Question

If the frequency of a wave is increased by 25%, then the change in its wavelength is (medium not changed):

• A. \( 20\% \) increase
• B. \( 20\% \) decrease
• C. \( 25\% \) increase
• D. \( 25\% \) decrease

Hint:

When the frequency of a wave increases while the medium remains unchanged, the wavelength decreases proportionally. Use the formula \( v = f\lambda \) and apply percentage change calculations to determine the new wavelength.

Answer 1

The Correct Option is B

To determine the change in wavelength when the frequency of a wave is increased by 25%, we start with the wave equation relating speed \(v\), frequency \(f\), and wavelength \(\lambda\):

\( v = f \cdot \lambda \)

Since the medium is unchanged, the speed \(v\) remains constant. Therefore, any change in frequency will inversely affect the wavelength.

Let the initial frequency be \(f\) and the initial wavelength be \(\lambda\). If the frequency is increased by 25%, the new frequency becomes:

\( f_{\text{new}} = f + 0.25f = 1.25f \)

Since the speed \(v\) is constant, we use the wave equation for the new situation:

\( v = f_{\text{new}} \cdot \lambda_{\text{new}} = 1.25f \cdot \lambda_{\text{new}} \)

Equating the constant speed in both situations, we get:

\( f \cdot \lambda = 1.25f \cdot \lambda_{\text{new}} \)

Solving for the new wavelength \(\lambda_{\text{new}}\):

\( \lambda_{\text{new}} = \frac{f \cdot \lambda}{1.25f} = \frac{\lambda}{1.25} = 0.8\lambda \)

This shows the new wavelength is 80% of the original wavelength, indicating a decrease.

The decrease in wavelength is therefore:

\( \text{Percentage decrease} = \left(1 - 0.8\right) \times 100\% = 20\% \)

Hence, when the frequency is increased by 25%, the wavelength decreases by 20%.

Correct Option: \( 20\% \) decrease

Answer 2

Step 1: Understanding the Wave Relation The relationship between the speed \( v \), frequency \( f \), and wavelength \( \lambda \) of a wave is given by the equation: \[ v = f \lambda \] Since the medium is unchanged, the wave speed \( v \) remains constant. 
Step 2: Finding the Change in Wavelength Rearranging the equation for wavelength: \[ \lambda = \frac{v}{f} \] If the frequency increases by 25\%, then the new frequency is: \[ f' = 1.25 f \] Since wave speed is constant, the new wavelength becomes: \[ \lambda' = \frac{v}{1.25 f} = \frac{\lambda}{1.25} \] 
Step 3: Calculating Percentage Change in Wavelength The percentage change in wavelength is: \[ \left( \frac{\lambda' - \lambda}{\lambda} \right) \times 100 = \left( \frac{\frac{\lambda}{1.25} - \lambda}{\lambda} \right) \times 100 \] \[ = \left( \frac{\lambda - 1.25\lambda}{1.25\lambda} \right) \times 100 \] \[ = \left( \frac{1 - 1.25}{1.25} \right) \times 100 \] \[ = \left( \frac{-0.25}{1.25} \right) \times 100 \] \[ = -20\% \] Thus, the wavelength decreases by 20%.