ya mwendo. Kama kukumbuka majaribio yetu na kubuni resized ya hewa kuwa na zaidi au chini ya nafasi inapatikana, utaona kwamba hali sawa sana hutokea hapa ndani ya kila slice <unk> hii inatuambia kwamba sisi ni kushuhudia baadhi ya mabadiliko katika shinikizo. Hebu kuangalia hali hii kutoka upande, ambayo itafanya kuwa rahisi kuona jinsi mabadiliko haya shinikizo kueneza. Katika maonyesho yafuatayo, nina rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi ya rangi. Chembe hizo huchochea na kupanuka kuwa chembe katika kipande cha hewa kilicho karibu, ambacho kisha huchochea kwenye kipande kijacho na kadhalika. Yote hayo husababisha eneo lenye shinikizo la juu zaidi kuenea mbali na bamba hilo. Kwa upande mwingine, wakati sahani inahama upande wa kushoto, inaunda eneo la kupungua, au rarefaction, ambayo kipande jirani ya hewa rushes kujaza. Hii kisha hupunguza shinikizo katika ujirani wa kipande cha pili na kadhalika. Wakati vipande kusonga kuelekea sahani receding, usumbufu shinikizo bado anaendelea kusonga mbali na sahani. Kile tulichokiona wazi zaidi ni mawimbi ya shinikizo yanayosambaa hewani. Mawimbi hayo ya shinikizo yanapofika masikioni mwetu tunayaona kuwa sauti. Kwa kuhamisha sahani tumeunda kwa ufanisi msemaji mkali sana ambaye hutoa sauti wakati kipengele chake kinachohamisha hubadilisha nafasi. Ingawa maonyesho haya rahisi husaidia kujenga uelewa intuitive ya kuenea kwa mawimbi ya shinikizo, uhusiano kati ya vibration ya kitu na usumbufu shinikizo ni zaidi ngumu. Sisi hivi karibuni kuchunguza baadhi ya maelezo hayo, lakini kwa sasa tutaweza kuzingatia uhusiano kati ya harakati ya kwamba vibrating kipengele na sauti inazozalishwa. Hebu tuanze kwa kuanzisha ala rahisi ya muziki. Kwa kubonyeza ufunguo chini, unaweza kuomba sahani kubadilisha haraka nafasi yake. Unapoachilia ufunguo huo, sahani itarudi nyuma kwa kiasi hicho hicho. Kumbuka kwamba kila moja ya funguo tatu husogeza sahani kwa kiasi tofauti, na alama nyekundu inaonyesha offset ya jumla: Licha ya unyenyekevu wake, kifaa hiki kinatuwezesha kuchunguza mambo machache. Kwanza, kuruka ghafula kwa sahani hutokeza sauti ya kulipuka. Pia haina maana kama kitu imeweza kuruka mbele au nyuma <unk> athari inaonekana sawa. Pili, sauti ya pop hiyo inategemea ukubwa wa kuruka <unk> kubwa kuruka kwa sauti ya pop. Hatimaye, sahani hii inaweza kuruka mbele au nyuma kutoka nafasi yoyote <unk> unaweza kuthibitisha hii kwa kushikilia moja ya funguo na kisha kubonyeza funguo nyingine. Sauti ya pop hiyo ni sawa kabisa na ile inayotokezwa na kubonyeza funguo peke yake. Pops tatu za sauti tofauti si mengi ya kuunda muziki na kama hii tu inaruhusu sisi kucheza baadhi ya beats rahisi. Unaweza kuwa umejaribu kubonyeza vifungo haraka iwezekanavyo ili kuunda nyimbo zenye msisimko zaidi, lakini kuna mipaka kwa jinsi vidole vya binadamu vinavyoweza kubonyeza vifungo haraka. Kwa shukrani, tunaweza kuuliza kifaa wewe ni kutumia kurudia pops katika kiwango fulani. Katika maonyesho hapa chini, unaweza kuhamisha slider ya kwanza kubadilisha mara ngapi kwa sekunde sahani inapaswa kuruka mbele na nyuma. Unaweza pia kudhibiti jinsi mbali sahani inahama na kila kuruka kwa kutumia slider ya pili: Kuelezea mzunguko wa vibration, mimi kutumia kitengo cha hertz, mara nyingi abbreviated kama Hz, inaelezea jinsi nyingi mzunguko kamili ya mwendo kutokea ndani ya sekunde moja. Kwa mfano, frequency ya 2 Hz inamaanisha kwamba mzunguko kamili wa mzunguko ulifanyika katika sekunde moja, ambayo inamaanisha kwamba bamba lilihama na kurudi tena. Wakati katika masafa ya chini mifumo hii inasikika kama mfululizo wa pops haraka, katika masafa ya juu sauti hizi kuchanganyika pamoja katika noti inayoonekana. Jambo muhimu zaidi tunaloweza kuona ni kwamba kadiri frequency ya vibrating ya sahani inavyozidi kuwa kubwa, ndivyo sauti ya sauti inayoonekana inavyozidi kuwa kubwa. Labda umeona kwamba kwa viwango vya chini unaweza kufuata mwendo wa sahani ya kutetemeka, lakini kama mzunguko huongezeka, inakuwa haiwezekani kwa uhakika kufuatilia mwendo wake. Kwa bahati mbaya, sisi ni kugonga mipaka ya masafa tunaweza kwa urahisi kuonyesha juu ya skrini na kutambua na macho yetu, hivyo kutoka hatua hii juu ya mimi itabidi kwa kiasi kikubwa kupunguza mwendo wa sahani. Mifumo ya mwendo bado itakuwa sahihi, wao tu kutokea kwa kasi ya polepole sana hivyo kwamba tunaweza kweli kuona nini kinaendelea. Kwa kuzingatia hilo, hebu tufanye synthesizer yetu ya muziki kuunda wale kuruka mara kwa mara sana ya sahani. Sisi <unk>ll kugawa mzunguko tofauti kwa kila ufunguo <unk> unaweza kuona yao kuonyeshwa juu ya funguo. Mimi pia kuchora machoro tatu zinazolingana na kila ufunguo, na machoro ya ziada kuonyesha athari ya jumla ya harakati za mtu binafsi. Michoro hiyo, au maumbo ya mawimbi, huwakilisha mwendo wa sahani kama inavyoombwa na funguo. Kama plots kupita kupitia mstari wa kati wao kuonyesha nafasi ya sasa ya sahani, lakini kumbuka kwamba harakati tunaona hapa ni mia chache mara polepole kuliko wale halisi: Kumbuka kwamba wakati wewe bonyeza funguo mbili kwa wakati mmoja unaweza bado kusikia sauti zote mbili kucheza kupitia - sisi tayari wameona kwamba mwisho sahani harakati ni tu jumla ya harakati za mtu binafsi kwamba kila funguo maombi. Unaweza pia kuwa na alitambua kwamba I've shifted mstari wa kati ya sahani. Kile tunachokiona kuwa nafasi ya msingi ya kitu hicho ni cha hiari. Funguo tu kuomba sahani ya kusonga mbele au nyuma na wao hawana huduma kutoka ambapo ni kuanzia, lakini kuwa na kitu oscillate karibu na asili vizuri maalum itakuwa rahisi zaidi. Kama wewe kuangalia kwa karibu katika njama wakati funguo mbili ni kushikiliwa, unaweza pia kuwa na kugundua kwamba kuna baadhi ya maeneo ambapo moja ya funguo anataka sahani kusonga mbele, wakati mwingine anataka kusonga nyuma. Matokeo ni kwamba, sahani hiyo hubaki mahali pake. Kile ambacho kinaweza kushangaza kidogo ni kwamba hakuna kitu cha kichawi kuhusu sauti inayotolewa na sahani wakati funguo mbili zinaingiliwa <unk> sauti inayotolewa inategemea tu mwendo wa mwisho wa sahani. Katika mfano wa chini, nilifanya funguo ya tatu itoke na muundo sawa wa mwendo kama mchanganyiko wa funguo mbili za kwanza: Sauti inayotokezwa wakati funguo ya tatu inashikiliwa ni sawa kabisa na sauti inayotokezwa wakati funguo mbili za kwanza zinashikiliwa pamoja. Hii ni uchunguzi muhimu sana, kama inatuwezesha kuchunguza baadhi ya mifumo mingine ya mwendo kwamba funguo inaweza kusababisha. Tutapanua wazo hili kwa kubadilisha njia ambayo sahani vibrates. Badala ya kuiruhusu iruke-ruke haraka, tunaweza kumwomba bamba hilo liendelee kusonga kwa mwendo wa mara kwa mara, likifuata muundo wa pembetatu. Hebu tuangalie muundo huo kwa karibu. Kama maumbo mengine ya mara kwa mara, wimbi la pembetatu linaweza kuonyeshwa kwa amplitude yake, mzunguko, na awamu: Katika mifano hii ya kuona amplitude inadhibiti urefu wa wimbi. Mzunguko huamua upana wa muundo unaorudiwa, ambao huathiri jinsi marudio mengi ya pembetatu za juu na chini vinavyoweza kufaa ndani ya sekunde moja. Hatua hiyo hubadilisha tu umbo ili kubadilisha kile tunachokiona kuwa mwanzo wa muundo huo. Tayari tumeona jinsi amplitude ya mwendo wa sahani huathiri sauti inayoonekana, na mwendo huu wa pembetatu sio tofauti. Katika mfano wa chini, funguo zote tatu kusonga sahani katika muundo huo, tu amplitude yao ni tofauti: sawa na muundo wa awali mstatili, sisi kutambua masafa tofauti ya mwendo huu wa pembetatu kama pitches tofauti. Katika maonyesho yafuatayo, kila ufunguo una mzunguko tofauti kama ilivyoonyeshwa na kuonyesha katika sehemu ya juu ya kibodi. Kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti, kwa sababu ya hali ya juu ya sauti. Hebu kuona mchakato huu hatua kwa hatua kwa kutumia mifano michache ambayo unaweza kuchagua katika maonyesho hapa chini. Slider kufuatilia mchakato wa kuongeza nne tofauti umbo la pembetatu ya amplitude tofauti: Kumbuka kwamba kila waveform inaonyesha baadhi ya mabadiliko kutoka nafasi ya sasa. Hii offset inaweza kuwa chanya au hasi <unk> Mimi <unk>m visualizing yake kwa kutumia mishale ndogo. Kuongeza waveforms tunahitaji tu kutumia mishale hii kwa umbo zilizopo, ambayo tunaweza kufanya kwa kuweka mishale juu ya umbo hilo na kisha deforming umbo kama mishale mahitaji. Vipande vya manjano katika sehemu ya chini huwakilisha amplitude ya maumbo maalum ya mawimbi ya pembetatu katika masafa tofauti. Kumbuka kwamba ni kabisa faini kwa amplitude kuwa sifuri <unk> kwamba kipengele tu si kushiriki katika uumbaji wa mwisho wa jumla waveform. parameter ya mwisho ya kazi yoyote ya mara kwa mara ni awamu. Kwa yenyewe, awamu haina kufanya mengi kwa sauti, lakini inaweza kutupa baadhi ya matatizo wakati awamu-shifted umbo ni pamoja na umbo jingine la mzunguko huo huo. Katika maonyesho ya chini, wote tatu funguo ya synthesizer yetu vibrate katika mzunguko huo na amplitude, lakini kila moja ni kuhamishwa katika awamu kwa robo ya mzunguko: moja kwa moja, kila moja ya funguo tatu sauti sawa kabisa, lakini mara moja sisi kuchanganya funguo matatizo huanza. Wakati ufunguo wa katikati unapounganishwa na ufunguo wa kushoto au wa kulia, sauti inayotokezwa ni kubwa kidogo. Hata hivyo, wakati funguo ya kwanza na ya mwisho kuunganisha, waveforms yao kikamilifu kufuta kila mmoja nje kusababisha hakuna harakati sahani <unk> hakuna sauti ni iliyotolewa! Sisi ni uzoefu wa kuharibu kuingiliana ambapo vilele vya moja waveform kuishia kupata kufuta nje na mabonde ya waveform nyingine. Kazi za pembetatu zilikuwa ngumu zaidi kuliko kuruka rahisi, lakini hakuna kitu kinachotuzuia kutumia harakati za hiari kabisa. Katika sehemu ya juu ya maonyesho hapa chini, unaweza kuchora sura yako mwenyewe ya wimbi kwamba hufafanua vibration ya sahani. Kisha nitarudia muundo huu kwa mzunguko uliopendekezwa na kila kitufe cha kibodi, nikikuruhusu usikie matokeo kupitia wasemaji wako: Baada ya kucheza na maumbo tofauti unaweza kuwa umegundua kwamba wakati wana tabia tofauti, wote wanashiriki sauti sawa. Mifumo tofauti huishia kuunda sauti tofauti ambayo ni mali ya sauti ambayo inatuwezesha kutofautisha vyombo tofauti vinavyopiga noti moja. Hadi sasa tumekuwa kujenga waveforms ya kuongezeka utata, lakini ni wakati sisi kugeuka mwelekeo huu kwa kujadili mawimbi sinusoidal, ambayo fomu ya msingi zaidi ya kazi ya mara kwa mara. Hebu jaribu kuunda sine wimbi kutoka mwanzo. Katika onyesho hapa chini, unaweza kushuhudia mduara unaozunguka na nukta ndogo nyekundu iliyochorwa juu yake. Kama sisi kufuatilia nafasi ya wima ya kwamba hatua kwa muda sisi <unk>ll kuchora kamili sine wimbi. Wakati tunaweza kugawa amplitude, mzunguko, na awamu kwa kazi nyingine yoyote ya mara kwa mara, mali hizo vizuri inalingana na radius, kasi, na pembe ya awali ya mzunguko wa kuzunguka ambayo inazalisha wimbi hili: Hebu tuone jinsi sinusoidal vibration ya sahani sauti. Katika maonyesho hapa chini, unaweza uzoefu mbalimbali nzima ya mawimbi sine na masafa tofauti na amplitudes kucheza kupitia wasemaji wako. Baadhi ya sauti zinatarajiwa kuwa zisizoonekana, kwa hiyo usibadilishe kiasi cha sauti yako mpaka ufikie masafa ya kati: Labda umegundua kwamba masafa ya chini huanza kusikika, na huongezeka kwa sauti kadiri mzunguko unavyoongezeka. Hata hivyo, kadiri masafa yanavyozidi kuongezeka, sauti hiyo hupotea na kuwa kimya. Ingawa msemaji wako na mipangilio ya sauti inaweza kwa kiasi fulani kuchangia athari hii, ni vinginevyo kabisa asili - wanadamu wanaweza kusikia katika mbalimbali ndogo ya masafa na sauti kuonekana mabadiliko na mzunguko. Katika ulimwengu wa kimwili, sauti hizi za sinus safi hutengenezwa kwa urahisi kwa kupiga forks tofauti za kusawazisha, lakini kwenye tovuti hii tunapaswa kuamua kufanya synthesizer yetu kucheza baadhi ya noti hizi kwa masafa tofauti: Kwa mara nyingine tena, kama kila ufunguo unaomba sahani ili kutetemeka kwa njia fulani, mwendo wake wa mwisho ni jumla ya maombi hayo ya mtu binafsi. Kuongeza kunaweza wakati mwingine kuunda athari zisizotarajiwa wakati masafa ya sauti zinazochezwa ziko karibu, ambazo unaweza uzoefu katika onyesho hapa chini. Kumbuka masafa ya sauti kama inavyoonyeshwa juu ya funguo za kibodi: Unapobonyeza funguo mbili kwa wakati mmoja utasikia kiasi cha sauti ikiongezeka na kupungua katika matukio yanayojulikana kama beats. Si masikio yako kuwadanganya wewe <unk> ni athari halisi inayosababishwa na masafa mawili ya awali kuingiliana, ambayo unaweza kuchunguza kwenye njama. Madhara haya yanaweza kutokea na ishara zote za mara kwa mara na unaweza kusikia wakati wa kukaa kwenye ndege - kasi ya mzunguko wa injini mbili inaweza kuwa haijapatani kikamilifu na unaweza kusikia tofauti katika kiasi cha sauti inayotokana. Kama vile mawimbi ya pembetatu, tunaweza kwa urahisi kuona jinsi kuongeza mawimbi mengi ya sinus yenye amplitude tofauti, masafa, na awamu huunda maumbo magumu zaidi. Katika maonyesho hapa chini, mimi nina kuongeza kundi la mawimbi haya sine. Unaweza kutumia udhibiti hapa chini kuchagua umbo tunalounda: Angalia kwamba mimi pia ninatumia wimbi la <unk>special<unk> sine na mzunguko wa 0 Hz <unk> inaonyesha thamani ya mara kwa mara ambayo ni muhimu kuwakilisha maumbo ya wimbi ambayo ni, kwa wastani, si katikati karibu na msingi. Kwa kuzingatia muundo wa mawimbi haya, tunaweza kusema kwa urahisi kwamba muundo wa mwisho hutengenezwa na mawimbi sita tofauti ya sinus yenye amplitude, mzunguko, na awamu tofauti. Badala ya kuongeza hizi sinusoidal msingi vitalu kuunda kazi ngumu, tunaweza, labda kushangaza, kugeuka mchakato huu na kuharibu yoyote kurudia wimbi katika sine mawimbi yake ya kiungo. Hebu tuone hii katika hatua. Katika maonyesho hapa chini, unaweza kuchora umbo linalotakiwa katika sehemu ya juu na nitafanya bidii yangu kuelezea kwa kutumia jumla ya mawimbi ya 64 ya sinus na kuongezeka kwa masafa. Kwa uwazi, mimi itabidi kuruka uhuishaji wa mivuko ya kijani na nyekundu kuwa aliongeza: Katika sehemu ya chini mimi tena kuonyesha amplitudes ya mtu binafsi sine mawimbi kujenga umbo la mwisho, lakini tangu tuna wengi wao mimi tu pakiti yao amplitude baa karibu na kila mmoja ili kuunda mchoro wa jumla ya amplitudes. Kulingana na jinsi ishara ya awali ni kuonyeshwa, njia hii ya decomposing kazi ya mara kwa mara katika sines inajulikana kama mfululizo Fourier au Fourier kubadilisha. Hii inatuwezesha kuunda wimbi la wimbi lililoonyeshwa kama kazi ya wakati katika wimbi lililoonyeshwa kama mkusanyiko wa masafa ya mtu binafsi yaliyo ndani ya wimbi hilo. Tunaweza kuangalia muundo wowote unaorudiwa ama kama mkusanyiko wa maadili ya mtu binafsi kwa muda au kama mkusanyiko wa mawimbi tofauti ya sinus ambayo yanaongeza hadi muundo huo. Maoni yote mawili yanawakilisha umbo moja, yanaionyesha tu kwa njia tofauti. Kwa ujumla, Fourier kubadilisha inatupa wote amplitude na awamu ya kila kuchangia sine wimbi. Kama tulivyoona, hata hivyo, kusikia kwa binadamu ni karibu kabisa agnostic kwa mabadiliko ya awamu hivyo amplitudes kawaida kutoa habari ya kutosha kuelezea umbo la awali. "Kama wewe kuhamisha slider kwa mwisho utaona kwamba uharibifu wa waveform kwamba haraka mabadiliko ya maadili mwisho kuunda ripples katika kazi recreated katika athari inayojulikana kama ""Gibbs Phenomenon.""" Kumbuka, hata hivyo, kwamba kama sisi kuongeza zaidi sines ya mzunguko wa juu upana wa eneo hilo ripple hupungua. Jambo la muhimu zaidi ni kwamba, muundo huu wa mraba ambao hurudiwa mara moja kwa sekunde una mawimbi mengi ya sinus ya masafa ya juu zaidi. Hii inaelezea kwa nini tulikuwa na uwezo wa kusikia sahani kuruka nyuma na mbele mara moja kwa sekunde <unk> wakati hatuwezi kusikia wengi wa wale masafa ya chini, tunaweza kusikia sauti za juu safi. Wimbi la sinus kwa mzunguko wa 1Hz lina ishara ya 1Hz tu ambayo hatuwezi kusikia. Hebu tuangalie wigo wa vitu katika mazingira yako. Katika onyesho hapa chini, unaweza kubonyeza kifungo kuanza kutumia kipaza sauti yako. Ill kisha kuchora mchoro wa masafa tofauti na amplitudes yao kama kusikia na kifaa: Mimi kupendekeza whistling, au kuimba <unk>aaaaaa<unk> kutumia sauti ya chini na ya juu ya kuona masafa tofauti kuonekana. Ikiwa una vyombo vya muziki karibu au programu zinazoweza kuunda tena sauti zao, michoro inayotokana nayo katika visa vingi pia itaonyesha usambazaji wa kuvutia. Sauti ngumu zaidi kama vile hotuba au muziki zitabadilisha umbo lao haraka, jambo linalodokeza hali yao ya kubadilika-badilika. Wakati wa kuimba au kucheza noti zako rahisi kwenye kipaza sauti unaweza kuwa umegundua mifumo ya kurudia ya masafa <unk> masafa fulani ya msingi hupata ndugu mwenye nguvu kwa mara mbili, mara tatu, na mara nyingi zaidi ya masafa ya awali. Mifumo hii ya kurudia inaweza kutokea kwa sababu tofauti, na katika sehemu inayofuata nitaelezea jinsi wanavyoundwa na baadhi ya vitu rahisi vya kutetemeka. Kuchunguza vibrations hizi, tutaanza kwa kujenga mfumo rahisi inayojumuisha wingi mdogo kushikamana na spring mbili. Katika maonyesho unaweza kusonga kwamba wingi juu na chini na slider: Katika amplitudes ndogo ya mwendo sisi ni kuzingatia hapa, mfumo oscillates sinusoidally na baadhi ya asili mzunguko kwamba inategemea wingi wa kitu na rigidity ya springs. Mvutano kati ya sehemu zinazozunguka na upinzani wa hewa husababisha miendo hiyo ipoe baada ya muda. Katika simulation hii, mzunguko wa mzunguko unaenda juu na chini tu, lakini kwa kanuni inaweza pia kwenda kushoto na kulia. Hata hivyo, harakati hizi za usawa hazitakuwa na matumizi mengi kwetu, kwa hiyo tutazingatia tu harakati za wima. Hebu kuongeza wingi mwingine na spring kwa mfumo huu: Hata na hii rahisi kuanzisha tabia ya mfumo huu inaweza kupata ngumu kabisa, ikiwa ni pamoja na baadhi ya mifumo ya kawaida ya mwendo, lakini matukio haya yote inaweza kweli kugawanywa katika jumla ya mbili rahisi zaidi mwendo. Katika mwendo wa kwanza wingi husonga pamoja katika mwelekeo uleule, na katika mwendo wa pili wingi husonga katika mwelekeo kinyume. Katika maonyesho ya chini, wewe si kudhibiti nafasi ya wingi, lakini badala ya amplitudes ya modes hizo mbili tofauti: Kwa mpangilio makini, unaweza recreate miundo ya mwendo ya kuvutia kwamba mfumo huu inaweza kuunda. Kumbuka kwamba hali ya pili oscillates haraka kuliko hali ya kwanza. Hii inapaswa kufanya maana intuitive tangu spring kati ni kupata kunyoosha, ambayo anaongeza ziada kuvuta nguvu kwa mfumo. Mfano wowote wa mwendo wima wa wale watu wawili sawa na chemchemi tatu inaweza kugawanywa katika modes mbili kawaida kwamba sisi tu nimeangalia. Wazo hilo linaweza kutumika katika mifumo migumu zaidi. Katika mfano hapa chini, unaweza kuona njia tano za kawaida za mfumo unaojumuisha mizigo mitano na springs sita: Tunapoongeza idadi ya mizigo na springs kwa kiwango cha juu, kila sehemu huanza kufanana na sehemu ndogo ya kamba ambayo ina uzito na elasticity. Conceptually, kamba rahisi inaweza kufikiriwa kama mfumo wa kiasi kikubwa cha ndogo sana spring-na-wingi sehemu <unk> unaweza kuona moja katika maonyesho yafuatayo. Hebu kuangalia baadhi ya kwanza ya hali ya kawaida ya mfumo huu: Tangu mwisho wa kamba haiwezi kusonga, tu baadhi ya mifumo ya wimbi ni kuruhusiwa <unk> hakuna maumbo wimbi ambayo kusababisha mwisho wiggle juu na chini inaweza kuelezwa na kuanzisha hii. Mzunguko wa hali ya kwanza unajulikana kama mzunguko wa msingi. Kila hali ya juu ina mzunguko ambao ni integer mara nyingi ya kwamba mzunguko wa msingi <unk> kwa mfano hali ya tatu ina mara tatu kama mzunguko wa juu kama hali ya msingi. Njia hizo za kutetemeka huitwa harmoniki. Kama ilivyo kwa mifumo ya awali, mwendo wowote wa kamba iliyounganishwa na ncha mbili unaweza kugawanywa katika jumla ya mwendo huu wa mtu binafsi, kila mmoja na amplitude tofauti. Dhana hizi ni sawa sana na shughuli za kubadilisha Fourier tulifanya kwa maumbo ya wimbi, lakini wakati huu zinatumika kwa umbo la kamba. Hii yote inamaanisha kwamba mwendo wa kamba ya vibration ni jumla ya modes harmonic mtu binafsi, kila mmoja na amplitude arbitrary, na mzunguko maalum ambayo ni mara nyingi ya mzunguko wa msingi. Hebu tuone na kusikia jinsi hii inaonekana katika mazoezi. Katika maonyesho hapa chini, unaweza kuondoa kamba kwa kubonyeza na kuhamisha juu na mouse yako na kidole chako: Unaweza kuwa umegundua kwamba mahali ambapo wewe kuondoa kamba hubadilisha sauti inazozalishwa - ni matokeo tu ya uwiano tofauti wa harmoniki iliyoundwa na umbo la kamba iliyochukuliwa kabla ya kutolewa. Kiwango ambacho kamba hii inaweza kuvutwa ni kupita kiasi hapa na kuifanya ionekane isiyo ya kweli kidogo, lakini kukamata polepole kwa kamba ya gitaa hufunua tabia sawa sana. Mzunguko wa msingi wa mtetemeko wa kamba iliyoinuka hutegemea urefu wake, uzito, na mvutano. Kadiri urefu ulivyo mfupi, ndivyo mzunguko ulivyo juu <unk> hii ndiyo sababu kuu kwa nini fidla ina sauti ya juu zaidi kuliko bass ya mara mbili. Jambo la muhimu zaidi ni kwamba kwa kuweka kidole kwenye kamba na hivyo kubadilisha urefu wake halisi, mwimbaji anaweza kucheza noti tofauti kwa kutumia seti ndogo ya kamba. Katika maonyesho ya chini, unaweza kusikia tofauti katika pitch kama wewe kuhamisha slider kubadilisha eneo la kidole kushikilia chini ya kamba: The thicker kamba zaidi inertia ina na vigumu zaidi ni kuhamisha karibu, hivyo mzunguko wa sauti hupungua kama kamba inakuwa nzito, au, hasa, kama wiani wake linear huongezeka. Kwa kutumia kamba za unene tofauti au vifaa, vyombo vya muziki vinaweza kufikia sauti tofauti kwenye kamba za urefu sawa - unaweza kucheza na parameter hiyo katika onyesho hapa chini: Mwishowe, kamba ilizidi kuwa ngumu, ilizidi kurudi nyuma, kwa hivyo mvutano wa juu huongeza mzunguko - kwa kugeuza pini za kusawazisha za chombo cha kamba mtu anaweza kurekebisha mzunguko wa msingi wa kila kamba. Kwa yenyewe, kamba inayotetemeka haisogezi hewa nyingi na haisemi kwa sauti kubwa sana. Ili sauti iwe na sauti kubwa zaidi, mwendo wa kamba huhamishwa kwenye eneo kubwa ili kuongeza eneo la kutetemeka na kiasi cha sauti inayotokezwa. Umbo na vifaa kutumika katika ujenzi wa mwili wa chombo kwa nguvu kuchangia tofauti sauti masafa kupata amplified au absorbed. Mifano yetu ya springs na misa pia inaenea kwa vipengele vingine vibrating. Vipuli vya ngoma hufanyiza gridi ya vipengele vinavyotetemeka katika vipimo viwili. Chembe za vitu imara zilizofungwa kwa nguvu huunda lattice ya pande tatu - sehemu zake pia huingiliana kwa nguvu, ambayo kwa kiwango kikubwa husababisha cymbals, gongs, na kengele kutetemeka na masafa mbalimbali ya asili ambayo hutegemea umbo na ujenzi wa vyombo hivi. Vifaa vingine kama vile filimbi au ala za muziki hutegemea safu ya hewa inayosafiri haraka mbele na nyuma kupitia miili yao kama bomba, na kuunda njia nyingine ya kufanya hewa inayozunguka vibre kwa viwango fulani. Tunapoimba, kamba za sauti hufunguka na kufungwa kwa mpangilio na kutokeza mvuke wa hewa - hizo pia husababisha usumbufu wa shinikizo ambao kisha husafiri. Katika sehemu ya mwisho ya makala hii tutajifunza jinsi mawimbi ya shinikizo yanavyotokana na vitu vinavyotetemeka na jinsi yanavyoishia kuenea kupitia hewa na kuingiliana na mazingira. Hebu kuleta nyuma sahani vibrating ndani ya bomba kwamba tunaweza kudhibiti moja kwa moja na slider: Wakati rahisi kuelewa, asili zilizomo ya mfano huu hufanya ni kidogo rahisi kama si mechi tabia ya vyombo ya kawaida au wasemaji. Ili kuchunguza hali halisi zaidi ya kuenea kwa sauti, kwanza tutatumia sferia rahisi ambayo inaweza kupanua na kupanua kwa haraka kubadilisha nusu-mwisho wake - unaweza kufikiria kama aina fulani ya puto lililojaa maji ambalo tunaweza kuongeza au kuondoa kioevu haraka. Katika maonyesho hapa chini, unaweza kuona mfano wa kwamba rahisi pulsing dunia na slider kudhibiti mzunguko wa vibration yake: Hebu kuangalia shinikizo karibu uso wa dunia hii karibu kutumia mbinu yetu ya slicing hewa, wakati huu katika shells nyembamba kuzunguka dunia hii. Katika masafa ya juu ya vibration tabia ya hewa kuzunguka dunia ni sawa na kwamba ya sahani rahisi katika bomba, ambayo unaweza uzoefu katika maonyesho yafuatayo. Slider inasimamia kasi ya uhuishaji, ambayo inakuwezesha kuona maelezo, lakini mviringo huo unatikisika kwa mzunguko sawa wakati wote: Kama mviringo huu unamalizia kupanuka, unaunda eneo la shinikizo kubwa karibu na eneo lake. Hata hivyo, katika masafa ya chini ya vibration hali ni tofauti kidogo <unk> unaweza kuona hii katika simulation chini. Angalia kwamba wakati mpira unapoisha kupanuka, shinikizo hupungua. Mimi ni exaggerating mabadiliko ya ukubwa wa dunia kufanya hivyo zaidi inayoonekana: Nini sisi ni kushuhudia hapa ni hewa ya inertia. Upanuzi wa awali wa mviringo huo huongeza mwendo wa hewa iliyo karibu. Wakati mviringo unapoacha kukua, hewa hiyo bado ina kasi fulani kwa hivyo inahama kutoka kwa mviringo, ikiunda rarification ya ndani. Kwa mfano, wakati mpira unapopungua, hewa inayozunguka huanza kuvutwa kuelekea upande huo, na inaendelea kukusanyika hata baada ya mpira kufikia ukubwa wake mdogo. Utafiti wa tabia hii ni ngumu sana, lakini mwishowe kiwango cha masafa ambayo tabia hizi tofauti hutokea inategemea ukubwa na umbo la kitu kinachotetemeka. Sphere ni mfano rahisi tu wa msemaji, mfano halisi zaidi ni sahani ya mviringo inayozungukwa na ukuta mkubwa sana, sawa na sferia, uhusiano kati ya nafasi ya sahani hii na shinikizo linalotokana na sauti pia hutegemea mzunguko wa sauti. Hata hivyo, dunia hiyo ilikuwa na umbo la usawa kamili, na hilo lilifanya nguvu za mawimbi ya shinikizo yaliyotokezwa ziwe huru na mwelekeo. Kwa sahani vibrating baadhi ya symmetries ni kupotea na nguvu ya usumbufu wa shinikizo pia inategemea mwelekeo kuhusiana na chanzo: Kama wewe kuongeza mzunguko wa sauti, mawimbi ya shinikizo kuzingatia katika mwelekeo wa mbele. Kwa kawaida, masafa ya chini yanayotokezwa na wasemaji ni ya kuenea zaidi kuliko masafa ya juu. Kumbuka kwamba mimi pia kusimamishwa kuchora mistari nyembamba kutenganisha sehemu ya mtu binafsi ya hewa <unk> sisi sasa <unk>ll zaidi realistically kudhani kwamba shinikizo mabadiliko bila hatua yoyote tofauti. Kama tulivyoona, usambazaji wa shinikizo kwamba kitu vibrating inajenga inaweza kupata kidogo ngumu, lakini, shukrani, katika umbali busara, utulivu sinusoidal oscillation ya chanzo inajenga utulivu sinusoidal tofauti katika shinikizo kwamba kusafiri kupitia hewa. Unaweza kuona mtazamo wa upande wa tofauti hiyo ya shinikizo kwenye mstari wa dotted katika sehemu ya chini ya simulation ifuatayo: Hii inamaanisha kwamba kila kitu kinachohusiana na vibrations sinusoidal ya sahani kama vile amplitude, mzunguko, na awamu ya mwendo wao pia inatumika kwa mawimbi ya shinikizo wenyewe. Wakati michoro hii ya rangi ilifanya iwe rahisi kuona jinsi mabadiliko ya shinikizo yanavyoenea kupitia hewa, kwa bahati mbaya ni ya kudanganya kidogo. Kwanza, mimi nina kupunguza mambo chini kwa kiasi kikubwa kufanya kuenea kwa mawimbi kuonekana, vinginevyo wao 'd tu zip juu ya skrini yako katika sehemu ya pili. Katika hali ya joto ya chumba, mwendo wa sauti katika hewa ni karibu 1125 ft (767 mph) na 343 m s (1233 km h). Isitoshe, tofauti halisi ya shinikizo katika mawimbi ya sauti ni ndogo sana. Wakati mtu anazungumza kwa kiasi cha kawaida, shinikizo la hewa huongezeka na kupungua kwa karibu 0.02 pascal au karibu 0.002% ya shinikizo la anga. Rangi nyekundu na bluu hufanya ionekane kama shinikizo linabadilika sana, lakini katika mazoezi tofauti hizi ni ndogo sana - ni ya kushangaza sana kwamba masikio yetu hupata tofauti ndogo sana. Mimi pia haja ya kumbuka kwamba simulations shinikizo sisi tu wameona walikuwa haki rahisi. Tabia ya msemaji wa kawaida ni ngumu zaidi na njia ambayo inaathiri hewa inayozunguka inategemea ukubwa wa msemaji, umbo, na ujenzi wa jumla. Yote ya tabia hizi tata inaweza kuficha baadhi ya matukio mengine ambayo tutazungumza, hivyo kwa mifano ifuatayo mimi itabidi kurahisisha mambo kidogo kwa kutumia msingi, omnidirectional chanzo cha sauti, sawa na kupanua dunia tumekuwa kutumika. Katika mfano hapa chini, icon ya bluu ya msemaji inaonyesha chanzo hicho na icon ya manjano ya sikio inaonyesha mahali ambapo unaweza kusikia vitu - unaweza kuvuta mwisho kuzunguka kubadilisha mahali ulipo. Kwanza, unapoondoka kwenye chanzo cha sauti, huenda sauti ikachukua muda ili ikufikie. Athari hii ni matokeo tu ya mwendo wa sauti wenye mwisho. Wakati katika umbali mdogo hatuoni kwamba kuchelewesha, ni kwa urahisi kuchunguzwa wakati umeme hits na sauti anapata kwetu na lag noticeable. Katika maonyesho haya umbali pia ni kubwa sana ambayo unaweza kuona juu ya kiwango katika kona ya chini ya kushoto. Pili, sauti tunayoisikia hutegemea umbali kutoka kwa chanzo chake. Kwa chanzo rahisi kama hiki mawimbi ya shinikizo husafiri mbali kwa radial. Ingawa mawimbi hayo huonekana kama duru katika maonyesho haya ya pande mbili, katika ulimwengu wa kimwili hufanyiza mviringo unaopanuka. Amplitude ya p-kiwango cha mawimbi ni inversely uwiano na umbali kutoka chanzo, ambayo ina maana kwamba wakati sisi mara mbili umbali kutoka chanzo, amplitude ya mawimbi ya sauti ya chini ya nusu. Kama tulivyoona mara chache na sahani vibrating, mabadiliko katika amplitude ya mwendo wao, na hivyo mabadiliko ya amplitude ya mawimbi ya shinikizo yanayotokana, huathiri jinsi kubwa sauti sauti. Tofauti ya ukubwa wa mabadiliko ya shinikizo kati ya sauti kubwa na sauti ya chini inaweza kuwa kubwa sana - amplitude ya mawimbi ya shinikizo la sauti kutoka kwa injini ya ndege ya kuruka ni karibu mara 10,000 kubwa kuliko ile ya kelele dhaifu. Kulinganisha maadili kwa kutumia idadi kubwa hivyo itakuwa isiyofaa na maneno yasiyoeleweka kama "nyoyo dhaifu" hayajafafanuliwa vizuri. Ili kutatua matatizo haya dhana ya kiwango cha shinikizo la sauti au SPL ni kuletwa: Hebu unpack equation hii. Kwa sababu ya kiwango cha juu cha sauti, kiwango cha juu cha sauti kinaweza kuwa 20 mPa (micropascal) au zaidi, na kwa sababu ya kiwango cha juu cha sauti, kiwango cha juu cha sauti kinaweza kuwa 20 mPa (micropascal). Kwa mfano, logarithm ya msingi ya 10 ni sawa na idadi ya tarakimu katika uwakilishi wa decimal wa idadi, na kuongeza tarakimu moja ya msingi ya 10 kwa 1 ni sawa na kuongeza tarakimu moja ya msingi ya 10 kwa 1 na kuongeza thamani ya logarithm ya msingi ya 10. Nini sisi ni kuhesabu hapa ni logarithm ya uwiano wa shinikizo kipimo p kwa shinikizo rejea. "Hii ni kwa sababu ya ""dB"" (dB) ambayo ni kipimo cha decibel, ambayo mara nyingi hufupishwa kama ""dB.""" Hebu tuone jinsi usawa huu unaweza kutumika katika mazoezi. Katika mfano wa chini, unaweza kubadilisha thamani ya shinikizo na kuona jinsi inavyoathiri thamani ya kiwango cha shinikizo la sauti: Kumbuka kwamba umbali wa mara mbili hupunguza shinikizo kwa nusu, ambayo kwa ufanisi inamaanisha kwamba kila mara mbili ya umbali hupunguza kiwango cha shinikizo la sauti kwa decibels 6. Kwa upande mwingine, kutumia wasemaji wawili badala ya mmoja pia huongeza SPL kwa decibels 6 tu. Kwa mfano, ikiwa sauti mbili zinatofautiana kwa dB 20, basi amplitude ya moja ni mara 10 kubwa kuliko ile ya nyingine. Uwezo wa sauti unaotambuliwa unategemea sana kiwango cha shinikizo la sauti, hata hivyo, kwamba sauti pia inategemea mzunguko wa sauti - umepata hii wakati wa kurekebisha sahani inayoenda kwa mwendo wa sinusoidal. Ili kuelezea hili, thamani ya amplitude ya shinikizo katika masafa tofauti mara nyingi ni uzito hivyo kwamba maadili decibel mechi ya binadamu utambuzi zaidi karibu. Pia ni thamani ya kutaja wingi mwingine kuhusiana na <unk>kiasi<unk> ya sauti <unk> sauti nguvu I ambayo hupima nguvu ya sauti kwa eneo. Nguvu ya wimbi lolote inategemea mraba wa amplitude yake, hivyo kwa mawimbi rahisi mviringo nguvu ya sauti ni sawa na mraba inverse ya umbali katika sheria ya mraba inverse, kama ilivyoelezwa hapo awali katika blog hii, sawa na shinikizo, mtu anaweza pia kufafanua kiwango cha nguvu ya sauti, ambayo inaelezea uwiano wa nguvu na nguvu ya kumbukumbu. Kipengele kingine kinachoathiri kuenea kwa sauti ni kupunguzwa kwa sauti kunasababishwa na, kwa mfano, Viscosity ya hewa na conductivity ya joto kati ya molekuli zake Maingiliano haya kusababisha kupungua kwa ziada katika amplitude ya mawimbi ya shinikizo kama wao kusonga mbali na chanzo. Kupungua kunategemea joto na unyevu wa hewa na pia hutofautiana na mzunguko wa sauti. Athari hizi ni ngumu sana hivyo mimi si akaunti kwa ajili yao hapa. Maonyesho ya mwisho yalikuwa na kipengele kimoja cha mwisho kilichofichwa ambacho unaweza kuwa umepata wakati wa kushinikiza funguo za kibodi na kubadilisha nafasi yako ya kusikia ikilinganishwa na msemaji. "Hii ni athari ya kawaida inayopatikana wakati chanzo cha sauti kinazunguka kwa jamaa na mpokeaji, kama katika maonyesho hapa chini, ambapo unaweza kudhibiti kasi ya msemaji kwa kutumia slider: ""Kumbuka kwamba wakati msemaji anakwenda kwa mpokeaji, sauti huongezeka, na wakati anasafiri mbali na mpokeaji, sauti hupungua katika jambo linalojulikana kama athari ya Doppler." Athari hii ni rahisi kuelewa kwa kuangalia kilele cha mtu binafsi wa mawimbi ya sauti. Katika onyesho hapa chini, unaweza kudhibiti mtiririko wa muda na kasi ya msemaji. Njama katika nyimbo ya chini wakati kilele cha wimbi la sauti ni iliyotolewa, na wakati wao ni kusikia katika kituo cha icons yao husika: Msemaji emits kilele katika cadence fulani fasta, ambayo inategemea sauti mzunguko. Wakati kilele cha kwanza hutolewa, itachukua muda fulani ili kifikie mpokeaji. Hata hivyo, wakati msemaji anapoelekea msikilizaji, kilele kinachofuata kitatolewa kutoka umbali wa karibu kuliko cha awali, na itachukua muda mfupi kufika kwa mpokeaji - mpokeaji ataona kilele hicho kikija kwa kasi zaidi, au mzunguko wa juu, ambao tunatambua kama kiwango cha juu. Wakati msemaji anapoondoka mbali na mpokeaji, hali ni kinyume - kila kilele kipya lazima kusafiri umbali mrefu kidogo kupata msikilizaji, na kufanya kilele zaidi kuenea nje kwa mpokeaji kusababisha mzunguko na pitch kushuka. Unaweza kuona athari zote mbili katika njama katika sehemu ya chini ya maonyesho <unk> baa za manjano kupata bunched juu au loosened juu kuhusiana na baa bluu, ambayo ina maana kuongezeka au kupungua kwa mzunguko wa sauti kupokea. Msemaji hajui kabisa kinachoendelea - anaendelea tu kutoa sauti kwa mzunguko uleule. Hata hivyo, kadiri mwangaza unavyoongezeka, mviringo unaofanana na kilele cha mawimbi huongezeka na kuunda hali ya shinikizo kubwa zaidi. Wakati kitu cha kutokeza sauti kinapozidi mwendo wa sauti kitatokeza mlipuko wa sauti. Ingawa mwendo wa jamaa ulitokeza athari zenye kupendeza, hata hali zisizo na mwendo zaweza kuishia kutokeza matukio yasiyotazamiwa. Katika onyesho lililo hapa chini, noti safi zinazochezwa kwenye kibodi hutolewa na wasemaji wawili kwa wakati mmoja. Unaweza kuhamisha nafasi yako ili kuona jinsi mwingiliano wa mawimbi ya shinikizo unavyoathiri kile unachosikia: Kulingana na eneo la mpokeaji, kiwango cha sauti huongezeka na kupungua, ambayo pia tunaweza kuona katika kupungua kwa nguvu ya mabadiliko ya shinikizo katika maeneo hayo. Athari hii husababishwa na kuharibu kuingiliana kwa mawimbi ya shinikizo kutoka kwa wasemaji wawili <unk> vilele kutoka kwa msemaji mmoja hufuta mabonde kutoka kwa mwingine. Kama wewe kuwa na upatikanaji wa mbili msemaji mfumo unaweza kweli kujaribu kuiga majaribio kwamba. Kwa bahati nzuri, masafa tofauti-tofauti yaliyochezwa na funguo tofauti-tofauti huishia kuwa na sehemu zilizokufa katika maeneo tofauti-tofauti, pia. Kwa kuwa sauti nyingi tunazosikia zina masafa mengi tofauti ambayo hubadilika daima, hatujapatia tofauti hii ya sauti katika hali za vitendo zaidi. Ni muhimu kutambua kwamba kuingiliana si kudumu kuharibu mawimbi kama usumbufu tu kupita kwa kila mmoja. Hii inapaswa kuwa na maana ya intuitive <unk> wakati watu wachache ni wakati huo huo kuzungumza katika chumba kimoja hotuba yao haina kuwa garbled au faded na bado tunaweza kuelewa nini wanasema. Kwa urahisi kabisa, majaribio yetu yote hadi sasa yalitokea katika nafasi kubwa ya wazi isiyo na vizuizi vyovyote. Hebu tuone nini kinatokea kama sisi kuweka ukuta katika mazingira yetu ya majaribio: Wakati wewe kutoa msukumo mfupi wa sauti wakati amesimama mbali ya kutosha kutoka ukuta, utasikia awali iliyotolewa sauti na baadhi ya kuchelewa katika athari ya kawaida inayojulikana kama echo. Angalia umbali unaohusika hapa kama unavyoonyeshwa na kipimo katika kona ya chini ya kushoto. Wakati wa kusimama karibu na ukuta, kuchelewesha kati ya sauti ya awali na iliyoonyeshwa inakuwa fupi sana - wakati ni ndogo kuliko karibu 50 ms, mfumo wetu wa kusikia hauwezi kutofautisha sauti hizi mbili tofauti. Huenda uliona kwamba wimbi linalorudiwa ni la utulivu zaidi. Wakati kuongezeka kwa umbali wa kusafiri akaunti kwa baadhi ya kupunguzwa kwamba, ni pia ukuta yenyewe kwamba sehemu hufyonza nishati ya usumbufu shinikizo. Kwa ajili ya maonyesho ya mwisho katika makala hii tutaweza kuhamia nafasi kabisa imefungwa kuruhusu mawimbi ya sauti bounce nyuma na mbele kutoka kuta. Unaweza kubadilisha ukubwa wa chumba hiki kwa kutumia slider na kushuhudia kutafakari kwa sauti kutoka kwa kuta: Hata baada ya kuacha kubonyeza funguo inaweza kuchukua muda kwa sauti kufa, kwa sababu mawimbi huendelea kutafakari kutoka kwa kuta, kupoteza tu sehemu ya nishati yao kwa kila bounce. "Huenda umesikia kuhusu hali inayoitwa ""reverb"" katika chumba tupu kisicho na fanicha, mapazia, au nguo ambazo hufyonza mawimbi ya shinikizo na kupunguza sana mwangaza wa sauti." Sayansi ya Sauti ni kiwango cha kuingia, bado kitabu kina sana kilichowekwa wakfu kwa mada nyingi zinazohusiana na sauti. Waandishi watatu hutumia kiasi kidogo cha hisabati na kuzingatia zaidi maelezo ya kiwango cha juu cha matukio tofauti, ikiwa ni pamoja na utambuzi wa sauti, mizani ya muziki, na vifaa vya sauti. Ubaya pekee ni bei, lakini kwa bahati nzuri mtu aweza kupata nakala za bei nafuu zilizotumiwa mtandaoni. Kwa ajili ya uchunguzi zaidi wa Fourier Transform na usindikaji wa ishara mimi sana kupendekeza The Scientist na Mhandisi 'Mwongozo wa Digital Signal Processing na Steven W. Smith. Mimi binafsi kweli kama mtindo wa mwandishi wa uvumilivu kueleza mambo hatua kwa hatua na mifano mingi. Juu ya hayo, kitabu chote kinapatikana mtandaoni bila malipo kwenye tovuti ya mwandishi. Mawimbi ya sauti si tu kwa gesi <unk> wao kuenea kwa njia ya kioevu na imara pia. Hata hivyo, katika maisha ya kila siku, karibu kila kitu tunachosikia ni tofauti ndogo za shinikizo la hewa, ambazo mwishowe ni mabadiliko tu katika idadi na nguvu ya mgongano wa chembe za hewa. Mimi kupata ni ya kuvutia kwamba kelele zaidi irritating na muziki zaidi kuchochea ni kuendeshwa na matukio sawa - ni tu maumbo ya msingi na ukubwa wa mawimbi yao shinikizo kwamba kufanya sauti yao hivyo distinctively tofauti.