16.3 Wave speed on a stretched string  (Page 2/4)

Speed of compression waves in a fluid

The speed of a wave on a string depends on the square root of the tension divided by the mass per length, the linear density. In general, the speed of a wave through a medium depends on the elastic property of the medium and the inertial property of the medium.

The elastic property describes the tendency of the particles of the medium to return to their initial position when perturbed. The inertial property describes the tendency of the particle to resist changes in velocity.

The speed of a longitudinal wave through a liquid or gas depends on the density of the fluid and the bulk modulus of the fluid,

Here the bulk modulus is defined as ${\rm B}=-\frac{\text{Δ}P}{\raisebox{1ex}{$\text{Δ}V$}\!\left/ \!\raisebox{-1ex}{$V_0$}\right.},$ where $\text{Δ}P$ is the change in the pressure and the denominator is the ratio of the change in volume to the initial volume, and $\rho \equiv \frac{m}{V}$ is the mass per unit volume. For example, sound is a mechanical wave that travels through a fluid or a solid. The speed of sound in air with an atmospheric pressure of $1.013\times {10}^5\text{Pa}$ and a temperature of $20\text{°}\text{C}$ is $v_{\text{s}}\approx 343.00\text{m/s}.$ Because the density depends on temperature, the speed of sound in air depends on the temperature of the air. This will be discussed in detail in Sound .

Summary

• The speed of a wave on a string depends on the linear density of the string and the tension in the string. The linear density is mass per unit length of the string.
• In general, the speed of a wave depends on the square root of the ratio of the elastic property to the inertial property of the medium.
• The speed of a wave through a fluid is equal to the square root of the ratio of the bulk modulus of the fluid to the density of the fluid.
• The speed of sound through air at $T=20\text{°}\text{C}$ is approximately $v_{\text{s}}=343.00\text{m/s}.$

Conceptual questions

Shown below are three waves that were sent down a string at different times. The tension in the string remains constant. (a) Rank the waves from the smallest wavelength to the largest wavelength. (b) Rank the waves from the lowest frequency to the highest frequency.

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