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The problem revolves around the Sound Pressure Level (SPL) and how it changes over time, based on the given diagram. The SPL is measured under different conditions: fast and slow weighting.

### Problem Understanding:
1. The graph shows that the rms pressure of the sound meter changes from 0 Pa to 1 Pa suddenly at \( t = 0 \), then holds at 1 Pa for 1 second before dropping back to 0 Pa.
2. We are asked to calculate the SPL reported by the sound meter at \( t = 1.5 \) seconds under both fast and slow weighting, and compare the time when the SPL under fast weighting is 20 dB less than the SPL under slow weighting.

### Key Information:
- **SPL** is calculated using the following formula:
  \[
  \text{SPL (in dB)} = 20 \log_{10} \left( \frac{p_{\text{rms}}}{p_{\text{ref}}} \right)
  \]
  Where:
  - \( p_{\text{rms}} \) is the root mean square sound pressure.
  - \( p_{\text{ref}} \) is the reference sound pressure (usually \( 20 \mu Pa \) in air).
  
- **Fast and Slow Weighting** are different time-averaging methods:
  - **Fast weighting** has a time constant of 125 milliseconds.
  - **Slow weighting** has a time constant of 1 second.

### Part a: SPL at \( t = 1.5 \) seconds under fast and slow weighting
- The pressure drops to 0 Pa at \( t = 1 \) second. Thus, at \( t = 1.5 \) seconds, the pressure is 0 Pa.
- However, due to the different time constants of fast and slow weighting, the SPL at \( t = 1.5 \) seconds will depend on how the sound meter averages the change in pressure over time.

#### Fast Weighting Calculation:
- Fast weighting responds more quickly (125 ms). By \( t = 1.5 \), the pressure has been at 0 Pa for 0.5 seconds, which is a sufficient time for the fast weighting to effectively measure the pressure close to 0 Pa. Thus, the SPL under fast weighting will be near 0 dB at \( t = 1.5 \) seconds.

#### Slow Weighting Calculation:
- Slow weighting averages the pressure over a longer period (1 second). At \( t = 1.5 \) seconds, slow weighting is still averaging the pressure change over the previous 1 second (from \( t = 0.5 \) to \( t = 1.5 \)). Since the pressure was 1 Pa for part of that period, the SPL will be higher than for fast weighting.

### Part b: Time when fast weighting is 20 dB less than slow weighting
- To find the time when the fast weighting SPL is 20 dB lower than the slow weighting SPL, we need to compare how the fast and slow weightings respond to the sudden drop in pressure at \( t = 1 \) second. 
- Since the fast weighting responds quickly, its SPL will decrease faster than the slow weighting.
- The exact time when the difference is 20 dB requires solving the differential equations governing the exponential decay of the SPL for each weighting, but typically this occurs within a fraction of a second after the drop (likely in the range of 1.2 to 1.3 seconds).

Would you like me to go into more detail about the equations or assumptions made in the process?

### Follow-up Questions:
1. How do fast and slow weightings mathematically model sound pressure changes?
2. What is the exact SPL at \( t = 1.5 \) seconds under slow weighting?
3. How does the time constant influence the decay rate of SPL for fast and slow weightings?
4. Can we approximate the SPL difference between fast and slow weighting after the pressure drop analytically?
5. How does the SPL calculation change if the pressure changes linearly instead of suddenly?

**Tip:** SPL weighting constants are crucial when measuring sound in environments with quick changes in pressure.

The rms pressure at the location of a sound meter in air changes suddenly from 0 Pa to 1 Pa at time = 0 sec before returning to 0 Pa 1 second later. What is the SPL reported by the sound meter at t=1.5 seconds under both fast and slow weighting? At what time is the SPL measured under fast weighting 20 dB less than the SPL measured under slow weighting for the same time point?

This problem involves calculating the Sound Pressure Level (SPL) of a sudden pressure change from 0 Pa to 1 Pa under fast and slow weighting. Learn how to calculate SPL using logarithmic formulas and understand the differences between fast and slow response times when handling pressure changes. Suitable for university-level students studying physics or acoustics.

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