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  • jalves61 11:44 pm on February 19, 2020 Permalink | Reply
    Tags: Acoustics   

    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.

     
  • jalves61 4:15 pm on February 16, 2020 Permalink | Reply
    Tags: Acoustics, ECE551   

    Acoustic and Electromagnetic Waves – Problems (Part I) 

    a1a2a3a4a5

     
  • jalves61 12:00 am on January 13, 2020 Permalink | Reply
    Tags: Acoustics   

    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

     

     
  • jalves61 12:00 am on January 6, 2020 Permalink | Reply
    Tags: Acoustics   

    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

     
  • jalves61 12:00 am on December 30, 2019 Permalink | Reply
    Tags: Acoustics   

    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

     
  • jalves61 12:00 am on December 25, 2019 Permalink | Reply
    Tags: Acoustics   

    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

     
  • jalves61 11:02 pm on December 23, 2019 Permalink | Reply
    Tags: Acoustics   

    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.

     
  • jalves61 12:00 am on December 20, 2019 Permalink | Reply
    Tags: Acoustics   

    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.

     
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