Tag Archives: Physics

The Acoustic Guitar – Intro

We will consider our study of sound by briefly analyzing the acoustic guitar: an instrument that uses certain physical properties to “amplify” (not really true as no energy is technically added) sound acoustically rather than through electromagnetic induction or piezoelectric means (piezoelectric pickups are common on acoustic-electric guitars however). A guitar can be tuned many ways but standard (E standard) tuning is E-A-D-G-B-E across the six strings from top to bottom, or thickest string to thinnest. The tuning is something that can be changed on the fly, which differentiates the guitar from something like a harp which the tension of the string cannot be adjusted.

Just like the tuning pegs on a guitar can be loosened or tighten to change the tension, the fretting hand can also be used to change the length of the string. Both of these affect the frequency or perceived pitch. In fact, two other qualities of the string (density and thickness) also effect the frequency. These can be related through Mersenne’s rule:

unnamed

As shown, the length and density of the string are inversely proportional to the pitch. The tension is proportional, so tightening the string will tune the string up.  The frequency is inversely proportional to string diameter.

The basic operation of the guitar is that plucking or strumming strings will cause a disturbance in the air, displacing air particles and causing buildups of pressure “nodes” and “antinodes”. This leads to the creation of a longitudinal pressure wave which is perceived by the human ear as sound. However, a string on its own does not displace much air, so the rest of the guitar is needed. The soundboard (top) of the guitar acts as an impedance matching network between the string and air by increasing the surface area of contact with the air. Although this does not amplify the sound since no external energy is applied, it does increase the sound intensity greatly. So in a sense the soundboard (typically made of spruce or a good transmitter of sound) can be thought of as something like an electrical impedance matching transformer. The acoustic guitar also employs acoustic resonance in the soundhole. As with the soundboard, the soundhole also vibrates and tends to resonate at lower frequencies. When the air in the soundhole moves in phase with the strings, sound intensity increases by about 3 dB. So basically, the sound is being coupled from the string to the soundboard, from the soundboard to the soundhole and from both the soundhole and soundboard to the external air. The bridge is the part of the guitar that couples the string vibration to the soundboard. This creates a reasonably loud pressure wave.

In terms of wood, the typical wood used for guitar making has a high stiffness to weight ratio. Spruce has an excellent stiffness to weight ratio, as it has a high modulus of elasticity and moderately low density. Rosewood tends to be used for the back and sides of a guitar. The main thing to note hear is the guitar is made of wood.. because wood does not carry vibrations well. As a result the air echos within the guitar instead, creating a sound that is pleasant to the ear. Another factor, of course is cost.

Strings are comprised of a fundamental frequency as well as harmonics and overtones, which lead to a distinct sound. If you fret a string at the twelfth fret, this is the halfway part of the string. This would be the first overtone with double the frequency. It is important to note that the frets of a guitar taper off as you go towards the bridge. This distance can be calculated since c = fλ is a constant. Each successive note is 1.0595 higher in pitch so the first fret is placed 1.0595 from the bridge. This continues on and on with 1.0595 being raised to a higher and higher power based on what fret is being observed.

Linearity of Quantum Mechanics, Schödinger’s Equation

Linearity

A linear function follows two properties:

  1. If a function is a solution and each variable is scaled or multiplied by the same number, then this is also a solution.
    lin
  2. If two solutions to a function are found, then a third solution of the function is the summation of each variable in the function.
    lin-1.png

 

Given a linear operator L and an unknown variable u, the following properties apply:

linw2

Here is an example:

linw23

 

Linearity as related to Quantum Mechanics

A linear system is far less complicated than a non-linear system.

Maxwell’s equation is linear, for instance. Newton’s equations are not linear.

Consider the example below that explains a particular scenario in which Newton’s equations are shwon to be non-linear:

newton

Quantum Mechanics is linear. Schroedinger’s Equation, devised in 1925 for a dynamic variable, ψ termed the wavefunction.

Captureschr

H_hat is the Hamiltonian, a linear operator as was L in the previous example for Linearity [link]. This means that in Quantum Mechanics, solutions can easily be scaled and added together. Thus, it is proven that Quantum Mechanics is actually simpler than classical mechanics. i is the complex number operator equal to the square root of negative one and h_bar is Planck’s constant.

You might ask what the wavefunction is about, if there are any units, for instance. Interestingly, Schroedinger was not sure what the wavefunction referred to exactly. Max Born later proposed that it had to do with probability.

 

Complex Numbers in Quantum Physics

The complex operator i at the front of the formula notes a significant departure from classical mechanics in which almost all systems are primarily real. In the case of Quantum Mechanics, complex numbers are essential.

Euler’s formula, e^(i*x) = cos(x) + i*sin(x) also proves useful in Quantum Mechanics.

180px-Euler's_formula

Below is some review relevant to Quantum Mechanics:

tt

 

 

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

The Oscillator

The oscillator is an important concept used in a variety of applications. One basic use of an oscillator is that of signal generation.

An oscillator is a system with a gain and positive feedback. The gain must be greater than the loss in the feedback system, so that each time the signal goes through the aplifier in the system, a net gain is produced. The phase shift of a single round trip in the gain-feedback loop must also be a multiple of 2*pi so that a pure signal is repeatedly amplified.

When these conditions are satisfied, the system is unstable and oscillation begins. Eventually, the amplifier gain becomes saturated and rather than a further increase of amplification, the added gain only compensates for system losses.oscillator

Since the system is dependen upon a 2*pi phase shift (the period), an oscillator may be designed for a specific frequency. An oscillator generate a signal from noise by repeatedly amplifying the noise periodically.

Although there are many applications for oscillators, a laser is fundamentally an optical oscillator, an optical signal generator. The maser, which stands for microwave amplification by stiumulated emission of radiation was used before the laser. The saser is an acoustic version of the laser, in which instead of emitting a beam of photons or electromagnetic radiation, an acoustic beam or signal is generated.

The following outlines the operation of a laser; an optical amplifier placed inside of a resonator with a partially transmitting mirror as the output of the system.

laser2

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

 

Doppler Effect

RF/Photonics Lab
November 2019
Michael Benker

 

Doppler Effect

                The Doppler Effect is an important principle in communications, optics, RADAR systems and other systems that deal with the propagation of signals through space. The Doppler Effect can be summarized as the resultant change to a signal’s propagation due to movement either by the source or receiving end of the signal. As the distance between two objects changes, so does the frequency. If, for instance, a signal is being propagated towards an object that is moving towards the source, the returning signal will be of a higher frequency.

260px-Dopplerfrequenz

The Doppler Effect is also applied to rotation of an object in optics and RADAR backscatter scenarios. A rotating target of a radar or optical system will return a set of frequencies which reflect the distances of each point on the target. If one side of the target is moving closer while the other side is moving away, there will be both a higher and lower frequency component to the return signal.

Dopplereffectsourcemovingrightatmach0.7

Interferometry – Introduction

RF/Photonics Lab
Jared Alves
November 2019

Interferometry – Introduction

                Interferometry is a family of techniques in which waves are superimposed for measurement purposes. These waves tend to be radio, sound or optical waves. Various measurements can be obtained using interferometry that portray characteristics of the medium through which the waves propagate or properties of the waves themselves. In terms of optics, two light beams can be split to create an interference pattern when the waves combine (superimpose). This superposition can lead to a diminished wave, an increased wave or a wave completely reduced in amplitude. In an easily realizable physical sense, tossing a stone into a pond creates concentric waves that radiate away from where the stone was tossed. If two stones are thrown near each other, their waves would interfere with each other creating the same effect described previously. Constructive interference is the superposition of waves that results in a larger amplitude whereas destructive interference diminishes the resultant amplitude. Normally, the interference is either partially constructive or partially destructive, unless the waves are perfectly out of phase. The following image displays total constructive and destructive interference.

interferrometry1

A simple way to explain the operation of an interferometer is that it converts a phase difference to an intensity. When two waves of the same frequency are added together, the result depends only on the phase difference between them, as explained previously.

interferrometry2The image above shows a Michelson interferometer which uses two beams of light to measure small displacements, refractive index changes and surface irregularities.  The beams are split using a mirror that is not completely reflective and angled so that one beam is reflected, and one is not. The two beams travel in separate paths which combine to produce interference. Whether the waves combine destructively or constructively depends on distancing between the mirrors. Because the device shows the difference in path lengths, it is a differential device. Generally, one leg length is kept constant for control purposes.