The Physics of Music

Sound is a Mechanical Wave

Sound and music are parts of our everyday sensory experience. Just as humans have eyes for the detection of light and color, so we are equipped with ears for the detection of sound. We seldom take the time to ponder the characteristics and behaviors of sound and the mechanisms by which sounds are produced, propagated, and detected. The basis for an understanding of sound, music and hearing is the physics of waves. Sound is a wave that is created by vibrating objects and propagated through a medium from one location to another. In this unit, we will investigate the nature, properties and behaviors of sound waves and apply basic wave principles towards an understanding of music.
As discussed in the previous unit of The Physics Classroom Tutorial, a wave can be described as a disturbance that travels through a medium, transporting energy from one location to another location. The medium is simply the material through which the disturbance is moving; it can be thought of as a series of interacting particles. The example of a slinky wave is often used to illustrate the nature of a wave. A disturbance is typically created within the slinky by the back and forth movement of the first coil of the slinky. The first coil becomes disturbed and begins to push or pull on the second coil. This push or pull on the second coil will displace the second coil from its equilibrium position. As the second coil becomes displaced, it begins to push or pull on the third coil; the push or pull on the third coil displaces it from its equilibrium position. As the third coil becomes displaced, it begins to push or pull on the fourth coil. This process continues in consecutive fashion, with each individual particle acting to displace the adjacent particle. Subsequently the disturbance travels through the slinky. As the disturbance moves from coil to coil, the energy that was originally introduced into the first coil is transported along the medium from one location to another
The Human Ear
Understanding how humans hear is a complex subject involving the fields of physiology, psychology and acoustics. In this part of Lesson 2, we will focus on the acoustics (the branch of physics pertaining to sound) of hearing. We will attempt to understand how the human ear serves as an astounding transducer, converting sound energy to mechanical energy to a nerve impulse that is transmitted to the brain. The ear's ability to do this allows us to perceive the pitch of sounds by detection of the wave's frequencies, the loudness of sound by detection of the wave's amplitude and the timbre of the sound by the detection of the various frequencies that make up a complex sound wave.
The ear consists of three basic parts - the outer ear, the middle ear, and the inner ear. Each part of the ear serves a specific purpose in the task of detecting and interpreting sound. The outer ear serves to collect and channel sound to the middle ear. The middle ear serves to transform the energy of a sound wave into the internal vibrations of the bone structure of the middle ear and ultimately transform these vibrations into a compressional wave in the inner ear. The inner ear serves to transform the energy of a compressional wave within the inner ear fluid into nerve impulses that can be transmitted to the brain. The three parts of the ear are shown below.
The outer ear consists of an earflap and an approximately 2-cm long ear canal. The earflap provides protection for the middle ear in order to prevent damage to the eardrum. The outer ear also channels sound waves that reach the ear through the ear canal to the eardrum of the middle ear. Because of the length of the ear canal, it is capable of amplifying sounds with frequencies of approximately 3000 Hz. As sound travels through the outer ear, the sound is still in the form of a pressure wave, with an alternating pattern of high and low pressure regions. It is not until the sound reaches the eardrum at the interface of the outer and the middle ear that the energy of the mechanical wavebecomes converted into vibrations of the inner bone structure of the ear.
The middle ear is an air-filled cavity that consists of an eardrum and three tiny, interconnected bones - the hammer, anvil, and stirrup. The eardrum is a very durable and tightly stretched membrane that vibrates as the incoming pressure waves reach it. As shown below, a compression forces the eardrum inward and a rarefaction forces the eardrum outward, thus vibrating the eardrum at the same frequency of the sound wave.
Being connected to the hammer, the movements of the eardrum will set the hammer, anvil, and stirrup into motion at the same frequency of the sound wave. The stirrup is connected to the inner ear; and thus the vibrations of the stirrup are transmitted to the fluid of the inner ear and create a compression wave within the fluid. The three tiny bones of the middle ear act as levers to amplify the vibrations of the sound wave. Due to a mechanical advantage, the displacements of the stirrup are greater than that of the hammer. Furthermore, since the pressure wave striking the large area of the eardrum is concentrated into the smaller area of the stirrup, the force of the vibrating stirrup is nearly 15 times larger than that of the eardrum. This feature enhances our ability of hear the faintest of sounds. The middle ear is an air-filled cavity that is connected by the Eustachian tube to the mouth. This connection allows for the equalization of pressure within the air-filled cavities of the ear. When this tube becomes clogged during a cold, the ear cavity is unable to equalize its pressure; this will often lead to earaches and other pains.
The inner ear consists of a cochlea, the semicircular canals, and the auditory nerve. The cochlea and the semicircular canals are filled with a water-like fluid. The fluid and nerve cells of the semicircular canals provide no role in the task of hearing; they merely serve as accelerometers for detecting accelerated movements and assisting in the task of maintaining balance. The cochlea is a snail-shaped organ that would stretch to approximately 3 cm. In addition to being filled with fluid, the inner surface of the cochlea is lined with over 20 000 hair-like nerve cells that perform one of the most critical roles in our ability to hear. These nerve cells differ in length by minuscule amounts; they also have different degrees of resiliency to the fluid that passes over them. As a compressional wave moves from the interface between the hammer of the middle ear and the oval window of the inner ear through the cochlea, the small hair-like nerve cells will be set in motion. Each hair cell has a natural sensitivity to a particular frequency of vibration. When the frequency of the compressional wave matches the natural frequency of the nerve cell, that nerve cell will resonate with a larger amplitude of vibration. This increased vibrational amplitude induces the cell to release an electrical impulse that passes along the auditory nerve towards the brain. In a process that is not clearly understood, the brain is capable of interpreting the qualities of the sound upon reception of these electric nerve impulses.

Echo vs. Reverberation
Sound is a mechanical wave which travels through a medium from one location to another. This motion through a medium occurs as one particle of the medium interacts with its neighboring particle, transmitting the mechanical motion and corresponding energy to it. This transport of mechanical energy through a medium by particle interaction is what makes a sound wave a mechanical wave.
As a sound wave reaches the end of its medium, it undergoes certain characteristic behaviors. Whether the end of the medium is marked by a wall, a canyon cliff, or the interface with water, there is likely to be some transmission/refraction, reflection and/or diffraction occurring. Reflection of sound waves off of barriers result in some observable behaviors which you have likely experienced. If you have ever been inside of a large canyon, you have likely observed an echo resulting from the reflection of sound waves off the canyon walls. Suppose you are in a canyon and you give a holler. Shortly after the holler, you would hear the echo of the holler - a faint sound resembling the original sound. This echo results from the reflection of sound off the distant canyon walls and its ultimate return to your ear. If the canyon wall is more than approximately 17 meters away from where you are standing, then the sound wave will take more than 0.1 seconds to reflect and return to you. Since the perception of a sound usually endures in memory for only 0.1 seconds, there will be a small time delay between the perception of the original sound and the perception of the reflected sound. Thus, we call the perception of the reflected sound wave an echo.
A reverberation is quite different than an echo. The distinction between an echo and a reverberation is depicted in the animation below.

A reverberation is perceived when the reflected sound wave reaches your ear in less than 0.1 second after the original sound wave. Since the original sound wave is still held in memory, there is no time delay between the perception of the reflected sound wave and the original sound wave. The two sound waves tend to combine as one very prolonged sound wave. If you have ever sung in the shower (and we know that you have), then you have probably experienced a reverberation. The Pavarotti-like sound which you hear is the result of the reflection of the sounds you create combining with the original sounds. Because the shower walls are typically less than 17 meters away, these reflected sound waves combine with your original sound waves to create a prolonged sound - a reverberation.

Vibration

Vibration is the source of all sound. Vibrating objects push against the air (or other medium- you can substitute water, jello, or whatever) around them, creating little zones of compressed air (or water, or jello). The zone of compressed air pushes against the air around it, which pushes against the air around that, and so on. Between compression pulses the air "springs" out past the pressure where it began, creating a zone of less pressure, or rarefaction. You end up with zones of compression and rarefaction that travel outward from the sound source, one after another at a rate equal to the rate of that source's vibration. These, friends, are sound waves.

Sound waves

Before you learn how sound equipment works it's very important to understand how sound waves work. This knowledge will form the foundation of everything you do in the field of audio.

Sound waves exist as variations of pressure in a medium such as air. They are created by the vibration of an object, which causes the air surrounding it to vibrate. The vibrating air then causes the human eardrum to vibrate, which the brain interprets as sound.

The illustration on the left shows a speaker creating sound waves (click the button to show animation).

Sound waves travel through air in much the same way as water waves travel through water. In fact, since water waves are easy to see and understand, they are often used as an analogy to illustrate how sound waves behave.

It's important to remember that sound waves are compression waves. You can imitate a compression wave by stretching out a slinky (you do have a slinky, don't you?) and flicking your finger against a coil at the end. Sound waves are not like the waves on the ocean or the waves you get by waving a stretched-out rope.

Frequency and pitch

Sound waves have a frequency, which is the number of compression pulses that go past a fixed point in a given amount of time. The frequency of audible sound is measured in hertz, or cycles per second. Sound waves have a wavelength, which is the physical distance between compression pulses. The wavelength is inversely proportional to the frequency. Sound waves also have an amplitude, which is the amount of air (yes, water or jello) that gets moved with each pulse of pressure. Without going far into the physiology of it, you hear sounds when the pulses of compressed air of sound waves excite your eardrum, which excites your inner ear, which sends signals to your brain, which makes you dance, dance, dance! In general, you perceive the frequency of the wave as a particular pitch (see the pitch discussion, above). You perceive the amplitude of the wave as loudness.

Resonance

Take a tuning fork (you do have a tuning fork, don't you?) and whack it on your knee. What do you hear? Unless you hold the tuning fork right next to your ear, you won't hear much of anything. This is because a small tuning fork can't push very much air around. Now take the same tuning fork, whack it on your knee again, and touch the non-forked end to a tabletop or other handy wooden surface. The sound should be a lot louder. This is because the vibrating tuning fork causes the tabletop to vibrate. The tabletop can push much more air around than the fork alone. If you touched the end of the tuning fork to a hollow box or, say, the body of a guitar, the sound would be even louder. This is because the vibrations get transferred to the air inside the box, which vibrates as well. If the dimensions of the inside of the box are a multiple of the wavelength of the sound, some of the sound waves will reinforce each other for even more volume. If a vibration or sound wave can excite another object into vibrating, the second object is said to resonate. This phenomenon is called resonance.

Overtones

The vibrating and resonating parts of musical instruments (and almost everything else that makes sounds) don't produce sound waves of just one frequency. This is because the vibrating body (e.g. string or air column) does not just vibrate as a whole; smaller sections vibrate as well. In the case of musical instruments, these additional frequencies are usually even multiples of the vibration frequency of the whole string, air column, bar, etc. For example, suppose you squeeze your accordion (the most sublime of all musical instruments) and press the key that lets the air out past a reed which, due to certain physical properties, vibrates 440 times per second. The vibrating reed will generate sound waves with a frequency of 440 Hz. (cycles per second), which happens to correspond to the A above Middle C. Because of other physical properties of the reed and the accordion, the instrument will also generate waves with a frequency of 880 Hz. (2 x 440), 1,320 Hz (3 x 440), 1,760 Hz. (4 x 440), etc. These extra frequencies are called overtones. Amazingly enough, when the overtones are close to even multiples of the fundamental frequency, our brains interpret the whole conglomeration of frequencies as a single pitch. Different instruments differ in the relative strengths of the various overtones, and that is what gives the instruments different timbres. This is also what makes your voice sound different from someone else's, even when you sing the exact same pitch. In the case of cymbals, gongs, snare drums, and the other indefinite-pitch percussion instruments, there are so many frequencies and overtones all at the same time that our brains don't pick out a definite pitch. You might notice, though, that the sound of a drum or woodblock can still be "higher" or "lower" than the sound of another.

Most oscillators, from a guitar string to a bell (or even the hydrogen atom or a periodic variable star) will naturally vibrate at a series of distinct frequencies known as normal modes. The lowest normal mode frequency is known as the fundamental frequency, while the higher frequencies are called overtones. Often, when an oscillator is excited by, for example, plucking a guitar string, it will oscillate at several of its modal frequencies at the same time. So when a note is played, this gives the sensation of hearing other frequencies (overtones) above the lowest frequency (the fundamental).

Timbre is the quality that gives the listener the ability to distinguish between the sound of different instruments. The timbre of an instrument is determined by which overtones it emphasizes. That is to say, the relative volumes of these overtones to each other determines the specific "flavor" or "color" of sound of that family of instruments. The intensity of each of these overtones is rarely constant for the duration of a note. Over time, different overtones may decay at different rates, causing the relative intensity of each overtone to rise or fall independent of the overall volume of the sound. A carefully trained ear can hear these changes even in a single note. This is why the timbre of a note may be perceived differently when played staccato or legato.

A driven non-linear oscillator, such as the human voice, a blown wind instrument, or a bowed violin string (but not a struck guitar string or bell) will oscillate in a periodic, non-sinusoidal manner. This generates the impression of sound at integer multiple frequencies of the fundamental known as harmonics. For most string instruments and other long and thin instruments such as a trombone or bassoon, the first few overtones are quite close to integer multiples of the fundamental frequency, producing an approximation to a harmonic series. Thus, in music, overtones are often called harmonics. Depending upon how the string is plucked or bowed, different overtones can be emphasized.

ECHO

Sound is a mechanical wave which travels through a medium from one location to another. This motion through a medium occurs as one particle of the medium interacts with its neighboring particle, transmitting the mechanical motion and corresponding energy to it. This transport of mechanical energy through a medium by particle interaction is what makes a sound wave a mechanical wave.

As a sound wave reaches the end of its medium, it undergoes certain characteristic behaviors. Whether the end of the medium is marked by a wall, a canyon cliff, or the interface with water, there is likely to be some transmission/refraction, reflection and/or diffraction occurring. Reflection of sound waves off of barriers result in some observable behaviors which you have likely experienced. If you have ever been inside of a large canyon, you have likely observed an echo resulting from the reflection of sound waves off the canyon walls. Suppose you are in a canyon and you give a holler. Shortly after the holler, you would hear the echo of the holler - a faint sound resembling the original sound. This echo results from the reflection of sound off the distant canyon walls and its ultimate return to your ear. If the canyon wall is more than approximately 17 meters away from where you are standing, then the sound wave will take more than 0.1 seconds to reflect and return to you. Since the perception of a sound usually endures in memory for only 0.1 seconds, there will be a small time delay between the perception of the original sound and the perception of the reflected sound. Thus, we call the perception of the reflected sound wave an echo.

A reverberation is quite different than an echo. The distinction between an echo and a reverberation is depicted in the animation below.

A reverberation is perceived when the reflected sound wave reaches your ear in less than 0.1 second after the original sound wave. Since the original sound wave is still held in memory, there is no time delay between the perception of the reflected sound wave and the original sound wave. The two sound waves tend to combine as one very prolonged sound wave. If you have ever sung in the shower (and we know that you have), then you have probably experienced a reverberation. The Pavarotti-like sound which you hear is the result of the reflection of the sounds you create combining with the original sounds. Because the shower walls are typically less than 17 meters away, these reflected sound waves combine with your original sound waves to create a prolonged sound - a reverberation.

The Physics of Music

Monday, February 28, 2011

References

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Sunday, February 27, 2011

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GUIDE CARD

Sound is a Mechanical Wave

The Human Ear

Echo vs. Reverberation

Vibration