The ratio of the sound intensity at the first point to that at the second point is 16, since intensity is proportional to the square of pressure amplitude. The difference in sound intensity levels in decibels is 12 dB, which is calculated using the logarithmic relationship between intensity and decibels.
Sound Intensity and Decibel Level Calculations
The question pertains to the relationship between pressure amplitude and sound intensity and the difference in sound levels in terms of decibels (dB). The amplitude of oscillations at the first point is four times that of the second point, indicating a difference in sound pressure levels at these points.
(a) The intensity of a sound wave is proportional to the square of its pressure amplitude. Therefore, if the amplitude of oscillations at the first point is four times that of the second point, the ratio of the intensities of the sound at the first and second points is the square of that ratio:
Ratio of intensities = (Amplitude1 / Amplitude2)^2
Ratio of intensities = (4 / 1)^2
Ratio of intensities = 16
(b) The difference in sound intensity levels in dB can be calculated using the formula:
ΔL = 10 × log10(I1 / I2)
Where ΔL is the difference in sound levels, I1 is the intensity at the first point, and I2 is the intensity at the second point. Plugging in the ratio of intensities, we get:
ΔL = 10 × log10(16) = 10 × 1.2 = 12 dB
Thus, the difference in intensities at the first and second points is 12 dB.