Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

The loudness of sound is measured in decibels in honor of Alexander Graham Bell (1847-1922), inventor of the telephone. If the variation in pressure is \(P\) pounds per square inch, then the loudness \(L\) in decibels is $$ L=20 \log _{10}(121.3 P) $$ Find the variation in pressure caused by music at 115 decibels.

Step-by-Step Solution

Answer

The pressure variation is approximately 4636.26 pounds per square inch.

Step 1: Understand the Equation

The given equation for the loudness \(L\) is \(L = 20 \log_{10}(121.3P)\), where \(P\) represents the pressure variation in pounds per square inch. We need to find \(P\) when \(L = 115\) decibels.

Step 2: Substitute the Loudness Value

Substitute \(L = 115\) into the equation: \[115 = 20 \log_{10}(121.3P)\].

Step 3: Isolate the Logarithmic Expression

Divide both sides of the equation by 20 to isolate the logarithmic term: \[\frac{115}{20} = \log_{10}(121.3P)\].Simplifying gives \[5.75 = \log_{10}(121.3P)\].

Step 4: Solve the Logarithmic Equation

To eliminate the logarithm, rewrite the equation in exponential form: \[10^{5.75} = 121.3P\].

Step 5: Solve for Pressure Variation

Divide both sides by 121.3 to solve for \(P\): \[P = \frac{10^{5.75}}{121.3}\].Calculate \(10^{5.75}\) and divide by 121.3 to find \(P\).

Step 6: Calculate the Numerical Value

Calculate \(10^{5.75} \approx 562341.33\), then divide by 121.3:\[P \approx \frac{562341.33}{121.3} \approx 4636.26\] pounds per square inch.

Key Concepts

Sound Pressure Level

Sound pressure level (SPL) is a measure of the intensity or loudness of sound, expressed in decibels (dB). This measure is crucial when expressing how much a sound can be heard or felt by humans. The concept of SPL is based on the pressure variations that sound waves cause in the air.

Understanding SPL is essential for defining how sound energy moves through the air. The pressure variation, indicated by a change in pressure from silence, is usually measured in units such as pounds per square inch (psi).

Key points about sound pressure level include:

• It is proportional to the amplitude of the sound wave.
• Higher SPL values indicate louder sounds.
• It uses a reference pressure, typically the threshold of hearing, which acts as a baseline measurement.

SPL is a fundamental concept in acoustics, connecting the physical aspects of a sound wave with its perceived loudness.

Logarithmic Scale

A logarithmic scale is a nonlinear scale used to represent a wide range of values compactly. This is particularly useful in measuring anything with large variations, like sound and earthquakes.

Sound levels in decibels (dB) are expressed on a logarithmic scale because the human ear perceives sound intensity logarithmically. This means each increase of 10 dB represents a tenfold increase in the intensity of sound pressure.

Why use a logarithmic scale for sound measurements?

• It matches the human ear's sensitivity to changes in sound levels.
• It allows us to measure very quiet to very loud sounds conveniently on the same scale.
• Simplifies calculations of multiplication into addition (e.g., finding combined sound levels).

Logarithmic scales make it manageable to work with the large numerical range of sound intensities found in various real-world environments.

Decibel Measurement

Decibel (dB) measurement is a relative measure of sound intensity. This unit of measurement compares a sound's intensity to a reference level, which is typically the quietest sound humans can hear.

Decibels are used because they effectively express both very small and very large numbers in a manageable way. For example, the equation for calculating loudness, \( L = 20 \log_{10}(121.3P) \), uses the logarithmic nature of decibels to express how much more intense a sound pressure is than the reference level.

Key elements of decibel measurement include:

• Based on a ratio; it compares two quantities.
• Expressed in a logarithmic form to compress the range of numbers.
• Allows easy manipulation and comparison of sound levels.

This system allows us to quickly make sense of a sound's impact, whether it’s a whisper in a library or a rock concert.

Alexander Graham Bell

Alexander Graham Bell, the inventor of the telephone, profoundly influenced the field of acoustics and communication. Born in 1847 in Scotland, his lifelong work in sound and voice set the stage for many inventions, including the decibel system.

Bell's pioneering work in electrical sound transmission led to the invention of the first practical telephone in 1876, revolutionizing long-distance communication.

Why is Bell's name used for the decibel?

• To honor his contributions to the field of sound and communication.
• His invention of the telephone relied heavily on understanding sound waves and pressure variation.
• The "bel" was initially used as a unit of measurement in telecommunication, later becoming the "decibel" for practicality.

Bell's legacy lives on through these innovations, symbolizing the connection between scientific exploration and its real-world applications.