16.3.3 The Velocity of a Particle in the Medium

As a sinusoidal wave travels along the x-axis with speed v, the particles of the medium oscillate back and forth in SHM. For a transverse wave, such as a wave on a string, the oscillation is in the y-direction. For a longitudinal sound wave, the particles oscillate in the x-direction, parallel to the propagation. We can use the displacement equation, Equation 16.14, to find the velocity of a particle in the medium.

At time t, the displacement of a particle in the medium at position x is

D (x, t) = A sin (k x - ω t + ϕ_{0})

(16.16)

The velocity of the particle—which is not the same as the velocity of the wave along the string—is the time derivative of Equation 16.16:

v_{particle} = \frac{d D}{d t} = - ω A cos (k x - ω t + ϕ_{0})

(16.17)

Thus the maximum speed of particles in the medium is $v_{max} = ω A$ . This is the same result we found for simple harmonic motion because the motion of the medium is simple harmonic motion. FIGURE 16.16 shows velocity vectors of the particles at different points along a string as a sinusoidal wave moves from left to right.

The figure illustrates a snapshot graph of a wave on a string with vectors showing the velocity of the string at various points.

FIGURE 16.16 Full Alternative Text

NOTE Creating a wave of larger amplitude increases the speed of particles in the medium, but it does not change the speed of the wave through the medium.