A sound wave of frequency 300 Hz has an intensity of 1.00 μW/m2 . What is the amplitude of the air oscillations caused by this wave?
The correct answer and explanation is:
Answer:
The amplitude of air oscillations (sms_m) caused by a sound wave is given by the equation: I=12ρvω2sm2I = \frac{1}{2} \rho v \omega^2 s_m^2
where:
- I=1.00 μW/m2=1.00×10−6W/m2I = 1.00 \, \mu W/m^2 = 1.00 \times 10^{-6} W/m^2 (Intensity),
- ρ\rho is the density of air (1.21 kg/m31.21 \, kg/m^3),
- vv is the speed of sound in air (343 m/s343 \, m/s),
- f=300 Hzf = 300 \, Hz is the frequency,
- ω=2πf=2π×300=1884.96 rad/s\omega = 2\pi f = 2\pi \times 300 = 1884.96 \, rad/s.
Rearranging for sms_m: sm=2Iρvω2s_m = \sqrt{\frac{2I}{\rho v \omega^2}}
Substituting the values: sm=2(1.00×10−6)(1.21)(343)(1884.96)2s_m = \sqrt{\frac{2(1.00 \times 10^{-6})}{(1.21)(343)(1884.96)^2}} sm=2.00×10−61.21×343×3.55×106s_m = \sqrt{\frac{2.00 \times 10^{-6}}{1.21 \times 343 \times 3.55 \times 10^6}} sm=2.00×10−61.47×109s_m = \sqrt{\frac{2.00 \times 10^{-6}}{1.47 \times 10^9}} sm=1.36×10−15s_m = \sqrt{1.36 \times 10^{-15}} sm≈1.17×10−7 m=0.117 μms_m \approx 1.17 \times 10^{-7} \, m = 0.117 \, \mu m
Explanation:
The amplitude of air oscillations is the maximum displacement of air molecules due to the sound wave. The equation for intensity incorporates the medium properties (density, speed of sound) and frequency of the wave.
Sound waves transport energy, and intensity measures the energy flow per unit area. Since intensity depends on the square of the amplitude, a small change in amplitude results in a significant change in intensity. For audible sound waves, the displacement amplitude is usually on the micrometer scale.
For a 300 Hz sound wave with 1 μW/m² intensity, the calculated amplitude of air oscillations is approximately 0.117 μm.
Now, I will generate an image illustrating the oscillations of air molecules caused by a sound wave.
Here is an illustration of the oscillation of air molecules due to a sound wave, showing the wave propagating through air with alternating compressions and rarefactions. Let me know if you need any modifications!
