Physical Chemistry Third Edition

E Classical Mechanics 1269

String

xx

xx 1 ∆x

z

F 1

F 2

 1

 2

Figure E.1 The Position of a Flexible String and the Forces on a Segment of the

String.This arbitrary conformation of the string as a function of length along the string is

used to describe the forces on a small segment of the string.

α 1 andα 2. For small displacements, the net force on the string segment will lie in the

zdirection:

FzF 2 z+F 1 zTsin(α 2 )−Tsin(α 1 )≈T[tan(α 2 )−tan(α 1 )]

≈T

[(

∂z

∂x

)∣

∣

∣

∣

x′+∆x

−

(

∂z

∂x

)∣

∣

∣

∣

x′

]

(E-7)

where the subscripts on the derivatives denote the positions at which they are evaluated.

We have used the fact that the sine and the tangent are nearly equal for small angles,

which corresponds to our case of small displacements. We have also used the fact that

the first derivative is equal to the tangent of the angle between the horizontal and the

tangent line.

The mass of the string per unit length isρand the mass of the segment isρ∆x. Its

acceleration is (∂^2 z/∂t^2 ), so that from Newton’s second law

T

[(

∂z

∂x

)∣∣

∣

∣

x+∆x

−

(

∂z

∂x

)∣∣

∣

∣

x

]

ρ∆x

∂^2 z

∂t^2

(E-8)

We divide both sides of this equation byρ∆xand then take the limit as∆xis made to

approach zero. In this limit, the quotient of differences becomes a second derivative:

lim

∆x→ 0

⎡

⎢

⎢

⎢

⎣

(

∂z

∂x

)∣∣

∣

∣

x+∆x

−

(

∂z

∂x

)∣∣

∣

∣

x

∆x

⎤

⎥

⎥

⎥

⎦



∂^2 z

∂x^2

(E-9)

This equation is substituted into Eq. (E-8) to obtain the classical wave equation:

∂^2 z

∂x^2



1

c^2

∂^2 z

∂t^2

(E-10)

wherec^2 T/ρ. We show in Chapter 14 thatcis the speed of propagation of a traveling

wave in the string.