bei48482_FM

x

y

v

A

y = A cos ω(t–x/v)

Figure 5.1Waves in the xyplane traveling in thexdirection along a stretched string lying on the

xaxis.

a standing wave in a string fastened at both ends, and so on. All solutions must be

of the form

yFt (5.4)

where Fis any function that can be differentiated. The solutions F(tx) represent

waves traveling in thexdirection, and the solutions F(tx) represent waves trav-

eling in the xdirection.

Let us consider the wave equivalent of a “free particle,” which is a particle that is

not under the influence of any forces and therefore pursues a straight path at constant

speed. This wave is described by the general solution of Eq. (5.3) for undamped (that

is, constant amplitude A), monochromatic (constant angular frequency ) harmonic

waves in the xdirection, namely

yAei(tx) (5.5)

In this formula yis a complex quantity, with both real and imaginary parts.

Because

eicos i sin 

Eq. (5.5) can be written in the form

yA cos  t iA sin t (5.6)

Only the real part of Eq. (5.6) [which is the same as Eq. (3.5)] has significance in the case

of waves in a stretched string. There yrepresents the displacement of the string from its

normal position (Fig. 5.1), and the imaginary part of Eq. (5.6) is discarded as irrelevant.

Example 5.1

Verify that Eq. (5.5) is a solution of the wave equation.

Solution

The derivative of an exponential function euis

(eu)eu

The partial derivative of ywith respect to x(which means tis treated as a constant) from Eq. (5.5)

is therefore

 y

i



y

x

du

dx

d

dx

x



x



x



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