A wave disturbance is created by a source. The source of a wave might be a rock thrown into water, your hand plucking a stretched string, or an oscillating loudspeaker cone pushing on the air. Once created, the disturbance travels outward through the medium at the wave speed v. This is the speed with which a ripple moves across the water or a pulse travels down a string.

As an example, we’ll prove in Section 16.4 that the wave speed on a string stretched with tension $T_{\text{s}}$ is

where $\mu$ is the string’s linear density, its mass-to-length ratio:

The SI unit of linear density is $\text{kg}\text{/}\text{m}$. A fat string has a larger value of $\mu$ than a skinny string made of the same material. Similarly, a steel wire has a larger value of $\mu$ than a plastic string of the same diameter. We’ll assume that strings are uniform, meaning the linear density is the same everywhere along the length of the string.

Equation 16.1 is the wave speed, not the wave velocity, so $v_{\text{string}}$ always has a positive value. Every point on a wave travels with this speed. You can increase the wave speed either by increasing the string’s tension (make it tighter) or by decreasing the string’s linear density (make it skinnier).

EXAMPLE 16.1 Measuring the linear density

A 2.00-m-long metal wire is attached to a motion sensor, stretched horizontally to a pulley 1.50 m away, then connected to a 2.00 kg hanging mass that provides tension. A mechanical pick plucks a horizontal segment of wire right at the pulley, creating a small wave pulse that travels along the wire. The plucking motion starts a timer that is stopped by the motion sensor when the pulse reaches the end of the wire. What is the wire’s linear density if the pulse takes 18.0 ms to travel the length of the wire?

MODEL Model the pulse as a traveling wave and the pulley as frictionless. The wave speed depends on the wire’s linear density.

VISUALIZE FIGURE 16.2 is a pictorial representation. The free-body diagram is for the hanging mass.

FIGURE 16.2 A wave pulse on a wire.

FIGURE 16.2 Full Alternative Text

SOLVE The wave speed on a wire is determined by the wire’s linear density $\mu$ and tension $T_{\text{s}}$. The hanging mass is in equilibrium, with no net force, so we see from the free-body diagram that the tension throughout the wire (because the pulley is frictionless) is $T_{\text{s}}=F_{\text{G}}=mg=19.6\text{N}$. Because the wave pulse travels 1.50 m in 18.0 ms, its speed is

$$v={\displaystyle \frac{1.50\text{m}}{0.0180\text{s}}}=83.3\text{m}\text{/}\text{s}$$

Thus, from Equation 16.1, the wire’s linear density is

$$\mu ={\displaystyle \frac{T_{\text{s}}}{v^2}}={\displaystyle \frac{19.6\text{N}}{(83.3{\text{m/s)}}^2}}=2.82\times {10}^{-3}\text{kg/m}=2.82\text{g}\text{/}\text{m}$$

Linear densities of strings are often stated in g/m, although these are not SI units. You must use kg/m in any calculations.

REVIEW A meter of thin wire is likely to have a mass of a few grams, so a linear density of a few g/m is reasonable. Note that the total length of the wire was not relevant.

The wave speed on a string is a property of the string—its tension and linear density. In general, the wave speed is a property of the medium. The wave speed depends on the restoring forces within the medium but not at all on the shape or size of the pulse, how the pulse was generated, or how far it has traveled.