The velocity of mechanical wave depends on elasticity and inertia of the medium.
Velocity of wave in different media :
1.Velocity of a transverse wave along a stretched string
2.Velocity of longitudinal waves in an elastic medium
3.Newton?s formula for the velocity of sound waves in air

*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.v^{2} ) / 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
v^{2} / R = T.dx / R

v^{2}
= T/m

v=root(T/m) ???.(3)

**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 ? 10^{3}
? 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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