The frequency of a sound wave propagating in air is 3000 Hz. The waves are propagating from a region where T = `30^(@)`C to the region where T = `20^(@)C`. What will be the percentage change of wavelength ?

Answer

To solve the problem of finding the percentage change in wavelength of a sound wave propagating from a region of 30°C to 20°C, we can follow these steps:

### Step 1: Understand the relationship between temperature, velocity, and wavelength
The speed of sound in air is affected by temperature. The formula for the speed of sound is given by:
\[ v = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( v \) = speed of sound
- \( \gamma \) = adiabatic index (constant for air)
- \( R \) = universal gas constant (constant for air)
- \( T \) = absolute temperature in Kelvin
- \( M \) = molar mass of air (constant for air)

Since \( \gamma \), \( R \), and \( M \) are constants, the speed of sound is proportional to the square root of the temperature:
\[ v \propto \sqrt{T} \]

### Step 2: Convert temperatures from Celsius to Kelvin
- Initial temperature \( T_1 = 30°C = 30 + 273 = 303 K \)
- Final temperature \( T_2 = 20°C = 20 + 273 = 293 K \)

### Step 3: Calculate the initial and final velocities
Using the relationship derived earlier:
- Initial velocity \( v_1 = \sqrt{T_1} = \sqrt{303} \)
- Final velocity \( v_2 = \sqrt{T_2} = \sqrt{293} \)

### Step 4: Calculate the initial and final wavelengths
The wavelength \( \lambda \) is given by:
\[ \lambda = \frac{v}{f} \]
where \( f \) is the frequency of the sound wave. Since the frequency remains constant at 3000 Hz, we can express the wavelengths as:
- Initial wavelength \( \lambda_1 = \frac{v_1}{f} \)
- Final wavelength \( \lambda_2 = \frac{v_2}{f} \)

### Step 5: Calculate the percentage change in wavelength
The percentage change in wavelength can be calculated using the formula:
\[ \text{Percentage Change} = \frac{\lambda_2 - \lambda_1}{\lambda_1} \times 100 \]

Substituting the expressions for wavelengths:
\[ \text{Percentage Change} = \frac{\frac{v_2}{f} - \frac{v_1}{f}}{\frac{v_1}{f}} \times 100 \]
This simplifies to:
\[ \text{Percentage Change} = \frac{v_2 - v_1}{v_1} \times 100 \]

### Step 6: Substitute the velocities
Using the relationship \( v_1 \) and \( v_2 \):
\[ \text{Percentage Change} = \left( \frac{\sqrt{T_2} - \sqrt{T_1}}{\sqrt{T_1}} \right) \times 100 \]

### Step 7: Plug in the values
Now substituting the temperatures:
\[ \text{Percentage Change} = \left( \frac{\sqrt{293} - \sqrt{303}}{\sqrt{303}} \right) \times 100 \]

### Step 8: Calculate the numerical value
Calculating the square roots:
- \( \sqrt{293} \approx 17.1 \)
- \( \sqrt{303} \approx 17.4 \)

Now substituting these values:
\[ \text{Percentage Change} = \left( \frac{17.1 - 17.4}{17.4} \right) \times 100 \]
\[ \text{Percentage Change} = \left( \frac{-0.3}{17.4} \right) \times 100 \approx -1.72\% \]

### Final Answer
The percentage change in wavelength is approximately **-1.72%**.

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