In a sound wave, we find a location and time when the density fluctuations are at a maximum.

At this same location and time ...

a) Pressure fluctuations and displacement are minimum.

b) Pressure fluctuations and displacement are maximum.

c) Pressure fluctuations are maximum and displacement is minimum.

d) Pressure fluctuations are minimum and displacement is maximum.

e) There is no relation between density fluctuations, pressure fluctuations and displacement.

To determine the relationship between density fluctuations, pressure fluctuations, and displacement at a location and time when the density fluctuations are at a maximum in a sound wave, we can refer to the properties of sound waves.

Sound waves are longitudinal waves, which means they propagate by the compression and rarefaction of particles in the medium through which the wave is traveling.

In a sound wave, the density fluctuations represent the variations in the density of the medium caused by compressions and rarefactions. When the density fluctuations are at a maximum, it means that the compression and rarefaction of the particles in the medium are at their highest point.

At this same location and time, pressure fluctuations are also at a maximum. The compressions and rarefactions in the medium directly affect the pressure exerted by the sound wave. When the density fluctuations are at their peak, it corresponds to the highest pressure fluctuations.

Displacement refers to the extent to which particles in the medium are displaced from their equilibrium positions due to the wave motion. However, at the location and time when the density fluctuations are at a maximum, the displacement is actually at its minimum. This is because the particles in the medium are most compressed or rarefied at this point, resulting in minimum displacement from their equilibrium positions.

Therefore, the correct answer is:

c) Pressure fluctuations are maximum, and displacement is minimum.