Bounds on Sound Intensity Decibels using Logarithmic Inequalities

Problem: Bounds on Sound Intensity Decibels using Logarithmic Inequalities

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

where ##I## is the measured sound intensity and ##I_0## is the reference intensity (threshold of hearing). When intensity increases from ##I_1## to ##I_2## (where ##I_2 > I_1##), the change in sound level ##\Delta\beta## is given by:

Given the fractional increase in intensity ##\delta = \dfrac{I_2 - I_1}{I_1}##, derive an analytical upper bound for ##\Delta\beta## using the inequality ##\ln(1 + x) \le x## for ##x > -1##.

Common Sound Intensity Levels

Worked Solution & Step-by-Step Explanation

To establish the upper bound, we must transform the base-10 logarithmic expression into a natural logarithmic form to utilize the provided inequality.

**Step 1: Express the intensity ratio in terms of ##\delta##.**

The fractional increase ##\delta## is defined as:

Rearranging this, we find the ratio of intensities:

**Step 2: Substitute the ratio into the decibel change formula.**

Substituting ##1 + \delta## into the expression for ##\Delta\beta##:

**Step 3: Perform a change of base for the logarithm.**

Using the property ##\log_b(a) = \dfrac{\ln(a)}{\ln(b)}##, we convert ##\log_{10}## to the natural logarithm:

**Step 4: Apply the logarithmic inequality.**

We are provided with the inequality ##\ln(1 + \delta) \le \delta##. Substituting this into our equation:

**Step 5: Calculate the numerical constant.**

The natural logarithm of 10 is approximately ##2.302585##. Therefore, the constant factor is:

Thus, we arrive at the final upper bound inequality:

Illustrative Examples: Actual Δβ vs. Upper Bound

Physical Interpretation and Significance

This result is mathematically significant for experimental physics. It demonstrates that for small fractional changes in intensity (##\delta \ll 1##), the change in decibels is approximately linear.

**Key Takeaways for JEE/NEET:**

1. **Linearization:** The inequality proves that ##\Delta\beta \approx 4.343 \delta## is a valid approximation for small variations, often used in error analysis.

2. **Logarithmic Behavior:** Because the logarithm grows slower than a linear function, the linear bound ##4.343\delta## is always greater than or equal to the actual decibel increase.

3. **Dimensional Consistency:** Note that ##\delta## is dimensionless (ratio of intensities), and ##\Delta\beta## is expressed in dB, confirming that the constant ##4.343## carries the units of decibels per unit fractional change.