Physical Chemistry Third Edition

14.3 Classical Waves 633

whereB,D,F, andGare constants. The product of the coordinate factorφand the

time factorηis a function that represents the displacement of the string as a function

of position and time. We callz(x,t)awave function. Our present wave function is not

necessarily a general solution to the original wave equation, since solutions can exist

that are not the product of two factors.

Exercise 14.5

Show that the product of the factors in Eq. (14.3-12) and (14.3-13) satisfies Eq. (14.3-3).

The wave functionz(x,t) does not yet apply to our case thatzvanishes atx0 and

atxL. We call this condition aboundary conditionsince it pertains to the ends of the

string. Ifz(x,t) vanishes at these points, the coordinate factorφ(x) must also vanish at

these points, since it contains all of thexdependence ofz. The condition thatφ(0) 0

requires thatB0, since sin(0)0 and cos(0)1. The sine function vanishes if its

argument is an integral multiple ofπ(90◦) so that the conditionφ(L)0 requires that

κLnπ (14.3-14)

wherenis an integer. We can write a different coordinate factor for each value ofn:

φ(x)φn(x)Dsin

(nπx

L

)

(14.3-15)

The coordinate factorφhas been determined partly by the differential equation and

partly by the boundary conditions.

The values of the constantsD,F, andGare now chosen to match our initial condi-

tions. A second-order differential equation requires two initial conditions. We choose as

the first initial condition that the string is passing through its equilibrium position (z 0

for allx) at timet0. This requires thatF0, since sin(0)0 and cos(0)1. The

time factorηis now determined by Eq. (14.3-14) and our initial conditions. The wave

function can now be written as

z(x,t)zn(x,t)φn(x)ηn(t)DGsin

(nπx

L

)

sin

(nπct

L

)

Asin

(nπx

L

)

sin

(nπct

L

)

(14.3-16)

Where we replace the constant productDGby a single symbol,A.

We now choose the second initial condition to be that the maximum value ofzis

equal toz 0 :

z(x,t)zn(x,t)z 0 sin

(nπx

L

)

sin

(nπct

L

)

(14.3-17)

This is a set of solutions, one for each value ofn. Each solution corresponds to a

different standing wave. The differential equation, the boundary conditions, and the

initial conditions have completely determined the set of wave functions.

The velocity is given by

νz

∂z

∂t

z 0

(nπc

L

)

sin

(nπx

L

)

cos

(nπct

L

)

(14.3-18)