Question

Two tuning forks of frequencies $320\text{ Hz}$ and $480\text{ Hz}$ are sounded together to produce sound waves. The velocity of sound in air is $320\text{ ms}^{-1}$. The difference between wavelengths of these waves is nearly

• A. $48\text{ cm}$
• B. $16.5\text{ cm}$
• C. $33\text{ cm}$
• D. $42\text{ cm}$

Hint:

When the velocity match exactly equals one of the frequencies ($v = f_1 = 320$), its matching wavelength calculates instantly to a perfect unit integer ($1\text{ m}$). This trivial cancellation allows you to immediately pinpoint the subtraction as $1 - \frac{2}{3} = \frac{1}{3}\text{ m} \approx 33\text{ cm}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Two distinct acoustic tuning forks vibrate at known constant frequencies ($f_1 = 320\text{ Hz}$ and $f_2 = 480\text{ Hz}$).
They produce simultaneous longitudinal sound waves propagating through a shared medium where the local velocity of sound is given as $v = 320\text{ ms}^{-1}$.
We are tasked with computing the absolute linear difference between their respective spatial wavelengths ($\Delta \lambda = |\lambda_1 - \lambda_2|$).

Step 2: Key Formula or Approach:
The physical relationship linking wave speed, frequency, and spatial wavelength is defined by the standard wave equation:
$$v = f \lambda \implies \lambda = \frac{v}{f}$$ The absolute difference between the two wavelengths is:
$$\Delta \lambda = \lambda_1 - \lambda_2 = \frac{v}{f_1} - \frac{v}{f_2}$$

Step 3: Detailed Explanation:
Let's calculate the explicit individual wavelength value for each tuning fork.
For the first fork with frequency $f_1 = 320\text{ Hz}$:
$$\lambda_1 = \frac{320\text{ ms}^{-1}}{320\text{ Hz}} = 1\text{ m}$$ For the second fork with frequency $f_2 = 480\text{ Hz}$:
$$\lambda_2 = \frac{320\text{ ms}^{-1}}{480\text{ Hz}} = \frac{32}{48} = \frac{2}{3}\text{ m} \approx 0.667\text{ m}$$ Now evaluate the spatial difference ($\Delta \lambda$) between these two calculated states:
$$\Delta \lambda = 1\text{ m} - \frac{2}{3}\text{ m} = \frac{1}{3}\text{ m}$$ Convert the resulting value from standard SI meters into centimeters to match the option formatting:
$$\Delta \lambda = \frac{1}{3} \times 100\text{ cm} \approx 33.33\text{ cm}$$ The closest matching approximation given among the choices is $33\text{ cm}$.

Step 4: Final Answer:
The difference between the wavelengths is nearly $33\text{ cm}$, which maps directly to option (C).