Acoustics: Sound Intensity and the Decibel

Acoustics: Sound Intensity and Decibels

Key Concepts: Logarithmic Scale

• A logarithmic scale is used to represent numerical data over a wide range of values in a compact manner.

• Unlike a linear scale (e.g., 1, 2, 3, 4), a logarithmic scale compresses large numbers and expands small numbers.

• It helps in making sense of very big numbers by using a special kind of "ruler" where numbers are not evenly spaced.

• Useful for displaying data that spans several orders of magnitude; the decibel scale is an example.

• The distance between 0.1 and 1 is the same as the distance between 1 and 10, or 10 to 100, and so on.

• Marks like 0.1, 1, 10 are known as orders of magnitude.

Logarithmic Function

• The logarithm function starts at one, and as the horizontal axis (x) increases, the vertical axis (y) flattens.

• The curve is steepest near one and gradually flattens out.

• It is the inverse of an exponential function.

Exponents

• Exponents are used to efficiently write very large and very small numbers.

• For example, 1,000 can be written as $10^3$.

• The exponent indicates the number of zeros in the number.

• Example: $4^2 = 4 \times 4 = 16$

• $0.0001$ can be written as $10^{-4}$.

Logarithm Defined

• A logarithm is the inverse operation of exponentiation.

• It answers the question: to what power must a base number be raised to get a particular result?

• The most common logarithm used is with base 10.

• For example, logarithm with base 10 for 1,000 asks: $10^? = 1000$, and the answer is 3.

• If a logarithm doesn't have a specified base, it is assumed to be base 10.

Properties of Sound: Frequency and Intensity

• Two main properties of sound:

  • Frequency: physical property measured in Hertz (cycles per second), perceived as pitch (high, medium, low).

  • Intensity: physical property, perceived as loudness or volume.

• Intensity is measured in decibels (dB).

• Frequency (Hertz) is the inverse of a period.

• Intensity (decibels) relates to amplitude, which is measured in Pascals (1 Pascal = 1 Newton per square meter).

• Intensity relates to power transfer per second, measured in watts per square meter.

The Decibel (dB)

• A "bell" is a unit of measurement for sound intensity that compares the power of two sounds; it's a logarithm of an intensity ratio.

• A decibel is one-tenth of a bell (0.1 bell = 1 decibel).

• The decibel is crucial in audiology to describe the softest sound a person can hear at a certain frequency.

• It's a logarithmic unit representing the relative magnitude of a sound compared to a reference measure of sound.

• The human ear can respond to a wide range of intensities (up to $10^{14}$), so the decibel scale simplifies this range.

Examples of Decibel Levels

• Threshold of hearing: 0 dB

• Normal conversation: 60 dB

• Jackhammer: 130 dB

Audiogram

• An audiogram represents frequency and intensity.

• Low-intensity sounds are displayed at the top, and high-intensity sounds at the bottom.

• Frequency is on the horizontal axis (low frequencies on the left, high frequencies on the right).

• Example: A vacuum cleaner is 50-60 dB at around 3,000 Hertz.

Decibel Formulas

• Bell: $log<em>{10}(\frac{I</em>x}{I<em>r})$, where $I</em>x$ is the intensity of interest and $I_r$ is the reference intensity.

• Decibel (dB): $10 \times log<em>{10}(\frac{I</em>x}{I_r})$

Reference Intensity

• The decibel is ambiguous unless the reference intensity is specified.

• The conventional reference intensity is $10^{-12}$ watts per square meter, which relates to the softest sound humans can detect at 1,000 Hertz.

• This reference intensity creates a pressure of 20 microPascals, also known as the reference pressure.

Decibel Intensity Level (dBIL) and Decibel Sound Pressure Level (dBSPL)

• dBIL is obtained using the formula $10 \times log<em>{10}(\frac{I</em>x}{I_r})$ with the reference intensity of $10^{-12}$ watts per square meter.

• dBSPL is calculated using $20 \times log<em>{10}(\frac{P</em>x}{P<em>r})$ with the reference pressure of 20 microPascals, where $P</em>x$ is the pressure of interest and $P_r$ is the reference pressure.

Examples and Conversions

• When the reference intensity is equal to the intensity of interest, the result is 0 dB.

• A tenfold increase in intensity corresponds to a 10 dB increase.

• Doubling the intensity results in a 3 dB increase.

• Doubling the intensity,$10^{-6}$ corresponds to 63 dBIL.

• Doubling the pressure results in a 6 dB increase.

• Halving the intensity results in a 3 dB decrease.

• Decreasing the pressure, it's a six dB decrease in sound pressure level.

Example Calculations

• If $I_x$ doubles, the increase is $3 dB$. Logarithm of 2 is 0.3. 0.3 multiplies by 10 is 3 dB.

• Logarithm of 10 is 1 because $10^1 = 10$. So, 1 multiplied by 10 is 10.

Example problem: Calculate the increase in dB. Reference intensity of $10^{-12}$. The reference intensity of the sound is increasing 10 times.

• $10 \times log_{10} (\frac{10^{-5}}{10^{-12}})$

• Apply subtraction to exponents: $10 \times log_{10} (10^{-5 -(-12)})$

• $10 \times log_{10} (10^7) = 10*7 = 70 dB IL$

Combining Sound Intensities

• When combining intensities, adding 60 dB + 60 dB will not result in 120 dB.

Equal Sound Intensities

• $dB<em>n = dB</em>i + 10\times log<em>{10}(n)$, where $dBn$ is the final result,$dB_i$ is the intensity of one of the sources, and n is the number of sources.

• Example: Two sources each produce 100 dB SPL. Result = 100 dB + $10<em>log_{10}(2) =100 + 10</em>0.3 = 103 dB SPL$

• Example: Three sources which produce 100 dB SPL, total dB SPL = 100 + $10<em>log_{10}(3) = 100+10</em>0.48 = 104.8 dB SPL$

Exercise 1: 1000 sources that are 80 db.

• $80 + 10<em>log{10}(1000) = 80 + 10 log</em>{10}(10^3) = 80 + 10 * 3 = 80 + 30 = 110 dB SPL$

Exercise 2: log eight times that by 10 then add whatever that number into plus 91.2 . The overall total was a 100.2. db

• Final result: 100.2 dB

Unequal Sound Intensities

• Three steps:

  • Calculate the intensity of each source in watts per square meter.

  • Add these intensities (E).

  • Calculate the dBIL from the final intensity.

  • dB SPL is equal to dB IL.

  • $80 dB = 10 \times log<em>{10}(\frac{I</em>x}{I_r})$

  • $10^8 = 10^{\frac{I<em>x}{I</em>r}}$

  • $I_x = 10^8 \times 10^{-12} = 10^{-4}$.

Calculating dBs With 80 DB and 83 DB Spl.

• 80 dB IL = an intensity of $10^{-4}$

• $83 dB SPL = 2*10^{-4}$, because the double of intensity increase is an additional three dB.

• Combined intensities: $3*10^{-4}$

Apply Formula To Calculate/Convert TO dB

• $10 \times log_{10} (\frac{3*10^{-4}}{10^{-12}})$

• $10 \times log_{10} (3*10^8) = 84.8 dB IL$

• This total from these two sources produces what final result?

• It's also tricky because this equation, for unequal, can expand from what can, at times, be 2 or 3, up to 16 intensities that you need to combine. As shown, it may not work and be doable based on such quick methods, but to instead make all of them to what is intensity is, and go from there.

• You can do such if fully remembering and knowing what all levels are, or at times when or if quick estimations are needed.

Another Problem

• Total SPL is based on two sources (70 db + 80 db).

  • Follow the example in one of the slide.

  • First, find their respective, equivalent db, which we can see (as provided in one table) that each is equivalent there for both.

  • To calculate and sum all of them properly (when not using all the fast math tricks), we need the actual intensities.

Key Equations and Values

• 70 dBIL is equal to an intensity of $10^{-5}$.

• 80 dBIL is equal to an intensity of $10^{-4}$.

So Apply Formula Again:

• $\frac{1.1* 10^{-4}}{10^{-12}} = 1.1 * 10^8$.

• $ {10 * log_{10} (1.1 {10^8}) = 80.4 db IL}$.

The Decibel Chart

• If there are charts and plots, it can quickly check.

• Is where it shows the differences between the intensities on the horizon axis.

• Then, if there can then be more output to then finally show the exact db increase (horizontal to vertical axis).

• There is a horizontal axis and plotting, the graph goes on a vertical axis, to measure and then determine a key number and plotting there for all measurements.

dBIL, dBSPL, and dBHL

• The human year is not able to equate any equal sensitivity is not what others here/ can.

• These different contours, at various level of what, for just an example shown needs to be tested for all that are various things tested, or on others shown.

• Equal Loudness Contours: At 1000, it's 10db at equal loud, need and the more from all of that for other details that they can here.

• 4,000 frequency is typically, what can be most sensitive to, which those key weeks are or about to fully be, here.

MAP (Minimum Audible Pressure) and MAF (Minimum Audible Field) Curves

• MAP is when at earphone and then MAF when there is one using a loudspeaker.

Types of Decibels

• DBA, DBC are based on what those, such as loudness, are what could there be if there read upon their intensities.

• DBHL = what would be the average that is clinical or could otherwise be for standard audio tests.

• Audiogram: in horizontal axis, what the HL could be in frequency.

• From dbspls, those will need additional norms, but at tone audio, they can calibrated using what is from loud contours, and from minimal press from different testing (such as when listening through earphones).

• The main idea, is that here ears best can take to, as its the most sensitive, high frequencies such as that from 1000 hertz or 7000, what some may need to know as what they can normalize the sound pressure to what could be the DB here, such dbhls need their own come about.