|MadSci Network: Physics
Dear Sunil: Thanks for your question. Lets begin with some definitions. In http://www.britannica.com/eb/article? eu=117553&tocid=0&query=acoustics we find: Acoustics: the science concerned with the production, control, transmission, reception, and effects of sound. The term is derived from the Greek akoustos, meaning “hearing.” Beginning with its origins in the study of mechanical vibrations and the radiation of these vibrations through mechanical waves, acoustics has had important applications in almost every area of life. It has been fundamental to many developments in the arts—some of which, especially in the area of musical scales and instruments, took place after long experimentation by artists and were only much later explained as theory by scientists. For example, much of what is now known about architectural acoustics was actually learned by trial and error over centuries of experience and was only recently formalized into a science. Additionally, I found: http://www.britannica.com/eb/article ?eu=117553&tocid=64047&query=acoustics The origin of the science of acoustics is generally attributed to the Greek philosopher Pythagorus (6th century BC), whose experiments on the properties of vibrating strings that produce pleasing musical intervals were of such merit that they led to a tuning system that bears his name. Aristotle (4th century BC) correctly suggested that a sound wave propagates in air through motion of the air—a hypothesis based more on philosophy than on experimental physics; however, he also incorrectly suggested that high frequencies propagate faster than low frequencies—an error that persisted for many centuries. Vitruvius, a Roman architectural engineer of the 1st century BC, determined the correct mechanism for the transmission of sound waves, and he contributed substantially to the acoustic design of theatres. In the 6th century AD, the Roman philosopher Boethius documented several ideas relating science to music, including a suggestion that the human perception of pitch is related to the physical property of frequency. The modern study of waves and acoustics is said to have originated with Galileo Galilei (1564–1642), who elevated to the level of science the study of vibrations and the correlation between pitch and frequency of the sound source. His interest in sound was inspired in part by his father, who was a mathematician, musician, and composer of some repute. Following Galileo's foundation work, progress in acoustics came relatively rapidly. The French mathematician Marin Mersenne studied the vibration of stretched strings; the results of these studies were summarized in the three Mersenne's laws. Mersenne's Harmonicorum Libri (1636) provided the basis for modern musical acoustics. Later in the century Robert Hooke, an English physicist, first produced a sound wave of known frequency, using a rotating cog wheel as a measuring device. Further developed in the 19th century by the French physicist Félix Savart, and now commonly called Savart's disk, this device is often used today for demonstrations during physics lectures. In the late 17th and early 18th centuries, detailed studies of the relationship between frequency and pitch and of waves in stretched strings were carried out by the French physicist Joseph Sauveur, who provided a legacy of acoustic terms used to this day and first suggested the name acoustics for the study of sound. And: http://www.britannica.com/eb/article ?eu=117553&tocid=64049&query=acoustics Simultaneous with these early studies in acoustics, theoreticians were developing the mathematical theory of waves required for the development of modern physics, including acoustics. In the early 18th century, the English mathematician Brook Taylor developed a mathematical theory of vibrating strings that agreed with previous experimental observations, but he was not able to deal with vibrating systems in general without the proper mathematical base. This was provided by Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany, who, in pursuing other interests, independently developed the theory of calculus, which in turn allowed the derivation of the general wave equation by the French mathematician and scientist Jean Le Rond d'Alembert in the 1740s. The Swiss mathematicians Daniel Bernoulli and Leonhard Euler, as well as the Italian-French mathematician Joseph-Louis Lagrange, further applied the new equations of calculus to waves in strings and in the air. In the 19th century, Siméon- Denis Poisson of France extended these developments to stretched membranes, and the German mathematician Rudolf Friedrich Alfred Clebsch completed Poisson's earlier studies. A German experimental physicist, August Kundt, developed a number of important techniques for investigating properties of sound waves. One of the most important developments in the 19th century involved the theory of vibrating plates. In addition to his work on the speed of sound in metals, Chladni had earlier introduced a technique of observing standing-wave patterns on vibrating plates by sprinkling sand onto the plates—a demonstration commonly used today. Perhaps the most significant step in the theoretical explanation of these vibrations was provided in 1816 by the French mathematician Sophie Germain, whose explanation was of such elegance and sophistication that errors in her treatment of the problem were not recognized until some 35 years later, by the German physicist Gustav Robert Kirchhoff. So far, so good. Lets now talk a little about guitars. In the following link: http://webster .aip.org/radio/html/guitar_physics.html there is a transcript of a radio broadcast. Guitar Physics (SFX:CLASSICAL GUITAR PLAYING) PEOPLE HAVE BEEN PLAYING GUITARS FOR ALMOST 3,000 YEARS. OF COURSE, THE EARLY GUITARS DIDN'T LOOK EXACTLY LIKE THE ONES WE HAVE TODAY SINCE BUILDERS HAVE TINKERED AND TOYED WITH THE GUITAR OVER THE CENTURIES TRYING TO GET THE BEST POSSIBLE SOUND. EVEN TODAY, SAYS PHYSICAL CHEMIST MICHAEL KASHA FROM FLORIDA STATE UNIVERSITY, THERE'S STILL ROOM FOR IMPROVEMENT. Kasha: "When I began to get interested in this which is quite a long time ago, I looked inside with a bycicle mirror and siad it can't be right. . . that violates everything I know about vibrations mechanics." IT'S THE INSIDE OF THE GUITAR THAT MATTERS SAYS KASHA, SINCE THE STRINGS DON'T ACTUALLY MAKE THE SOUNDS. Kasha: "If the strings were mounted on a concrete wall. . . you'd hear nothing." BUT IN A GUITAR, THE STRINGS SET THE WOOD VIBRATING AND IT'S THE WOOD WHICH CREATES THE RICH SOUND THAT EVENTUALLY MAKES ITS WAY OUT OF THE SOUND HOLE--THAT'S THE BIG HOLE IN THE FRONT. DIFFERENT SOUNDS ARE CREATED WHEN THE WOOD VIBRATES AT DIFFERENT FREQUENCIES. AND THOSE FREQUENCIES DEPEND ON THE SHAPE OF THE GUITAR. CONVENTIONAL GUITARS DON'T VIBRATE PERFECTLY AT ALL FREQUENCIES, AND SO KASHA, IN CONJUNCTION WITH GUITAR BUILDER RICHARD SCHNEIDER, HAS RESHAPED THE GUITAR--MAKING PARTS OF IT LONGER AND REPOSITIONING THE SOUNDHOLE TO AN UPPER CORNER. KASHA THINKS HE DESIGNED A BETTER SOUNDING GUITAR Kasha: "It is unorthodox as if someone moved your eye into the upper right hand corner of your head. . . But the good musicians say I like the sound. If they like the sound, that's what counts." The guitar is the most common stringed instrument, and shares many characteristics with other stringed instruments. For example, the overtones potentially available on any stringed instrument are the same. Why, then, does a guitar sound so much different from, say, a violin? The answer lies in which overtones are emphasized in a particular instrument, due to the shape and materials in the resonator (body), strings, how it's played, and other factors. In the course of studying the overtones, or harmonics of a string fixed at both ends, we will uncover the overtone series for strings, which is the basis of Western harmony. Waves on a String A guitar string is a common example of a string fixed at both ends which is elastic and can vibrate. The vibrations of such a string are called standing waves, and they satisfy the relationship between wavelength and frequency that comes from the definition of waves: v = f, where v is the speed of the wave, f is the frequency (measured in cycles/second or Hertz, Hz) and is the wavelength. The speed v of waves on a string depends on the string tension T and linear mass density (mass/length) µ, measured in kg/m. Waves travel faster on a tighter string and the frequency is therefore higher for a given wavelength. On the other hand, waves travel slower on a more massive string and the frequency is therefore lower for a given wavelength. The relationship between speed, tension and mass density is a bit difficult to derive, but is a simple formula: v = T/µ Since the fundamental wavelength of a standing wave on a guitar string is twice the distance between the bridge and the fret, all six strings use the same range of wavelengths. To have different pitches (frequencies) of the strings, then, one must have different wave speeds. There are two ways to do this: by having different tension T or by having different mass density µ (or a combination of the two). If one varied pitch only by varying tension, the high strings would be very tight and the low strings would be very loose and it would be very difficult to play. It is much easier to play a guitar if the strings all have roughly the same tension; for this reason, the lower strings have higher mass density, by making them thicker and, for the 3 low strings, wrapping them with wire. From what you have learned so far, and the fact that the strings are a perfect fourth apart in pitch (except between the G and B strings in standard tuning), you can calculate how much µ increases between strings for T to be constant. String Harmonics (Overtones) If a guitar string had only a single frequency vibration on it, it would sound a bit. What makes a guitar or any stringed instrument interesting is the rich variety of harmonics that are present. Any wave that satisfies the condition that it has nodes at the ends of the string can exist on a string. The fundamental, the main pitch you hear, is the lowest tone, and it comes from the string vibrating with one big arc from bottom to top: fundamental (l = /2) The fundamental satisfies the condition l = /2, where l is the length of the freely vibrating portion of the string. The first harmonic or overtone comes from vibration with a node in the center: 1st overtone (l = 2/2) The 1st overtone satisfies the condition l = . Each higher overtone fits an additional half wavelength on the string: 2nd overtone (l = 3/2) 3rd overtone (l = 4/2) 4th overtone (l = 5/2) Since frequency is inversely proportional to wavelength, the frequency difference between overtones is the fundamental frequency. This leads to the overtone series for a string: overtone f/f0 freq/tonic approx interval fundamental 1 1=1.0 tonic 1st 2 1=1.0 tonic 2nd 3 3/2=1.5 perfect 5th 3rd 4 1=1.0 tonic 4th 5 5/4=1.25 major 3rd 5th 6 6/4=1.5 perf 5th 6th 7 7/4=1.75 dominant 7th 7th 8 1=1.0 tonic 8th 9 9/8=1.125 major 2nd 9th 10 10/8=1.25 major 3rd 10th 11 11/8=1.375 between 4th and dim 5th 11th 12 12/8=1.5 perfect 5th 12th 13 13/8=1.625 between aug 5th and maj 6th Most of the first 12 overtones fall very close to tones of the Western musical scale, and one can argue that this is not coincidence: it is natural to use a musical scale which incorporates the overtones of stringed instruments. The equal-tempered scale has 12 intervals (half- steps) making up an octave (factor of two). The ratio, r, of frequencies for a half-step therefore satisfies r12=2, which means r=1.0595. The top row shows the intervals of the major scale. The equal-tempered scale and overtone series don't match perfectly, of course, but the difference between, say, a major 3rd of the equal-tempered scale (1.2599) and the 4th overtone (1.2500) is pretty hard to hear. Guitar Overtones The thing that makes a guitar note "guitarry" is the overtone content and how the note rises and decays in time. This varies with how you play it, such as with a pick vs. a finger, or near the bridge vs. in the middle. (This, of course, isn't counting all the electronic methods for emphasizing different overtones such as the bass/treble control on electric guitars.) You've probably noticed that the frets on a guitar get closer together towards the bridge. From the the fact that each successive note is r=1.0595 higher in pitch, and the fact that v=f=constant on a given string, we can figure out the fret spacing. Let's say the open string length is l. Then the first fret must be placed a distance l/1.0595 from the bridge, the second fret a distance l/1.0595˛ from the bridge, and so on. The twelfth fret, which makes an octave, is at a distance l/1.059512=l/2 from the bridge. The diagram below shows the fret positions (as does the photo at the top of this page for that matter!). Additional information regarding guitars is in the excellent page: http://www.gmi.edu/~ drussell/guitars/index.html And more about sound could be find in: http://online .anu.edu.au/ITA/ACAT/drw/PPofM/INDEX.html Sound amplification is explained in: http://hyperphysics.phy- astr.gsu.edu/hbase/ph4060/p406ex6.html There are some nice sound demos in: http://www.physics.umd.edu/deptinfo/facilities/lecdem/h2-01.htm But, why the guitar sounds and the cone amplifies?. Here is the answer: http://www.monroeinstitute.org/voyagers/voyages/hsj-1997-fall- hssoundresonance-hayduk.html The simplest definition of resonance is the universal striving of objects to vibrate at the same rate. One of the best examples is a tuning fork which--once struck and put near a similar tuning fork--will cause the second fork to vibrate at the same rate. Resonance is also the principle involved in the amplification and elongation of the sound/vibrations experienced when a guitar string is plucked and the wood of the hollow vibrates in tune with that string. We instinctively grasp the idea of resonance and demonstrate our understanding by using terms such as in sync or out of sync and getting good or bad vibes about a person or situation. The phenomenon of resonance, while seemingly easy to understand, still holds many mysteries within its entraining capabilities. I hope this help Regards Jaime Valencia
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