It is important to understand the Lp is an RMS (Root Mean Squared) quantity and is therefore averaged within the constraints of an averaging time constant. These time constants are helpfully termed Slow, Fast and Impulse and are sometimes referred to as the time weighting.
The Lp therefore is not only affected by the frequency response weighting but by the time weighting of the instrument as well. Hence, by selecting Slow on the sound level meter a sharp rise in noise will be shown as a gradual rise in the Lp, alternatively a sharp fall in noise will be shown as a gradual decrease in the Lp.
By using the Fast weighting on the instrument in both of the above scenarios the Lp will rise and fall on a more rapid scale.
The Impulse response is not as commonly used as the above, it is used in situations of sharp impulsive noises, typically a piling operation may require an impulse time weighting. It has a very fast rise time and a very slow exponential fall time.
Peak responses are different from Slow, Fast and Impulse weightings, in that it is not an RMS quantity. It is simply the crest of the sound pressure level and it is shown as the highest peak achieved within the measuring period. The rise time of the peak is extremely sharp and it is not uncommon for the peak to be much higher than the Lp.