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Velocity of wave in different media
The velocity of mechanical wave depends on elasticity and inertia of the medium.
Velocity of a transverse wave along a stretched string
Let us consider a string fixed at one of its ends and tension be applied at the other end. When the string is plucked at a point, it begins to vibrate.
Consider a transverse wave proceeding from left to right in the form of a pulse when the string is plucked at a point as shown in Fig. . EF is the displaced position of the string at an instant of time. It forms an arc of a circle with O as centre and R as radius. The arc EF subtends an angle 2θ at O.
If m is the mass per unit length of the string and dx is the length of the arc EF, then the mass of the portion of the string is m dx.
∴ Centripetal force = ( m.dx.v2 ) / R
This force is along CO. To find the resultant of the tension T at the points E and F, we resolve T into two components Tcos θ and T sinθ.
T cosθ components acting perpendi- cular to CO are of equal in magnitude but opposite in direction, they cancel each other.
T sin θ components act parallel to CO. Therefore the resultant of the tensions acting at E and F is 2T sin θ. It is directed along CO. If θ is small, sinθ = θ and the resultant force due to tension is 2Tθ.
resultant force = 2Tθ
= 2T dx/2R
= T.dx/R ???.(2)
For the arc EF to be in equilibrium,
m.dx v2 / R = T.dx / R
v2 = T/m
Velocity of longitudinal waves in an elastic medium
Velocity of longitudinal waves in an elastic medium is
V= root(E/ρ ) ????(1)
where E is the modulus of elasticity, ρ is the density of the medium.
(i) In the case of a solid rod
v= root(q/p) ????..(2)
where q is the Young?s modulus of the material of the rod and ρ is the density of the rod.
(ii) In liquids, v = root(x/p) ??(3)
where k is the Bulk modulus and ρ is the density of the liquid.
Newton?s formula for the velocity of sound waves in air
Newton assumed that sound waves travel through air under isothermal conditions (i.e) temperature of the medium remains constant.
The change in pressure and volume obeys Boyle?s law.
PV = constant
Differentiating, P . dV + V .dP = 0
P. dV = ?V dP
P = -dP/(dV/V) = change in pressure / volume strain
P = k (Volume Elasticity
Therefore under isothermal condition, P = k
v= root(k/p) = root(P/p)
where P is the pressure of air and is the density of air. The above equation is known as Newton?s formula for the velocity of sound waves in a gas.
At NTP, P = 76 cm of mercury
= (0.76 ? 13.6 ? 103 ? 9.8) N m?2 = 1.293 kg m?3.
Velocity of sound in air at NTP is
v = 280 m s?1
The experimental value for the velocity of sound in air is 332 m s?1. But the theoretical value of 280 m s?1 is 15% less than the experimental value. This discrepancy could not be explained by Newton?s formula.
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