Understanding Wave Properties
When two waves interact, they can produce a phenomenon known as beats. The beat frequency is determined by the difference in frequencies of the two waves.
Given Data
- Wavelength of first wave (λ1) = 50 cm = 0.5 m
- Wavelength of second wave (λ2) = 51 cm = 0.51 m
- Beat frequency (f_beat) = 12 beats per second
Calculating Frequencies
The frequency (f) of a wave can be calculated using the formula:
- f = v / λ
Where:
- v = velocity of sound
- λ = wavelength
Using the wavelengths, we can find the frequencies of both waves:
- f1 = v / 0.5
- f2 = v / 0.51
Finding Frequency Difference
The beat frequency is given by the absolute difference in frequencies:
- f_beat = |f1 - f2| = 12
Substituting the expressions for frequencies, we get:
- |(v / 0.5) - (v / 0.51)| = 12
Simplifying the Equation
To solve for the velocity of sound (v), we can find a common denominator and simplify:
- |(0.51v - 0.5v) / (0.5 * 0.51)| = 12
- |0.01v / 0.255| = 12
Cross-multiplying gives:
- 0.01v = 12 * 0.255
- 0.01v = 3.06
- v = 3.06 / 0.01 = 306 m/s
Conclusion
The velocity of sound is approximately 306 m/s. This value encompasses the combined effects of the two wavelengths and their interaction through beats.