Physical Chemistry Third Edition

14.3 Classical Waves 637

We can find the speed of a traveling wave by following the motion of one of the nodes.

At timet0 there is a node in the wave function of Eq. (14.3-26) atx0. At timet

this node will be located at the point wherex−ct0. Thus

x(node)ct (14.3-27)

The node is moving toward the positive end of thexaxis with a speed equal toc.

From Eq. (14.3-4) the speed is determined by the tension forceTand the mass per unit

lengthρ:

c

√

T

ρ

(14.3-28)

Exercise 14.9

What change would have to be made in the mass per unit length to quadruple the speed of

propagation? What change would have to made in the tension force to double the speed of

propagation?

If the function of Eq. (14.3-26) is replaced by

z(x,t)Asin(κx+κct) (14.3-29)

the wave travels toward the negative end of thexaxis with speedc. This function

satisfies the same wave equation as the function shown in Eq. (14.3-26) and the speed

is the same.

Exercise 14.10

a.Show that the function of Eq. (14.3-29) satisfies Eq. (14.3-3).

b.Show that the speed of the wave is equal tocand that the wave moves to the left.

In one wavelength, the argument of theφfunction changes by 2πiftis fixed, so that

the same relationship occurs as in Eq. (14.3-21) for a standing wave:

κ

2 π

λ

(14.3-30)

The relationship between the frequency and the wavelength can be obtained by observ-

ing that in timet, the length of the wave “train” that passes a fixed point is

Lengthct (14.3-31)

wherecis the speed. The number of wavelengths in this wave train is equal to

Number

ct

λ

(14.3-32)

In timet, the number of oscillations is equal to

Numberνt (14.3-33)