Question

(a) What are the loudnesses in phons of sounds having frequencies of $200\mathrm{H}\mathrm{z}$, $1000\mathrm{H}\mathrm{z}$, $5000\mathrm{H}\mathrm{z}$, and $10,000\mathrm{H}\mathrm{z}$ if they are all at the same $60.0-\mathrm{d}\mathrm{B}$ sound intensity level? (b) If they are all at $110\mathrm{d}\mathrm{B}$? (c) If they are all at $20.0\mathrm{d}\mathrm{B}$?

Short answer

(a) For frequencies $200\mathrm{H}\mathrm{z}$, $1000\mathrm{H}\mathrm{z}$, $10000\mathrm{H}\mathrm{z}$, and sound level $60\mathrm{d}\mathrm{B}$ loudness will be $50\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},60\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},60\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},50\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$respectively

(b) For frequencies $200\mathrm{H}\mathrm{z}$, $1000\mathrm{H}\mathrm{z}$, $5000\mathrm{H}\mathrm{z}$, $10000\mathrm{H}\mathrm{z}$ and sound level $110\mathrm{d}\mathrm{B}$ loudness will be $110\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},110\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},120\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},110\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$respectively

(c) For frequencies $200\mathrm{H}\mathrm{z}$, $1000\mathrm{H}\mathrm{z}$, $5000\mathrm{H}\mathrm{z}$, $10000\mathrm{H}\mathrm{z}$ and sound level $60\mathrm{d}\mathrm{B}$ loudness will be $0\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},20\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},20\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s},10\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$respectively

Step-by-step solution

A concept

The greater the amplitude, the louder and more intense the sound.

Intensity is the number of incident photons per second per unit area (more precisely, it is energy per unit area per unit time), while frequency refers to the frequency of the photon when referred to as a wave, which is the number of waves in a second.

(a) Figure for sound level, intensity, and loudness at given frequencies

First, make vertical lines for all given frequencies and draw a horizontal line corresponding to these vertical lines then we will get our sound intensity levels in $\mathrm{d}\mathrm{B}$ and value of loudness in phons as shown in the figure.

Analysis of figure

Here the phons values that are crossings of vertical lines and horizontal line.

Now follow these points on a particular curve until they meet at phons values at a particular curve. Then we will get phons values for given frequencies.

Then you will get following:

The frequency of $200\mathrm{H}\mathrm{z}$ and $60\mathrm{d}\mathrm{B}$sound intensity level - line crosses at a red line of phons around $50\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency of $1000\mathrm{H}\mathrm{z}$ and $60\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $60\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency of $5000\mathrm{H}\mathrm{z}$ and $60\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $60\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency of $10000\mathrm{H}\mathrm{z}$ and $60\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $50\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

(b) Analysis of figure:

Follow the same process here for $110\mathrm{d}\mathrm{B}$ sound intensity level.

Then you will get following:

The frequency $200\mathrm{H}\mathrm{z}$ and $110\mathrm{d}\mathrm{B}$sound intensity level - line crosses at a red line of phons around $110\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency $1000\mathrm{H}\mathrm{z}$ and $110\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $110\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency $5000\mathrm{H}\mathrm{z}$ and $110\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $120\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency $10000\mathrm{H}\mathrm{z}$ and $110\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $110\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

(c) Analysis of figure

You will follow the same process here for $20\mathrm{d}\mathrm{B}$ sound intensity level.

The frequency $200\mathrm{H}\mathrm{z}$ and $20\mathrm{d}\mathrm{B}$sound intensity level - line crosses at a red line of phons around $0\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency $1000\mathrm{H}\mathrm{z}$ and $20\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $20\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency $5000\mathrm{H}\mathrm{z}$ and $20\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $20\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.

The frequency $10000\mathrm{H}\mathrm{z}$ and $20\mathrm{d}\mathrm{B}$ sound intensity level - line crosses at a red line of phons around $10\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{s}$.