Tag Archives: Acoustics

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.

Acoustics and Sound: The Vocal Apparatus

The study of modulation of signals for wireless transmission can, to some extent, be applied to the human body, In the RF wireless world, a “carrier” signal of a high frequency has a “message” encoded on it (message signal) in some form or fashion. This is then transmitted through a medium (generally air) as a radio frequency electromagnetic wave.

In a similar way, the vocal apparatus of the human body performs a similar function. The lungs forcibly expel air in a steady stream comparable to a carrier wave.  This steady stream gets encoded with information by periodically varying its velocity and pressure into two forms of sound: voiced and unvoiced. Voiced sounds produce vowels and are modulated by the larynx and vocal cords. The vocal chords are bands which have a narrow slit in between them which are flexed in certain ways to produce sounds. The tightening of the cords produces a higher pitch and loosening or relaxing produces a lower pitch. In general, thicker vocal cords will produce deeper voices. The relaxation oscillation produced by this effect converts a steady air flow into a periodic pressure wave. Unvoiced sounds do not use the vocal chords.

The tightness of the vocal cords produces a fundamental frequency which characterizes the tone of voice. In addition, resonating cavities above and below the larynx have certain resonant frequencies which also contribute to the tone of voice through inharmonic frequencies, as these are not necessarily spaced evenly.

Although the lowest frequency is the fundamental and most recognizable tone within the human voice, higher frequencies tend to be of a greater amplitude. Different sounds produced will of course have different spectrum characteristics. This is demonstrated in the subsequent image.

fuck

The “oo” sound appears to contain a prominent 3rd harmonic, for example. In none of these sounds is the fundamental of highest amplitude. The image also shows how varying the position of the tongue as well as the constriction or release of the larynx contributes to the spectrum.

It is interesting to note the difference between male and female voices: male voices contain more harmonic content. This is because lower multiples of the fundamentals are more represented in the male voice and are spaced closed to one another in the frequency domain.

 

Acoustics and Sound: Beating

Beating is a very important concept in musical instruments. This tremolo-like variation in sound intensity occurs when two pure tones of slightly different frequencies are sounded simultaneously. An experiment can be performed with two tuning forks (one regular and one wrapped with tight rubber bands on the prongs) struck at the same time. The resultant sound intensity will vary, rising and falling periodically. When the sound wave arrives at the ear, the waves initially appear out of phase (destructive interference) then appear in phase (constructive interference). The superposition of the two waves is shown to have a pulsation effect. The frequency of the pulsations is determined by the beat frequency, which is the difference in frequency of the pure tones.

beating

The pitch perceived by the ear is the average of the two pure tones. A demonstration can be done with two pipes in an organ (one adjustable pipe). The adjustable pipe is varied within a certain range of the other pipe’s frequency. If the second pure tone is greater than approximately 15 Hz away from the first tone, beating is no longer heard.

The Human Ear

The Human ear is important to the study of acoustics because it is inborn pressure sensor. It is one of the most sensitive parts of the human body and its job is to sense pressure changes in air and convert these to electrical signals that the brain can process as “sound”. Humans can hear roughly between 20 Hz to 20 kHz but this range decreases with age. The human ear can sense sound intensities from 1 W/sqm to 1 trillionth of a W/sqm. What most people intuitively perceive as music loudness, pitch and timbre roughly corresponds to amplitude (or sound intensity, which is proportional to the square of amplitude), frequency and waveform shape. Of course, these are not one to one relationships because if a tone is too high in frequency (ultrasound) or too low (infrasound) it will effect the perceived loudness because it will not be heard at all, for example.

The human ear consists of three main parts: inner ear, middle ear and outer ear. The outer ear consists of the pinna, auditory canal and eardrum. The pinna (the only visible part of the ear) serves as a guide to guide pressure waves into the ear canal. The ear canal is filled with air which is necessary because sound needs a medium such as air to transmit pressure waves. The waves reach the conically shaped eardrum, which vibrates and sends signals to the brain to process.

ear.png

The middle ear consists of several dense bones (ossicles) called the hammer, anvil and stirrup. These are elastically connected and serve to transmit and amplify sound from the outer to inner ear. These bones are necessary because the pressure waves are being transferred to a different medium (air to ear fluid called endolymph) and require an impedance matching network to transmit sound effectively. This is not unlike the soundboard of a guitar (for impedance matching to air) or an electrical impedance matching network design for maximum power transfer from a source to a load.

The inner ear contains the cochlea and the semicircular canals. The cochlea contains thousands of tiny hair cells that are stimulated by the vibrations of sound. The semicircular canals contribute to our sense of balance, but not the sensation of hearing. The inner ear fluid causes the hairs in the cochlea to bend, which are converted to electrical pulses and sent to the brain. These are sent to the auditory nerve and are interpreted as sound.

The following diagram depicts the human ear as a passive electrical circuit using the “impedance analogy”. The eardrum middle ear section is shown to be a transformer to match the outer ear to the middle ear. There could also be another transformer between the middle ear and the cochlea, as stated before. Without going into excruciating detail, it is important to show that the human ear is not all different from an electrical circuit in the sense that it impedance matches and transforms/transduces different forms of energy.

analogear.png

 

Sound – ADSR Envelope

In terms of sound and music, the ADSR envelope describes how sound changes over time. In terms of physics of wave, a general envelope outlines the extreme points (max and mins) of a wave through a smooth curve.

It is obvious to the human ear that when a musical instrument is played, its volume (amplitude) changes over time. For example when a guitar string is plucked, the string vibrates and the initial amplitude is high. After a brief period, the sound amplitude decays. Different musical instruments will have different ADSR envelopes to describe their sound characteristics.

The “Attack” phase of the sound refers to how quickly a sound reaches its maximum amplitude. This is the initial phase of the sound. For most instruments, this period is extremely short (almost instantaneous). The next phase is the “Decay” phase, or the time the note takes to drop to the sustain level. The “sustain” level is generally the longest portion of time, and this is when the amplitude envelope stays relatively constant. The “release” is the period of time the sound takes to go from the sustain level to zero amplitude and is generally short.

ADSR

Electroacoustic Transducers

An electroacoustic transducer converts energy from electrical to mechanical. Transducers in general convert energy from various forms. For example a “piezoelectric” (stress electric) transducer converts a mechanical force into a voltage. These transducers can be used to convert speech or music signals into electrical signals for processing or to serve as measuring instruments for acoustic quantities. A transducer can be modeled as a two port network relating electrical and mechanical properties. All of these values are RMS (effective) values.

Various electrical quantities can be transformed into mechanical quantities:

Voltage<—>Force

Current<—>Velocity

Inductance<—>Mass

Capacitance<—>Inverse of stiffness

Resistance<—>Mechanical Resistance or Damping

Transducers can be reciprocal or nonreciprocal. Crystal or ceramic electroacoustic transducers are considered reciprocal (Transduction coeffecients for electrical and mechanical are equivalent).

Two major types of electroacoustic transducers are the electromagnetic variety (which use the principle of Faraday’s Law of Induction to stimulate charge flow) or electrostatic transducers which store charge on capacitive plates which then vibrate to create changes in pressure. They can be further classified as “active” (not requiring external power) or “passive”. There are resistive, inductive, capacitive or light dependent transducers. Transducers can be characterized by their frequency response or directivity pattern. For example, a microphone with a circular directive pattern would be seen as “omnidirectional” or equal amplification in all directions. The following is the frequency response of a microphone that appears very flat. This means the microphone is very neutral and does not amplify certain frequencies more than others.

freqresponse.jpg

Reflection and Transmission of Sound

Similar to electromagnetics, sound waves that are incident upon a medium with different properties will experience reflection, transmission/refraction or absorption depending on multiple factors. The analysis of the transmission and reflection of sound is greatly simplified when the boundary between media and the incident wave are planar. The amount of transmission and reflection depends on each material’s acoustic or characteristic impedance (r = p*c) and the angle that the incident wave makes with the boundary.

Much like in electromagnetics, transmission and reflection and reflection coefficients can be defined as shown.

transreflect.PNG

The figure shows that the transmission coefficient is the ratio of the transmitted pressure wave to the incident pressure wave. The reflection coefficient is the ratio of the reflected pressure wave to the incident pressure wave. This is similar to the voltage reflection coefficient from transmission line theory.

 

The angle of the sound wave with the boundary can either be “normal” (at a 90 degree angle) or “oblique”. For normal incidence, the problems are greatly simplified, as there is no refracted wave, only a reflected component and a transmitted component. These are described by plane wave equations:

pi

pr

pt

The propagation vector (k) is dependent on the material. The transmitted wave has a different propagation vector value because it has surpassed the boundary into the second medium. This is due to the fact that a material’s characteristic impedance is dependent on the speed of sound.

normalincidence.PNG

In order for boundary conditions to be satisfied, both the normal component of velocity and pressure must be continuous. This means the acoustic pressure on both sides of the boundary must be equal, leaving no net force on the planar boundary separating the fluids. The fluids must also remain in contact, meaning the normal component of the velocity must be continuous.

boundary.PNG

These equations can be used to derive the reflection and transmission coefficients.

It is important to note that the reflection coefficient is always real. When the second medium (the medium the transmitted wave propagates into) has a greater acoustic impedance than the first, the reflection coefficient is positive. This makes sense because when a sound pressure wave comes in contact with a rigid boundary, sound echoing occurs. When r2 is much greater than r1, this defines the rigid boundary condition. When r1 is much greater than r2, the boundary is termed “pressure release” and there is an 180 degree phase shift between incident and reflected wave.

 

Having an incident angle other than 90 degrees complicates the solution process a bit. This is termed “oblique incidence”. The pressure equations become a bit more complex.

oblique

The angles made by an obliquely incident wave are shown below.

angles.PNG

Applying the same continuity of pressure boundary condition from before leads to Snell’s law.

snell.PNG

The critical angle can be defined as

critical.PNG

This angle, as well as the comparison of speeds of the materials determines the bending of the refracted wave.

rules.PNG

Introduction to Acoustic Waveguides/Cavities

Waveguides are of great importance to both electromagnetics (for example, guiding microwaves into the cooking chamber of a microwave oven to cook food) and the world of acoustics. Waveguides are hollow tubes that guide waves by reflecting them. Without waveguides, waves propagate spherically and decay with range. The waveguides restricts the propagation of the wave to one dimension. These devices can be rectangular or circular in shape. A major area of importance when studying waveguides is boundary conditions. For acoustic waveguides, the boundary conditions are governed by the linearized force equation:

force

This equation shows that the spacial derivative (gradient) of the pressure (a scalar field) is proportional to density and acceleration. The equation can be used to show that at a rigid boundary where the pressure is maximum, the velocity must be minimum. Calculus tells us that taking the derivative of a function and setting it to zero (and hence the velocity in this case) will yield maximum values. The opposite is also true for a “pressure release” boundary.

We will first consider the case of a rectangular boundaried cavity. A cavity is similar to a waveguide, however the dimensions of a cavity are comparable to each other whereas a waveguide will have one direction that is much longer than the others (to propagate the waves). Applying boundary conditions, it is apparent that if all boundaries are rigid, only standing waves can be contained within the cavity. A pressure equation can be derived from the boundary conditions through substitution into the wave equation. The wave equation will always be satisfied for any kind of wave, including pressure waves. The following equation is substituted into the wave equation:

pressure1

This results in an equation of three sinusoids in each direction. In this case, the boundary conditions result in cosine (which is maximal at zero (x=0,y=0,z=0)).

pressure2.PNG

Then, from the boundary conditions it must be true that these cosine functions be equal to zero at the boundaries x =Lx, y = Ly, z = Lz.

pressure3

This leads to solving for the cutoff frequency of the cavity. The wave number (k) is defined as w/c. Solving for w leads to

cutoff

For waveguides, it is best for the frequency of propagation to be much higher than the cutoff for a decrease in waveguide dispersion. Frequencies below cutoff produce evanescent waves, or waves that die off without propagating.

Simple Harmonic Oscillator

Any spring that obeys Hooke’s Law is appropriately named a “simple harmonic oscillator”. This law of physics describes the behavior of a mass-spring system disturbed from its equilibrium by pulling or pushing on the mass in such a way that the mass experiences a “restoring force” described by Hooke’s Law,  F = -sx where “F” is the restoring force, “s” is the spring constant (units – Newtons/meter) and “x” is the displacement of the mass. If x is positive, this refers to stretching the spring, and if x is negative this refers to compression. The spring constant determines how easily the spring is deformed (also known as stiffness). The negative sign refers to the fact that the force opposes the force applied (it is a restoring force), which aligns with Newton’s Third Law of Motion.

Another important Newtonian equation is F = ma which states that force is proportional to both an object’s mass and acceleration. If these two equations are brought together, using the relation that acceleration is the second derivative of position, a second order linear differential equation can be formed.

Mechanical-Oscillator

It is seen that the solution to this undamped mechanical system is sinusoidal in nature. This makes intuitive sense, because sinusoids are proportional to their second derivative, meaning if a sine or cosine is plugged into the differential equation, it can be shown to be a solution. The system’s “natural frequency” is obtained by equating the potential and kinetic energy of the system. This is intuitively satisfying because the kinetic energy dominates the system at frequencies below the resonance/undamped natural frequency and above this frequency, potential energy dominates.

For damped system, the equation becomes slightly more complex as a new force must be considered (the product of mechanical resistance and acceleration). This is a much more realistic approach because in a real mass spring system, air creates a frictional force effect on the spring as it oscillates, causing the oscillations to die out as mechanical/motional energy is lost to heat. The solution to this new differential equation contains a decaying exponential term.

damped

Three cases of damping are shown above: critical damping, overdamping and underdamping. These cases depend upon whether the resonance/undamped natural frequency of the system is equal to the real part of gamma (called temporal absorption coefficient) in which case the system would be critically damped. An underdamped system would occur when the absorption coefficient is lower than the resonance frequency and the opposite would be considered an overdamped system. As shown above, critically damped systems have no oscillations because the imaginary part of gamma is zero (no reactance). Underdamped systems experiences oscillations that decay to zero over time. Overdamped systems decay to equilibrium without oscillating, but not as quickly as critically damped systems.