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WAVES 285 the speed; Eq. (14.12) then relates wavelength to arising due to an external force). It does not frequency for the given speed. Of course, as depend on wavelength or frequency of the wave remarked earlier, the medium can support both itself. In higher studies, you will come across transverseandlongitudinalwaves,whichwillhave waves whose speed is not independent of differentspeedsinthesamemedium.Laterinthis frequency of the wave. Of the two parameters λ chapter, we shall obtain specific expressions for and ν the source of disturbance determines the the speed of mechanical waves in some media. frequency of the wave generated. Given the speed of the wave in the medium and the 14.4.1 Speed of a Transverse Wave on frequency Eq. (14.12) then fixes the wavelength Stretched String The speed of a mechanical wave is determined v λ = (14.15) by the restoring force setup in the medium when ν it is disturbed and the inertial properties (mass density) of the medium. The speed is expected to Example 14.3 A steel wire 0.72 m long be directly related to the former and inversely to has a mass of 5.0 ×10 –3kg. If the wire is under a tension of 60 N, what is the speed the latter. For waves on a string, the restoring of transverse waves on the wire ? force is provided by the tension T in the string. The inertial property will in this case be linear mass density µ, which is mass m of the string Answer Mass per unit length of the wire, divided by its length L. Using Newton’s Laws of 5.0×10 −3 kg Motion, an exact formula for the wave speed on µ = a string can be derived, but this derivation is 0.72m outside the scope of this book. We shall, = 6.9 ×10 kg m –1 therefore, use dimensional analysis. We already know that dimensional analysis alone can never Tension, T = 60 N yield the exact formula. The overall The speed of wave on the wire is given by dimensionless constant is always left T 60N −1 undetermined by dimensional analysis. v = µ = −3 −1 = 93ms ⊳ The dimension of µ is [ML ] and that of T is 6.9×10 kg m like force, namely [MLT ]. We need to combine these dimensions to get the dimension of speed 14.4.2 Speed of a Longitudinal Wave –1 (Speed of Sound) v [LT ]. Simple inspection shows that the quantity T/µ has the relevant dimension In a longitudinal wave, the constituents of the medium oscillate forward and backward in the MLT −2 2 −2 =  T  direction of propagation of the wave. We have [ML ] already seen that the sound waves travel in the form of compressions and rarefactions of small Thus if T and µ are assumed to be the only volume elements of air. The elastic property that relevant physical quantities, determines the stress under compressional T strainisthebulkmodulusofthemediumdefined v = C (14.13) by (see Chapter 8) where C is the undetermined constant of P B = − V/V (14.16) dimensional analysis. In the exact formula, it turms out, C=1. The speed of transverse waves Here, the change in pressure ∆P produces a on a stretched string is given by V volumetric strain . Bhasthesamedimension V v = T (14.14) as pressure and given in SI units in terms of µ pascal (Pa). The inertial property relevant for the Note the important point that the speed v propagation of wa–3 is the mass density ρ, with depends only on the properties of the medium T dimensions [ML ]. Simple inspection reveals and µ (T is a property of the stretched string that quantity B/ ρ has the relevant dimension: Reprint 2026-27