Converting sound intensity to dB

In summary, the sound intensity at a concert is measured to be 0.5 W/m^2, which corresponds to a sound level of 269.378 dB using the formula β = 10ln(I/I_0). However, using the definition of dB as 10*log10(I/I_0) gives a result of 116.99 dB. This discrepancy is due to using natural logarithms (ln) instead of base 10 logarithms (log) in the equation. To calculate the logarithm function on a TI-89 calculator, one can use the formula log(x) = ln(x)/ln(10).
  • #1
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Homework Statement



The sound at a concert is measured to be 0.5 W/m^2. How many decibels is this?

Homework Equations



β = 10ln(I/I_0)
I/I_0 = 10^(β/10)

The Attempt at a Solution



I_0 = 1*10^-12 W/m^2
I = 0.5 W/m^2

Therefore,
β = 10ln(0.5/10^-12)
β = 269.378 dB

However,
.5/10^12 = 10^(β/10)
When using a TI-89 calculator to solve for β the answer comes to 116.99 dB, what I expected.

I can't find anything wrong with my work, but the dB level seems way too high for the first equation. I expected it should be somewhere around 115 dB. Why does the formula involving natural logs give the wrong answer?
 
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  • #2
The sound intensity in dB is defined as 10*log10(I/I0).

ehild
 
  • #3
ehild said:
The sound intensity in dB is defined as 10*log10(I/I0).

ehild

Nevermind, I thought log(x) was another way to write ln(x)... some research cleared this up. However, is there an easier way to find the log( function in the TI-89 calculator other than scrolling through the catalog?
 
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  • #4
log(x)=ln(x)/ln(10). Calculate ln(I/I0) and divide by ln(10)=2.302.

ehild
 
  • #5


I would like to clarify that the formula for converting sound intensity to decibels is correct and there is no error in your calculation. However, the discrepancy in the results can be explained by the fact that there are two different ways of measuring sound intensity - one using the logarithmic scale (decibels) and the other using the linear scale (watts per square meter).

When using the formula β = 10ln(I/I_0), we are converting the linear scale of intensity (I) to the logarithmic scale of decibels (β). This formula is based on the assumption that the reference intensity (I_0) is equal to 1*10^-12 W/m^2. However, in reality, the reference intensity may vary depending on the measurement system being used.

On the other hand, when using the formula I/I_0 = 10^(β/10), we are converting the logarithmic scale of decibels (β) to the linear scale of intensity (I). This formula is based on the assumption that the reference intensity (I_0) is equal to 1 W/m^2. This is the standard reference intensity used in most sound measurement systems.

Therefore, the discrepancy in the results is due to the difference in the reference intensity used in the two formulas. In this case, the reference intensity used in the first formula is 1*10^-12 W/m^2, which is much lower than the standard reference intensity of 1 W/m^2 used in the second formula. This results in a much higher decibel value using the first formula compared to the second formula.

In conclusion, both formulas are correct and give accurate results. However, the reference intensity used in the first formula may vary depending on the measurement system, while the second formula uses a standard reference intensity of 1 W/m^2. It is important to take this into consideration when converting sound intensity to decibels.
 

1. What is sound intensity and how is it measured?

Sound intensity is a measure of the energy carried by sound waves. It is measured in watts per square meter (W/m²) and is used to quantify the loudness of a sound. Sound intensity can be measured using specialized equipment such as a sound level meter.

2. What is the unit of measurement for decibels (dB)?

Decibels (dB) is a unit used to express the intensity of a sound relative to a reference level. It is a logarithmic unit, meaning that the difference between two decibel values is not a simple arithmetic ratio, but a ratio of the logarithms of the two power quantities being compared.

3. How do you convert sound intensity into decibels (dB)?

To convert sound intensity to decibels (dB), you can use the following formula: dB = 10 * log (I/I0), where I is the sound intensity in watts per square meter and I0 is the reference intensity of 10^-12 W/m².

4. What is the reference level for sound intensity measurements?

The reference level for sound intensity measurements is 10^-12 watts per square meter (W/m²). This is equivalent to the threshold of hearing, which is the lowest sound intensity that a person with normal hearing can detect.

5. How is decibel (dB) scale used to measure sound intensity?

The decibel (dB) scale is used to measure sound intensity because it allows for a larger range of values to be represented in a more manageable scale. The decibel scale is logarithmic, which means that an increase of 10 dB represents a tenfold increase in sound intensity. This makes it easier to compare and quantify different levels of sound intensity.

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