Problem 29 - Applied Calculus

Question

A standard tone of 20,000 dynes/cm \(^{2}\) (about the loudness of a rock band) is assigned a value of 10 . A subject listened to other sounds, such as a light whisper, normal conversation, thunder, a jet plane at takeoff, and so on. In each case, the subject was asked to judge the loudness and assign it a number relative to 10 , the value of the standard tone. This is a "judgment of magnitude" experiment. The power law \(J=a l^{0.3}\) was found to model the situation well, where \(l\) is the actual loudness (measured in dynes/cm \(^{2}\) ) and \(J\) is the judged loudness. (a) What is the value of \(a\) ? (b) What is the judged loudness if the actual loudness is \(0.2\) dynes \(/ \mathrm{cm}^{2}\) (normal conversation)? (c) What is the actual loudness if judged loudness is \(20 ?\)

Step-by-Step Solution

Answer

(a) The value of \(a\) is approximately 0.4647. (b) For 0.2 dynes/cm², judged loudness is approximately 0.2716. (c) For judged loudness 20, actual loudness is approximately 797.68 dynes/cm².

Step 1: Understand the Power Law

The given power law formula is \(J = a \, l^{0.3}\) where \(J\) is the judged loudness, \(l\) is the actual loudness in dynes/cm², and \(a\) is a constant to determine. We are given a standard tone where \(l = 20000\) dynes/cm² and \(J = 10\). Our task is to first find \(a\).

Step 2: Solve for Constant 'a'

Substitute the known values of \(J = 10\) and \(l = 20000\) in the power law. The equation becomes:\[ 10 = a (20000)^{0.3} \]Calculate \( (20000)^{0.3} \) which is approximately \(21.5443\). Then solve for \(a\):\[ a = \frac{10}{21.5443} \approx 0.4647 \]

Step 3: Calculate Judged Loudness for 0.2 dynes/cm²

Now that we have \(a = 0.4647\), use the power law to find the judged loudness \(J\) when the actual loudness \(l = 0.2\) dynes/cm².\[ J = 0.4647 \, (0.2)^{0.3} \]Calculate \( (0.2)^{0.3} \) which is approximately \(0.5848\). Therefore,\[ J = 0.4647 \times 0.5848 \approx 0.2716 \]

Step 4: Determine Actual Loudness for Judged Loudness 20

Find the actual loudness \(l\) when the judged loudness \(J = 20\). Use the known value of \(a\):\[ 20 = 0.4647 \, l^{0.3} \]Solve for \(l^{0.3}\):\[ l^{0.3} = \frac{20}{0.4647} \approx 43.0423 \]Raise both sides to the power of \(\frac{10}{3}\) to get \(l\):\[ l = (43.0423)^{\frac{10}{3}} \approx 797.68 \text{ dynes/cm}^2 \]

Key Concepts

Power Law

The power law is an influential relation that captures how one quantity affects another in a specific and predictable way. In applied calculus, it is represented by a formula showcasing how variations in one physical property can lead to proportional changes in another. In the context of the problem, the power law is given by the equation \( J = a \, l^{0.3} \). Here, \( J \) symbolizes the judged loudness, and \( l \) stands for the actual loudness measured in dynes/cm².One of the key features of a power law is the exponent, in this case, \( 0.3 \), which dictates how the scaling occurs. The smaller the exponent, the less sensitive \( J \) is to changes in \( l \). The constant \( a \) plays a critical role since it adjusts the formula to fit the particular situation or experiment. Once \( a \) is known, we can predict \( J \) for any given \( l \). This kind of relationship is incredibly useful in fields like psychology, where subjective perceptions often have to be linked with objective measurements.

Judgement of Magnitude

The judgment of magnitude is an experiment designed to understand how humans perceive changes in a stimulus compared to its actual magnitude. It seeks to quantify a subjective response by allowing individuals to assign numerical values to the intensity of stimuli. In our scenario, a standard tone at 20,000 dynes/cm² is used as a reference point, and other sounds like whispers or thunder are compared against it.Participants were asked to judge each sound’s loudness and assign a number relative to the reference tone. This subjective judgment is \( J \) in the power law equation. The experiment highlights how perceptions aren't always linear. People may not feel a sound is twice as loud just because its intensity is doubled, reflecting the complexity of human senses.

Actual Loudness

Actual loudness refers to the physical measurement of sound intensity using specific units like dynes/cm². It signifies the objective reality or factual amplitude of a sound wave, absent of human interpretation. In the exercise, the actual loudness varies from a whisper to thunder, demonstrating the variety of sounds that can be analyzed.When calculating judged loudness \( J \), actual loudness \( l \) is raised to the power of \( 0.3 \), showing how this real magnitude is transformed through human perception. This part of the equation shows that, even if a sound has a high actual loudness, the judged loudness may not increase as significantly because of the small exponent.

Judged Loudness

Judged loudness, on the other hand, is the subjective evaluation of sound intensity by an individual. It is represented by \( J \) in the power law equation. Real-world factors, such as the listener's environment and mental state, may influence how loud a sound seems to them. This subjective measure can often differ from the actual loudness due to nonlinear perception.In the power law, the judged loudness is a result of combining actual loudness with the constant \( a \) and the power factor \( 0.3 \). When given values for actual loudness, the equation can be solved to find the judged loudness. For instance, even a sound with an actual loudness of 0.2 dynes/cm² might be perceived as quite soft when compared to a standard tone.