wo waves of the same pitch have amplitudes in the ratio 1:3. What will be the ratioof their (i) loudness and (ii) frequencies?

To address the question regarding the two waves with different amplitudes, we will break down the problem and use relevant concepts related to sound waves and their characteristics.

1. Break Down the Problem

• We have two waves:
  • Wave 1: Amplitude $A_1$
  • Wave 2: Amplitude $A_2$
• The ratio of their amplitudes is given as $\frac{A_1}{A_2} = \frac{1}{3}$.

2. Relevant Concepts

• Loudness: The loudness of a sound is proportional to the square of the amplitude of the wave. Thus, if $L$ is the loudness, it can be expressed as: $L \propto A^2$

• Frequency: The frequency of the sound waves is not dependent on the amplitude. Therefore, the frequencies of both waves will be constant and the same.

3. Analysis and Detail

• For the loudness ratio: $L_1 \propto A_1^2 \quad \text{and} \quad L_2 \propto A_2^2$ Given the amplitude ratio $\frac{A_1}{A_2} = \frac{1}{3}$, we find: $\frac{L_1}{L_2} = \frac{A_1^2}{A_2^2} = \left( \frac{1}{3} \right)^2 = \frac{1}{9}$

• For the frequency ratio, since both waves have the same pitch, we can denote their frequencies as: $f_1 = f_2$ Therefore, $\frac{f_1}{f_2} = 1$

4. Verify and Summarize

• The loudness ratio has been confirmed through the relation of amplitudes. The frequencies are equal, thus confirming the frequency ratio.

Final Answer

(i) The ratio of their loudness is $\frac{1}{9}$.

(ii) The ratio of their frequencies is $1$.