For longitudinal sound waves, why is the pressure a maximum when the displacement of the particles is zero? I can't quite wrap my head around that...please help?
To understand why the pressure is maximum when the displacement of particles is zero in a longitudinal sound wave, let's break it down step by step.
A longitudinal sound wave consists of compressions and rarefactions traveling through a medium, such as air. In a compression, the particles of the medium are densely packed together, while in a rarefaction, the particles are spread out.
When the sound wave reaches a particular point in the medium, the particles start to vibrate or oscillate back and forth along the direction the wave is traveling. These vibrations cause variations in the density of the medium, resulting in areas of high and low pressure.
Now, let's consider the case when the particles are at their equilibrium position—in other words, when they are not displaced from their rest or initial position. At this point, the particles are neither compressed nor rarefied; they are at the average density of the medium. Consequently, the pressure is also at its average value.
As the particles start to move away from their equilibrium position, either in the direction of compression or rarefaction, they experience a displacement. When the particles move towards compression (increasing density) or rarefaction (decreasing density), their displacement increases, causing a change in pressure from the average value.
When the particles are displaced the most—either at the maximum positive displacement (going towards compression) or at the maximum negative displacement (going towards rarefaction)—the change in density is at its maximum. This change in density corresponds to the extreme variation in pressure.
However, when the particles reach their maximum displacement, they momentarily come to rest before reversing direction. This brief moment when the particles are momentarily at rest corresponds to zero displacement. Since the particles are no longer moving, the density returns to its average value, resulting in a maximum pressure at that point.
In summary, at the point of zero displacement, the particles are momentarily at rest, and thus the density (and consequently the pressure) reaches its maximum value. As the particles continue to move away from this point, the displacement increases again, leading to variations in pressure until reaching another point of zero displacement.
To visualize this phenomenon, imagine a spring oscillating back and forth. The compression and rarefaction regions of a sound wave can be represented as the compressed and expanded states of the spring. At the extreme points of compression and rarefaction, the particles stop momentarily before changing direction. This momentary rest correspondingly leads to a maximum pressure.