Final answer:
The new intensity of a sound wave, when the amplitude is increased by 30%, is calculated by squaring the new amplitude, which is 130% of the original, and multiplying by the original intensity. Intensity is proportional to the square of the amplitude in the medium where the sound wave travels.
Step-by-step explanation:
The measurement of a microphone diaphragm's response to an acoustic waveform involves understanding the changes in sound intensity relative to the pressure amplitude of the wave. When a microphone receives a pure sound tone with an initial intensity of 2.00 × 10−5 W/m², and the amplitude increases by 30%, the new intensity can be calculated by leveraging the relationship between intensity (I) and the square of the pressure amplitude (Ap).
According to the physics of sound, intensity is directly proportional to the square of the pressure amplitude. Mathematically, this relationship is represented as I ∝ Ap². Therefore, when the amplitude changes, the corresponding intensity change is given by (ΔI/I) = 2(ΔAp/Ap), where ΔI is the change in intensity, and ΔAp is the change in pressure amplitude.
In this scenario, the amplitude increases by 30%, implying a new amplitude of 130% of the original. Substituting this into the intensity change equation, we get (ΔI/I) = 2(0.30), which simplifies to ΔI/I = 0.60. This signifies a 60% increase in intensity.
Applying this increase to the initial intensity of 2.00 × 10−5 W/m², the new intensity (I') can be calculated as 2.00 × 10−5 W/m² multiplied by (1 + 0.30)², resulting in I' = 2.00 × 10−5 W/m² * 1.60², yielding the new intensity value.
In conclusion, understanding the relationship between intensity and the square of pressure amplitude allows for the precise calculation of changes in sound intensity when the amplitude is altered. In this case, a 30% increase in amplitude corresponds to a 60% increase in intensity, providing valuable insights into the mic diaphragm's response to varying acoustic waveforms.