In wave physics, acoustics, and acoustic engineering, velocity (denoted as u) and speed of propagation of a wave (denoted as c) refer to different aspects of wave motion.
Velocity (u) typically refers to the particle velocity or the velocity of individual particles or elements within the medium that is carrying the wave. It represents the speed and direction at which these particles are moving as the wave passes through them. In the context of acoustics, the particle velocity is often measured in meters per second (m/s).
On the other hand, the speed of propagation of a wave (c) refers to the speed at which the wave as a whole travels through the medium. It represents the rate at which the disturbance or energy associated with the wave is transmitted through the medium. The speed of propagation depends on the properties of the medium itself, such as its density and elasticity. In acoustics, the speed of sound in a medium is often denoted as c and is commonly measured in meters per second (m/s).
It's important to note that in many cases, the velocity of individual particles within the medium may not be the same as the speed of propagation of the wave. This is particularly true for waves in which the particles oscillate back and forth in a different manner than the wave propagates. For example, in a water wave, the particles move in a circular or elliptical motion as the wave moves forward. In such cases, the particle velocity (u) and the speed of propagation of the wave (c) can have different magnitudes and directions.