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16.3 Wave speed on a stretched string  (Page 3/4)

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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.

| v | = elastic property inertial property

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,

v = Β ρ .

Here the bulk modulus is defined as Β = − Δ P Δ V V 0 , where Δ P is the change in the pressure and the denominator is the ratio of the change in volume to the initial volume, and ρ ≡ 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 × 10 5 Pa and a temperature of 20 ° C is v s ≈ 343.00 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 ° C is approximately v s = 343.00 m/s .

Conceptual questions

If the tension in a string were increased by a factor of four, by what factor would the wave speed of a wave on the string increase?

The wave speed is proportional to the square root of the tension, so the speed is doubled.

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Does a sound wave move faster in seawater or fresh water, if both the sea water and fresh water are at the same temperature and the sound wave moves near the surface? ( ρ w ≈ 1000 kg m 3 , ρ s ≈ 1030 kg m 3 , B w = 2.15 × 10 9 Pa , B s = 2.34 × 10 9 Pa )

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Guitars have strings of different linear mass density. If the lowest density string and the highest density string are under the same tension, which string would support waves with the higher wave speed?

Since the speed of a wave on a string is inversely proportional to the square root of the linear mass density, the speed would be higher in the low linear mass density of the string.

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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.

Figure shows three waves labeled A, B and C on the same graph. All have their equilibrium positions on the x axis. Wave A has amplitude of 4 units. It has crests at x = 1.5 and x = 7.5. Wave B has amplitude of 3 units. It has a crest at x = 2 and a trough at x = 6. Wave C has amplitude of 2 units. It has crests at x = 1 and x = 5.
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Electrical power lines connected by two utility poles are sometimes heard to hum when driven into oscillation by the wind. The speed of the waves on the power lines depend on the tension. What provides the tension in the power lines?

The tension in the wire is due to the weight of the electrical power cable.

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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