Sound

1. Sound What is sound?

2. Sound, Amplitude and Ears Sound is a complex phenomena involving physics and perception. sound always involves at least three things: � something moves something transmits the results of that movement; � and though this is philosophically debatable: � something (or someone) hears the results of that movement.

3. Sound All things that make sound move, and all things that move, make sound

4. Sound Sound is the vibration of any substance. The substance can be air, water, wood, or any other material. The only place in which sound cannot travel is a vacuum. When these substances vibrate, or rapidly move back and forth, they produce sound. Our ears gather these vibrations and allow us to interpret them.

5. Sound To be more accurate in our definition of sound, we must realize that the vibrations that produce sound are not the result of an entire volume moving back and forth at once. Instead, the vibrations occur among the individual molecules of the substance, and the vibrations move through the substance in sound waves. As sound waves travel through the material, each molecule hits another and returns to its original position. The result is that regions of the medium become alternately more dense, when they are called condensations, and less dense, when they are called rarefactions.

6. Condensations and rarefactions

7. Sound Waves

8. Wavelength and Period What is meant by wavelength and Period?

9. Wavelength and Period The wavelength is the horizontal distance between any two successive equivalent points on the wave. That means that the wavelength is the horizontal length of one cycle of the wave. The period of a wave is the time required for one complete cycle of the wave to pass by a point. So, the period is the amount of time it takes for a wave to travel a distance of one wavelength.

10. Wavelength and Period

11. Amplitude What is meant by the Amplitude of a sound?

12. Amplitude The amplitude of a sound is represented by the height of the wave. When there is a loud sound, the wave is high and the amplitude is large. A smaller amplitude represents a softer sound. A decibel is a scientific unit that measures the intensity of sounds. The softest sound that a human can hear is the zero point. Humans speak normally at 60 decibels. 

13. Amplitude

14. Frequency What is meant by the Frequency of a sound wave?

15. Frequency Every cycle of sound has one condensation, a region of increased pressure, and one rarefaction, a region where air pressure is slightly less than normal. The frequency of a sound wave is measured in hertz. Hertz (Hz) indicate the number of cycles per second that pass a given location. If a speaker's diaphragm is vibrating back and forth at a frequency of 900 Hz, then 900 condensations are generated every second, each followed by a rarefaction, forming a sound wave whose frequency is 900 Hz.

16. Pitch What is Pitch?

17. Pitch How the brain interprets the frequency of an emitted sound is called the pitch. We already know that the number of sound waves passing a point per second is the frequency. The faster the vibrations the emitted sound makes (or the higher the frequency), the higher the pitch. Therefore, when the frequency is low, the sound is lower.

18. What is meant by constructive and Destructive Interference of Sound Waves?

19. Constructive and Destructive Interference of Sound Waves Let's set up a situation: Two speakers are situated at the exact same distance (3 meters) away from you; and each speaker is emitting the same sound (each wave length is the same length) Most importantly, the speakers' diaphragms are vibrating in sync (moving outward and inward together). Since the distance from the speakers to you is the same, the condensations of the wave coming from one speaker are always meeting the condensations from the other at the same time. As a result, the rarefactions are also always meeting rarefactions.

20. Constructive and Destructive Interference of Sound Waves The combined pattern of the waves is the sum of the individual wave patterns. So, the pressure fluctuations where the two waves meet have twice the amplitude of the individual waves. An increase in amplitude results in a louder sound. When this situation occurs it is said to be "exactly in phase" and to exhibit "constructive interference".

21. Constructive and Destructive Interference of Sound Waves

22. Constructive and Destructive Interference of Sound Waves if we change one of the variables, the resulting sound is nearly the opposite of what it was. If we move one of the speakers 1/2 of the wavelength further away. This movement causes the condensations from one speaker to meet the rarefactions from the other sound wave and vice versa. The result is a cancellation of the two waves. The rarefactions from one wave are offset by the condensation from the other wave producing constant air pressure. A constant air pressure means that you can hear no sound coming from the speakers. This is called "destructive interference" where two waves are "exactly out of phase".

23. Beats What are beats?

24. Beats Now that we know what happens when two sound waves with the same frequency overlap. What happens when two sound waves with different frequencies overlap?

25. Beats Two instrument tuners are sitting beside each other, one is emitting a sound whose frequency is 440 Hz and the other is emitting a sound whose frequency is 438 Hz. If the two tuners (which have the same amplitude) are turned on at the same time, you will not hear a constant sound. Why is that?

26. Beats Instead, the loudness of the combined sound rises and falls. Whenever a condensation meets a condensation or a rarefaction meets a rarefaction, there is constructive interference and the amplitude increases. Whenever a condensation meets a rarefaction and vice versa, there is destructive interference, and you can hear nothing. These periodic variations in loudness are called beats.

27. Beats In this situation you will hear the loudness rise and fall 2 times per second because 440-438=2. So, there is a beat frequency of 2 Hz. Musicians listen for beats to hear if their instruments are out of tune. The musician will listen to a tuner that has the correct sound and plays the note on his instrument. If the musician can hear beats, then he knows that the instrument is out of tune. When the beats disappear, the musician knows the instrument is in tune.

28. Beats

29. Sound

30. Sound

31. Sound In the last image we can see the almost sample-by-sample movement of the waveform (we'll learn later what samples are). You can see, however, that sound is pretty much a symmetrical type of affair (compression and rarefaction) The above graphs/pictures/charts of sounds are often called functions. �

32. Sound as a function Sound, can be described as a function. And it's this that lies at the heart of things like compact disc players, cellular phones, and even radio broadcasts. Mathematicians take in numbers as raw material and from this input, produce another number, which we'll call the output. There are lots of different kinds of functions. Sometimes, our functions operate by some easily specified rule, like squaring. When a number is input into the squaring function, the output is the number squared, so the input of 2 produces an output of 4, the input 3 produces an output of 9, and so on.

33. Amplitude, Pressure In the graphs of sound waves shown earlier, time was represented on the x-axis, amplitude on the y-axis. As a function, time is the input, amplitude is the output. One way to think about sound is as a sequence of time varying amplitudes, or pressures, or more succinctly, as a function of time. The amplitude (y-) axis of pictures of sound represents the amount of air compression (above zero) or rarefation (below zero) caused by a moving object, like vocal chords. Note that zero is the "rest" position, or pressure equilibrium (silence). Looking at the changes in amplitude over time gives a good idea of the amplitude shape of the soundwave.

34. Amplitude, Pressure This amplitude shape might correspond closely to a number of things, including: �the actual vibration of the object the changes in pressure of the air, or water, or some other medium and even, and perhaps most importantly, the deformation (in or out) of the eardrum.

35. Amplitude, Pressure The picture of a sound wave, as amplitudes in time, is a nice visual metaphor for the idea of sound as a continuous sequence of pressure variations. When we talk about computers, this just becomes a picture of a list of numbers plotted against some variable (again, time).

36. Frequency Amplitude is just one mathematical, or acoustical characteristic of sound, just as loudness is only one of the perceptual characteristics of sounds. But sounds aren't just loud and soft... � People often describe sounds as being "high" or "low". A bird tweeting may sound "high", or tuba may sound "low". This is one of our fundamental verbal means of describing sounds. �

37. Frequency But what are we really saying? It turns out that there's a fundamental characteristic of these graphs of pressure in time, mainly if there is a repeating pattern and how fast it repeats. This is frequency! � When we say that the tuba sounds are low and the bird sounds are high, what we are really talking about is some result of the frequency of these particular sounds How fast some pattern in their picture repeats. In terms of waveforms, we can state that the rate at which the air pressure fluctuates (moves in and out) is the frequency of the sound wave.

38. How our Ears Work The ear is a complex mechanism that tries to make sense out of these arbitrary functions of pressure in time, and sends them to the brain. � Our eardrums, like microphones and speakers, are in a sense transducers � they turn one form of information or energy into another.

39. How our Ears Work When soundwaves reach our ears they vibrate our eardrums, transferring the sound energy through the middle to the inner ear, to a snail-shaped organ called the cochlea. The cochlea is filled with fluid and is bisected by an elastic, hair cell-covered partition called the basilar membrane. When sound energy reaches the cochlea, it produces fluid waves that form a series of peaks in the basilar membrane, the position and size of which depend on the frequency content of the sound.

40. How our Ears Work Different sections of the basilar membrane resonate (form peaks) at different frequencies: high frequencies cause peaks towards the front of the cochlea. low frequencies cause peaks towards the back.

41. How our Ears Work These peaks match up with and excite certain hair cells, which send nerve impulses to the brain via the auditory nerve. The brain interprets these signals as "sound� In short, the cochlea transforms sounds from their physical, time domain (amplitude v. time) form, to the frequency domain (amplitude v. frequency) form which our brains can understand.

42. How our Ears Work

43. Describing Sound How do you describe sound? Sound can be, and is, described in many ways. There are a lot of different words for sounds, different ways of speaking about them. To manipulate digital signals with a computer it is useful to have access to a different sort of description, So instead, we need to ask and answer the following kinds of questions:..

44. Describing Sound How loud is it? What is its pitch? What is its spectrum? What frequencies are present ? How loud are the frequencies? How does the sound change over time? Where is the sound coming from? What's a good guess as to the characteristics of the physical object that made the sound?

45. Amplitude and Loudness Amplitude and frequency are not independent they both contribute to our perception of loudness � that is, we use both of them together. But to describe what we call loudness, we need to first understand something about amplitude and another related quantity called intensity.

46. Amplitude and Loudness

47. Amplitude and Loudness The chart gives some sense of the way that the terminology for sound varies depending on whether we talk about direct physical measures (frequency, amplitude), or cognitive ones (pitch, loudness).

48. Amplitude and Loudness If one listens to a pure sine wave starting at 20 Hz (that�s cycles per second or the lowest possible sound the healthy human ear can perceive) and rising gradually to 20 kHz (that 20,000 cycles per second and the highest possible sound we can hear) over the course of 3 seconds and with no change in amplitude, one perceives certain areas as being louder than others. Why is this?

49. Amplitude and Loudness The amplitude of the sine wave does not change, but the perceived loudness changes as it moves through areas of greater frequency. In other words, how loud we hear something is mostly a result of amplitude, but also a result of frequency.

50. Digital Audio sampling and quantization

51. Digital Audio 1 sampling and quantization Analog sound versus digital sound compares the two ways in which sound is recorded and stored. Actual sound waves consist of continuous variations in air pressure. Electronic representations of these signals can be recorded in either digital or analog formats

52. Analog recording An analog recording is one where the original sound signal is modulated onto another physical medium or substrate such as the groove of a gramophone disc or the surface of a magnetic tape. A physical quality in the medium (e.g., the intensity of the magnetic field or the path of a record groove) is directly related, or analogous, to the physical properties of the original sound (e.g., the amplitude, phase, etc.)

53. Digital Representation of Sound Time series functions are examples of "continuous" functions. What we mean by this is that at any instant of time, the functions take on a value. They might also be called "analogue" functions.

54. Digital Representation of Sound At every instance in time, we could write down a number that is the value of the function at that instant whether it be how much your eardrum has been displaced or what the current temperature is. This is an infinite list of numbers (any one of which may have an infinite expansion, like = 3.1417...) and no matter how big your storage capacity is, you're going to have a pretty tough time fitting an infinite collection of numbers in it

55. Digital Representation of Sound How can we represent sound as a finite collection of numbers that can be stored efficiently, in a finite amount of space, on your computer, and played back, and manipulated at will. In short, how do we represent sound digitally?

56. Digital Representation of Sound computers basically store a list of numbers (which can then be thought of as a long list of 0s and 1s). When we are storing sound we have to come up with a finite list of numbers which does a good job of representing our continuous function. We do this by taking samples of the original function, at every few instants (of some predetermined rate, called the sampling rate) recording the value of the function. For example, maybe we only record a sample of the wheather every 5 minutes, but for sounds we need to go a lot faster, and we use a special device which grabs instantaneous amplitudes at rapid, audio rates (called an Analog to Digital converter, or ADC).

57. Digital Representation of Sound A continuous function is also called an analog function, we have to convert analog functions to lists of samples, or digital functions, the fundamental way that computers store information. We store these functions as lists of numbers in computer memory, and as we read through them we are basically creating a discrete function of time of individual amplitude values.

58. Digital Representation of Sound

59. Digital Representation of Sound Analog to digital (A>D) and digital to analog conversion (D>A). In A>D, continuous functions (air pressures, sound waves, voltages) are sampled, and stored as numerical values. In D>A, these numerical are interpolated by the converter to force some continuous system (such as amplifiers, speakers, and subsequently, the air and our ears) into a continuous vibration. Interpolation just means smoothly going between the discrete numerical values.

60. Digital Representation of Sound When a sound, image, or even a temperature reading is recorded digitally, we numerically represent it by storing information about it. A digital sound recording is just a numerical representation of a sound.

61. Digital Representation of Sound To convert sounds between analog and digital world of the computer, we use a device called an Analog to Digital Converter (ADC). A Digital to Analog Converter (DAC) is used to convert these numbers back to sound (or to make the numbers usable by an analog device, like a loudspeaker). An ADC takes smooth functions and returns a list of discrete values. A DAC takes a list of discrete values and returns a smooth, continuous function.

62. Analogue v. Digital The distinction between analogue and digital information is fundamental to the realm of computer music. While an analogue signal is continuous and theoretically made up of an infinite series of points, a digital signal is made up of a finite series of points or snapshots. We'll call these snapshots samples. The rate at which we obtain these samples is called the sampling rate.

63. Analogue v. Digital The faster we sample, the better chance we have of getting an accurate picture of the entire continuous path along which a wave is moving. That�s the main distinction between analog and digital representations of information: analog information is continuous, while digital information is not.

64. Analogue v. Digital

65. Analogue v. Digital The analog waveform has smooth and continuous changes. The digital version of the same waveform has a stairstep type look. The black squares are the actual samples taken by the computer. The "staircasing" effect is an inevitable result of converting an analog signal to digital form. All that the computer knows about are the discrete points marked by the black squares. There is nothing in between those points. The analog waveform is smooth, while the digital version is chunky. This "chunkiness" is called quantization..

66. Quantization Quantization is an artifact of the digital recording process, and illustrates how digitally recorded waveforms are only approximations of analog sources. They will always be approximations, in some sense, since it is theoretically impossible to store truly continuous data digitally. By increasing the number of samples taken each second (sample rate), as well as the accuracy of those samples (resolution), an extremely accurate recording can be made.

67. Quantization We can prove mathematically that we can get so accurate that, theoretically, there is no difference between the analog waveform and its digital representation, at least to our ears.

68. Sampling Theory We know that we need to sample a continuous waveform to represent it digitally. We also know that the faster we sample it, the better , but how often do we need to sample a waveform in order to record a good representation of it? The answer to this question is given by the Nyquist Sampling Theorem,

69. The Nyquist Sampling Theorem The Nyquist Sampling Theorem states that to represent a signal the sampling rate needs to be at least twice the highest frequency contained in the sound of the signal. �

70. Sampling Theory For example, to record a sine wave which is 8000 Hz we would need to sample the sound at a rate of 16000 Hz (16kHz) in order to reproduce the sound. That is, we would need to sample 16000 times a second.

71. The Nyquist Sampling Theorem It's a good idea to remember that since the human ear only responds to sounds up to about 20,000 Hz. We need to sample sounds at least 40,000 times a second, or at rate of 40,000 Hz, to represent these sounds

72. What happens when we undersample?

73. Undersampeiling

74. Under sampling We take samples (black dots) of a sinewave (the one with the shorter wavelength) at a certain interval (the sample rate). If the sinewave is changing too quickly (its frequency is too high) then we can't grab enough information to reconstruct the waveform from our samples. The result is that the high frequency waveform masquerades as a lower frequency waveform, or that the higher frequency is aliased to a lower frequency.

75. Under sampling

76. Aliasing � The most common standard sampling rate for digital audio (the one used for CDs) is 44.1kHz, giving us a Nyquist Frequency (defined as half the sampling rate) of 22.05kHz. If we use lower sampling rates, for example, 20kHz, we can�t represent a sound whose frequency is above 10KHz. In fact, if we try, we�ll get usually undesirable artifacts, called foldover or aliasing, in the signal. �

77. Aliasing In other words: if the sinewave is changing too quickly, we will get the same set of samples that we would have obtained had we been taking samples from a sinewave of lower frequency! As we said before, the effect of this is that the higher frequency contributions now act as impostors of lower frequency information. The effect of this is that there are extra, unanticipated and new low frequency contributions to the sound. Sometimes we can use this in interesting ways, and other times it just messes up the original sound. � So in a sense, these impostors are aliases for the low frequencies, and we say that the result of our undersampling is an aliased waveform at a lower frequency.