Re: how does a gramaphone horn amplify sound?

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