Use the Fletcher-Munson curves of equal loudness to answer the following questio

Question

Use the Fletcher-Munson curves of equal loudness to answer the following questions. Remember that the horizontal axis is marked as a logarithmic scale. The divisions between the far left and 100 represent 20, 30, 40, 50, 60, 70 ,80 and 90. The divisions between 100 and 1000 represent 200, 300, 400, 500, 600, 700, 800, and 900.

1. What is the minimum threshold of hearing in intensity level (decibels) for a tone at 50 Hz? (Do not enter units)

2. What is the loudness of a 60 db tone at 50 Hz? (Enter the answer in phons. Do not enter units.)

3. What is the intensity level required for a 400 Hz tone to have a loudness of 80 phons?

Explanation / Answer

For simple mechanical waves like sound, intensity is related to the density of the medium and the speed, frequency, and amplitude of the wave. This can be shown with a long, horrible, calculation. Jump to the next highlighted equation if you don't care to see the sausage being made below.

Start with the definition of intensity. Replace power with energy (both kinetic and elastic) over time (one period, for convenience sake).

Since kinetic and elastic energies are always positive we can split the time-averaged portion into two parts.

Mechanical waves in a continuous media can be thought of as an infinite collection of infinitesimal coupled harmonic oscillators. Little masses connected to other little masses with little springs as far as the eye can see. On average, half the energy in a simple harmonic oscillator is kinetic and half is elastic. The time-averaged total energy in then either twice the average kinetic energy or twice the average potential energy.

Let's work on the kinetic energy and see where it takes us. It has two important parts

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