A musical instrument ' P ' produces sound waves of frequency ' n ' and amplitude ' A '. Another musical instrument ' Q ' produces sound waves of frequency 4 n

Question

A musical instrument ' P ' produces sound waves of frequency ' n ' and amplitude ' A '. Another musical instrument ' Q ' produces sound waves of frequency 4 n ​ . The waves produced by ' P ' and ' Q ' have equal energies. If the amplitude of waves produced by ' P ' is ' A P ​ ', the amplitude of waves produced by ' Q ' will be

TestHub · Accepted Answer

The energy of a sound wave is directly proportional to the square of its frequency and the square of its amplitude.

$E \propto n^{2} A^{2}$

Given that the waves produced by 'P' and 'Q' have equal energies:

$E_{P} = E_{Q}$

$n_{P}^{2} A_{P}^{2} = n_{Q}^{2} A_{Q}^{2}$

Substituting the given values $n_{P} = n$ and $n_{Q} = \frac{n}{4}$ :

$n^{2} A_{P}^{2} = (\frac{n}{4})^{2} A_{Q}^{2}$

$n^{2} A_{P}^{2} = \frac{n ^{2}}{16} A_{Q}^{2}$

$A_{Q}^{2} = 16 A_{P}^{2}$

$A_{Q} = 4 A_{P}$

Answer: $4 A_{P}$

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