Question

(a) Ear trumpets were never very common, but they did aid people with hearing losses by gathering sound over a large area and concentrating it on the smaller area of the eardrum. What decibel increase does an ear trumpet produce if its sound gathering area is$900\mathrm{c}{\mathrm{m}}^2$and the area of the eardrum is$0.500\mathrm{c}{\mathrm{m}}^2$, but the trumpet only has an efficiency of$5.00\mathrm{\%}$in transmitting the sound to the eardrum? (b) Comment on the usefulness of the decibel increase found in part (a)

Short answer

(a) The sound increases by$139.5\mathrm{d}\mathrm{B}$

(b) It is not sustainable in the longer term but it will amplify the sound.

Step-by-step solution

Given Data

The area of the ear trumpet is$900\mathrm{c}{\mathrm{m}}^2$.

The area of the ear is$0.500\mathrm{c}{\mathrm{m}}^2$.

The efficiency of ear trumpet is$5\mathrm{\%}$.

Loudness of Sound

The increase in intensity increases the loudness of a sound. The loudness is unsustainable if it is greater than the threshold level.

Calculation of the intensity of the first sound

(a)

The intensity of the unaided hearing is,

$d_1=10\log \frac{I_0}{{10}^{12}}$

The change in intensity of sound is,

$Q_1=I_0\times 900$

$\begin{array}{c}{I}_{ear}=efficiency\times \frac{Q_1}{0.500}\\ I_{ear}=5.00\times I_0\times \frac{900}{0.500}\\ I_{ear}=90I_0\end{array}$

The intensity of the sound is,

$\begin{array}{c}d=10\log \frac{90I_0}{{10}^{-12}}\\ d=10\log \frac{I_0}{{10}^{-12}}+10\log \frac{90}{{10}^{-12}}\\ d=d_1+139.5\mathrm{d}\mathrm{B}\end{array}$

Hence, the intensity increases by$139.5\mathrm{d}\mathrm{B}$

Usefulness of the situation

(b)

The maximum limit for the intensity of sound is about$139.5\mathrm{d}\mathrm{B}$. The increase in the intensity level may damage the ear. The radius of the trumpet is close to $16\mathrm{c}\mathrm{m}$ which is not practical for holding the sound to the ear. This situation is not sustainable for a short period.