Physical Chemistry Third Edition

14.3 Classical Waves 631

wave nature of light was established experimentally when interference and diffraction

of light were observed.

The Flexible String

In order to illustrate the mathematics of classical waves, we now analyze the vibrations

of a flexible string, which is a model system designed to resemble a real vibrating

string. It is defined to have the following properties:

1. The string is uniform. All parts have the same mass per unit length, denoted byρ.


2. There is a tension force of magnitudeTpulling at each end of the string.


3. The string is perfectly flexible.


4. There is no friction.


5. The string undergoes only small displacements, so that the total length of the
   string remains nearly constant and the magnitude of the tension forceT is nearly
   constant.

The state of the string at timetis specified by giving the displacement and velocity at

each point of the string.

zz(x,t) (14.3-1)

νzνz(x,t)

∂z

∂t

(14.3-2)

The displacement and velocity are functions ofxas well as functions oft. The derivative

∂z/∂tis a partial derivative, taken with a fixed value ofx. The classical equation of

motion of the flexible string is derived in Appendix E from Newton’s second law. From

Eq. (E-10) the wave equation is

∂^2 z

∂x^2



ρ

T

∂^2 z

∂t^2



1

c^2

∂^2 z

∂t^2

(14.3-3)

where we let

c^2 T/ρ (14.3-4)

Since Eq. (14.3-3) contains partial derivatives, it is called apartial differential

equation. Its solution giveszas a function ofxandt.

Let the string be fixed atx0 andxLso that nodes occur at these locations.

We assume that at timet0 the string is displaced into some shape in thex−zplane

and released to vibrate freely in this plane. We obtain a solution of Eq. (14.3-3) by

separation of variables. We assume atrial solutionthat is a product of a function ofx

and a function oft:

z(x,t)φ(x)η(t) (14.3-5)

We substitute the trial solution into the differential equation and perform algebraic oper-

ations that produce an equation with terms that are functions of only one independent

variable. We substitute the trial function of Eq. (14.3-5) into Eq. (14.3-3):

η

d^2 φ

dx^2



1

c^2

φ

d^2 η

dt^2

(14.3-6)