Question 318: Waves

Question

A musical instrument X produces sound waves of frequency n and amplitude A. Another musical instrument $Y$ produces sound waves of frequency $\frac{n}{3}$. The waves produced by $x$ and $y$ have equal energies. The amplitude of waves produced by Y will be

Accepted Answer

To determine the amplitude of the waves produced by instrument $Y$, we need to consider the relationship between the energy carried by a wave, its frequency, and amplitude. The energy ($E$) of a wave is proportional to the square of its amplitude and the square of its frequency, which can be expressed as:
$$ E \propto A^2 f^2 $$
For instrument $X$, the frequency is $n$ and the amplitude is $A$. Therefore, the energy of instrument $X$ ($E_X$) is:
$$ E_X \propto A^2 n^2 $$
For instrument $Y$, the frequency is $\frac{n}{3}$, and we need to determine the amplitude (let's denote it as $A_Y$). The energy of instrument $Y$ ($E_Y$) is:
$$ E_Y \propto A_Y^2 \left(\frac{n}{3}\right)^2 = A_Y^2 \frac{n^2}{9} $$
Given that the energies of the waves produced by $X$ and $Y$ are equal ($E_X = E_Y$), we can set the two expressions equal to each other:
$$ A^2 n^2 = A_Y^2 \frac{n^2}{9} $$
To solve for $A_Y$, we can simplify this equation by canceling $n^2$ from both sides:
$$ A^2 = \frac{A_Y^2}{9} $$
Multiplying both sides by 9 gives:
$$ 9A^2 = A_Y^2 $$
Taking the square root of both sides, we find:
$$ A_Y = 3A $$
Thus, the amplitude of the waves produced by instrument $Y$ is 3A. Therefore, the correct option is:
Option A: 3A

Answer

3A

Answer

4A

Answer

2A

Answer

A

Waves

A musical instrument X produces sound waves of frequency n and amplitude A. Another musical instrument Y produces sound waves of frequency \frac{n}{3}. The waves produced by x and y have equal energies. The amplitude of waves produced by Y will be

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