Question

The ratio of amplitude of two waves is $3:4$. What is the ratio of their
(i) loudness? (ii) frequencies?

Answer 1

Hint: Ratio of their loudness can be known by using the expression, loudness is directly proportional to the square of the amplitude. Each and every object has its own amplitude independent frequency which is determined by its nature.

Formula Used:
By the expression, loudness is directly proportional to the square of the amplitude,
$\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{{a_1}^2}}{{{a_2}^2}}$
Where,
${l_1}$ and ${l_2}$ are the intensities of the two waves
${a_1}$ and ${a_2}$ are the amplitude of the two waves

Complete step by step solution:
(i) Loudness ratio:
By using the expression loudness is directly proportional to the square of the amplitude,
$\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{{a_1}^2}}{{{a_2}^2}}$
Substitute the given ratio of amplitude of two waves, where, ${a_1} = 3$ and ${a_2} = 4$,
Then,
$\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{{{\left( 3 \right)}^2}}}{{{{\left( 4 \right)}^2}}}$
Squaring the RHS side,
$\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{9}{{16}}$
Thus, the ratio of their loudness is $9:16$.

(ii) Frequency ratio:
Frequency refers to the number of waves formed per second. The frequency depends only on the time period and does not depend upon amplitude. So, the change in amplitude of a wave will not affect its frequency.
Thus, the ratio of the frequencies is independent of the ratio of amplitudes of the waves.

Additional information:
The relationship between the time period and frequency is expressed as,
$f = \dfrac{1}{T}$
Or,
$T = \dfrac{1}{f}$
Where,
$T$ is the time period
$f$ is the frequency

Note: The frequency is the time parameter it explains about the oscillation and vibrations like frequency waves, mechanical vibration, sound signals etc. The frequency is mainly classified into two types namely, angular frequency and spatial frequency.