Converting sound intensity to dB

[gmmstr827]

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?

 

[ehild]

The sound intensity in dB is defined as 10*log₁₀(I/I₀).

ehild

 

[gmmstr827]

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?

 

[ehild]

log(x)=ln(x)/ln(10). Calculate ln(I/I₀) and divide by ln(10)=2.302.

ehild

 

[rwisz]

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.