The beat frequency (\( f_{\text{beat}} \)) is determined by the difference between two frequencies:
\[f_{\text{beat}} = f_1 - f_2,\]
where \( f_1 \) and \( f_2 \) are the frequencies associated with wavelengths \( \lambda_1 \) and \( \lambda_2 \), respectively.
Step 1: Calculate the beat frequency.
Given the number of beats and the time elapsed:
\[f_{\text{beat}} = \frac{\text{Number of beats}}{\text{Time}} = \frac{40}{12} \, \text{Hz}.\]
Simplified, this is:
\[f_{\text{beat}} = \frac{10}{3} \, \text{Hz}.\]
Step 2: Relate wavelength to velocity and frequency.
The relationship between wave frequency (\( f \)), velocity (\( v \)), and wavelength (\( \lambda \)) is given by:
\[f = \frac{v}{\lambda}.\]
Using the provided wavelengths, \( \lambda_1 = 4.08 \, \text{m} \) and \( \lambda_2 = 4.16 \, \text{m} \), the frequencies are:
\[f_1 = \frac{v}{4.08}, \quad f_2 = \frac{v}{4.16}.\]
Step 3: Solve for the velocity \( v \).
The beat frequency is the difference between \( f_1 \) and \( f_2 \):
\[f_{\text{beat}} = f_1 - f_2 = \frac{v}{4.08} - \frac{v}{4.16}.\]
This can be rewritten as:
\[\frac{10}{3} = v \left(\frac{1}{4.08} - \frac{1}{4.16}\right).\]
The difference in the reciprocals of the wavelengths is calculated as:
\[\frac{1}{4.08} - \frac{1}{4.16} = \frac{4.16 - 4.08}{4.08 \cdot 4.16} = \frac{0.08}{16.9728}.\]
Substituting this back into the equation:
\[\frac{10}{3} = v \cdot \frac{0.08}{16.9728}.\]
Solving for \( v \):
\[v = \frac{\frac{10}{3} \cdot 16.9728}{0.08} = 707.2 \, \text{m/s}.\]
Final Answer:
\[\boxed{707.2 \, \text{m/s}}.\]