Bounds on Sound Intensity Decibels using Logarithmic Inequalities

Problem: Bounds on Sound Intensity Decibels Using Logarithmic Inequalities

The sound intensity level ##\beta## in decibels is defined by the logarithmic relation:

###\beta = 10\log_{10}\left(\frac{I}{I_0}\right)###

where ##I## is the measured sound intensity and ##I_0## is the reference intensity, usually taken as the threshold of hearing.

When the sound intensity increases from ##I_1## to ##I_2##, where ##I_2 > I_1##, the change in sound intensity level is:

###\Delta\beta = \beta_2 - \beta_1 = 10\log_{10}\left(\frac{I_2}{I_1}\right)###

Given the fractional increase in intensity

###\delta = \frac{I_2-I_1}{I_1}###

derive an analytical upper bound for ##\Delta\beta## using the logarithmic inequality:

###\ln(1+x) \leq x, \qquad x > -1###

Common Sound Intensity Levels

The table below gives a few common sound intensity levels. These values help us understand why the decibel scale is logarithmic rather than linear.

Worked Solution and Step-by-Step Explanation

To establish the upper bound, we first express the intensity ratio in terms of the fractional increase ##\delta##. Then we convert the base-10 logarithm into a natural logarithm so that the inequality ##\ln(1+x)\leq x## can be applied.

Step 1: Express the Intensity Ratio in Terms of ##\delta##

The fractional increase in sound intensity is defined as:

###\delta = \frac{I_2-I_1}{I_1}###

Splitting the fraction gives:

###\delta = \frac{I_2}{I_1} - \frac{I_1}{I_1}###

Therefore,

###\delta = \frac{I_2}{I_1} - 1###

Rearranging, we get:

###\frac{I_2}{I_1} = 1+\delta###

Since ##I_2 > I_1##, the fractional increase ##\delta## is positive. Hence, ##1+\delta > 1##.

Step 2: Substitute the Ratio into the Decibel Change Formula

The change in sound intensity level is:

###\Delta\beta = 10\log_{10}\left(\frac{I_2}{I_1}\right)###

Using ##\frac{I_2}{I_1}=1+\delta##, we get:

###\Delta\beta = 10\log_{10}(1+\delta)###

Step 3: Convert the Logarithm to Natural Logarithm

The given inequality involves the natural logarithm ##\ln##. Therefore, we use the change-of-base formula:

###\log_b a = \frac{\ln a}{\ln b}###

For base ##10##, this gives:

###\log_{10}(1+\delta) = \frac{\ln(1+\delta)}{\ln 10}###

Substituting this into the expression for ##\Delta\beta##:

###\Delta\beta = 10\cdot\frac{\ln(1+\delta)}{\ln 10}###

or

###\Delta\beta = \frac{10}{\ln 10}\ln(1+\delta)###

Step 4: Apply the Logarithmic Inequality

We are given the inequality:

###\ln(1+x) \leq x, \qquad x > -1###

Here, we take ##x=\delta##. Since ##\delta>0## for an increase in intensity, the condition ##x>-1## is automatically satisfied.

Therefore,

###\ln(1+\delta) \leq \delta###

Now substitute this into

###\Delta\beta = \frac{10}{\ln 10}\ln(1+\delta)###

Since ##\frac{10}{\ln 10}## is positive, the direction of the inequality remains unchanged:

###\Delta\beta \leq \frac{10}{\ln 10}\delta###

Step 5: Evaluate the Numerical Constant

The natural logarithm of ##10## is approximately:

###\ln 10 \approx 2.302585###

Therefore,

###\frac{10}{\ln 10} \approx \frac{10}{2.302585} \approx 4.3429###

Hence, the required analytical upper bound is:

###\Delta\beta \leq 4.343\delta###

Illustrative Examples: Actual ##\Delta\beta## vs Upper Bound

The table below compares the actual decibel increase with the linear upper bound ##4.343\delta## for different fractional increases in intensity.

Notice that the upper bound is always slightly greater than the actual value. This is expected because ##\ln(1+\delta)## grows more slowly than ##\delta##.

Physical Interpretation and Significance

The result is useful in experimental physics because it gives a quick estimate of how much the decibel level can increase when the sound intensity increases by a small fraction.

For small fractional changes, where ##\delta \ll 1##, the logarithmic expression behaves almost linearly:

###\Delta\beta = 10\log_{10}(1+\delta) \approx 4.343\delta###

The inequality shows that this linear expression is not just an approximation; it is also an upper bound:

###\Delta\beta \leq 4.343\delta###

Key Takeaways for JEE and NEET

1. Linearization: For small fractional changes in intensity, the logarithmic expression can be approximated as:

###\Delta\beta \approx 4.343\delta###

This is useful for quick numerical estimation and error analysis.

2. Logarithmic growth is slower than linear growth: Since ##\ln(1+\delta)\leq \delta##, the actual decibel increase is always less than or equal to the linear bound ##4.343\delta##.

3. The fractional increase is dimensionless: The quantity ##\delta## is a ratio of intensities, so it has no unit. The final value ##\Delta\beta## is expressed in decibels.

4. The bound is especially accurate for small changes: When ##\delta## is very small, the upper bound ##4.343\delta## becomes extremely close to the actual value of ##10\log_{10}(1+\delta)##.

Final Conclusion

Starting from the decibel change formula,

###\Delta\beta = 10\log_{10}\left(\frac{I_2}{I_1}\right)###

and using the fractional increase ##\delta=\frac{I_2-I_1}{I_1}##, we obtained:

###\Delta\beta = 10\log_{10}(1+\delta)###

Changing to natural logarithm gives:

###\Delta\beta = \frac{10}{\ln 10}\ln(1+\delta)###

Using ##\ln(1+\delta)\leq\delta##, we finally get:

###\boxed{\Delta\beta \leq 4.343\delta}###

This proves that the decibel increase caused by a fractional increase in intensity is bounded above by a simple linear expression.