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**Week 1: *Fundamentals of Sound *Sound Levels and the Decibel**

Acoustics Week 1: *Fundamentals of Sound *Sound Levels and the Decibel

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Fundamentals of Sound Sound can be defined as a wave motion in air or other elastic media (stimulus) or as that excitation of the hearing mechanism that results in the perception of sound (sensation). Frequency is a characteristic of periodic waves measured in hertz (cycles per second), readable on an oscilloscope or frequency counter. The ear perceives a different pitch for a quiet tone than a loud one. The pitch of a low-frequency tone goes down, while the pitch of a high-frequency tone goes up as intensity increases. We cannot equate frequency and pitch, but they are analogous. The same situation exists between intensity and loudness. The relationship between the two is not linear. Similarly, the relationship between waveform (or spectrum) and perceived quality (or timbre) is complicated by the functioning of the hearing mechanism.

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Sine Waves The sine wave is a basic waveform closely related to simple harmonic motion. Vibration or oscillation is possible because of the elasticity of the spring and the inertia of the weight. Elasticity and inertia are two things all media must possess to be capable of conducting sound.

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Sine Wave Language The easiest value to read is the peak-to-peak value (of voltage, current, sound pressure, etc.). If the wave is symmetrical, the peak-to-peak value is twice the peak value. Another common way to measure the sine wave is using RMS (Root Mean Square) values. The other two scales of measurement used are peak and average. The picture to the right shows four different ways that a sine wave’s amplitude can be measured and the equations used for conversions.

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Propagation of Sound If an air particle is displaced from its original position, elastic forces of the air tend to restore it to its original position. Because of the inertia of the particle, it overshoots the resting position. Sound is readily conducted in gases, liquids, and solids such as air, water, steel, etc., which are all elastic media. Without a medium, sound cannot be propagated. Outer space is an almost perfect vacuum; no sound can be conducted due to the absence of air. Particles of air propagating a sound wave do not move far from their undisplaced positions.

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**Forms of Particle Motion**

There is more than a million molecules in a cubic inch of air. The molecules crowded together represent areas of compression and the sparse areas represent rarefactions.

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Sound in Free Space The intensity of sound decreases as the distance to the source is increased. Doubling the distance reduces the intensity to 1/4 the initial value, tripling the distance yields 1/9, increasing the distance four times yields 1/16 the initial intensity. The inverse square law states that the intensity of sound in a free field is inversely proportional to the square of the distance from the source.

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**Wavelength, Period, & Frequency**

The wavelength is the distance a wave travels in the time it takes to complete one cycle. The period is the time it takes a wave to complete one cycle. The frequency is the number of cycles per second (Hertz).

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**Wavelength and Frequency**

Formulas for calculating wavelength and frequency. The speed of sound in air is about 1,130 feet per second at normal temperature and pressure. Two graphical approaches for an easy solution to the above equations.

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Complex Waves The sine wave with the lowest frequency (f1) is called the fundamental, the one with twice the frequency (f2) is called the second harmonic, and the one three times the frequency (f3) is the third harmonic. Harmonics are whole number multiples of the fundamental frequency.

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Phase Phase is the time relationship between waveforms. Each waveform is lagging the previous by 90 degrees.

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**Combinations of Waveforms**

Combinations of waveforms that are not in phase. The difference in waveshapes is due entirely to the shifting of the phase of the harmonics with respect to the fundamental.

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Harmonics and Octaves

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Spectrum The audio or frequency spectrum of the human ear is about 20 Hz to 20 kHz. The spectrum tells how the energy of the signal is distributed in frequency. For the ideal sine wave, all the energy is concentrated at one frequency. All other types of waveforms have more than one frequency present. The diagram shows various types of waveforms and their harmonic content. The sine, triangle, and square waves are known as period waves due to their cyclic pattern.

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**Sound Levels and the Decibel**

Levels in decibels make it easy to handle the extremely wide range of sensitivity in human hearing. The threshold of hearing matches the ultimate lower limit of perceptible sound in air. A level in decibels is a convenient way of handling the billion fold range of sound pressures to which the ear is sensitive without getting bogged down in a long string of zeros.

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Ratios vs. Difference Ratios of pressure seem to describe loudness changes better than difference in pressure. Ernst Weber (1834), Gustaf Fechner (1860), Hermann von Helmholtz (1873), and other researchers pointed out the importance of ratios. Ratios of stimuli come closer to matching up with human perception than do differences of stimuli. Ratios of powers, intensities, sound pressure, voltage, or anything else are unitless. This is important because logarithms can be taken only of unitless numbers.

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**Here are three different ways that numbers can be expressed**

Handling Numbers Here are three different ways that numbers can be expressed

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**Handling Numbers, cont’d.**

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**The study of acoustics requires a knowledge of some basic algebra.**

Back to Basics: Math The study of acoustics requires a knowledge of some basic algebra.

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More Math...

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Exponents: A Review

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Exponents, cont’d.

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Exponents, cont’d.

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Exponents, cont’d.

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Exponents, cont’d.

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Exponents, cont’d.

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Exponents, cont’d.

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Exponents, cont’d.

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Practice Problems

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Logarithms

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Logarithms

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Logarithms

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Practice Problems

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Practice Problems Theorem 1

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Decibels A level is a logarithm of a ratio of two like-power quantities. A level in decibels is ten times the logarithm to the base 10 of the ratio of two power quantities.

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**Decibels Sound pressure is proportional to (sound power)2.**

The squaring of the sound power results in the equation SPL = 20 log (p1/p2) instead of 10 log.

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**Sound Pressure Level (SPL)**

Sound pressure is usually the most accessible parameter to measure in acoustics, as voltage is for electronic circuits. For this reason, the Equation (2-3) form is more often encountered in day-to-day technical work.

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Reference Levels A sound level meter is used to read sound pressure levels. A sound level meter reading is a certain sound pressure level, 20 log (p1/p2). For sound in air, the standard reference pressure is 20 μPa (micropascal). The μPa is a very minute sound pressure and corresponds closely to the threshold of human hearing. The equations to the right show how to convert SPL (in dB) to μPa.

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**Log-to-Exponent Conversion**

Here’s another quick look at the math used to convert logs to exponents:

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**Acoustic Reference Levels**

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**Acoustic Reference Levels**

Greek prefixes used for the powers of 10:

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Pascals vs. Decibels Doubling of acoustic power is an increase of 3 dB (10 log 2 = 3.01). Doubling of acoustic pressure is an increase of 6 dB (20 log 2 = 6.02)

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Pascals vs. Decibels

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**Decibel Level Examples**

Here are some examples of decibel level conversions:

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**Decibel Level Examples**

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Examples, cont’d.

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Examples, cont’d.

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Reflected Sound The equations below show how to calculate Reflection Delay and Reflection Level.

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Examples, cont’d.

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Examples, cont’d.

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Examples, cont’d.

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Ratios and Octaves An octave is defined as a 2:1 ratio of two frequencies. The ratio 3:2 is a fifth and 4:3 is a fourth.

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Examples

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Examples, cont’d.

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**Measuring Sound Pressure Level**

Sound level meters usually offer a selection of weighting networks designated A, B, and C. Network selection is based on the general level of sounds to be measured. For SPLs of 20 – 55 dB, use network A. For 55 – 85 dB use network B, and for 85 – 140 dB, use network C.

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**Equal Loudness Contours**

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**The Precedence (Haas) Effect**

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**SPL Changes With Distance**

The Inverse Square Law says that the intensity of sound is inversely proportional to the square of the distance from the point source.

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Inverse Distance Law The Inverse Square Law of intensity becomes the Inverse Distance Law for sound pressure. Sound pressure level is reduced 6 dB for each doubling of distance.

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Critical Distance The Critical Distance is that distance at which the direct sound pressure is equal to the reverberant sound pressure.

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