Page 282 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 9.1 Vibrating String 269

Figure 9.1-1. String element in
lateral vibration.

In Fig. 9.1-1, a free-body diagram of an elementary length dx of the string is
shown. By assuming small deflections and slopes, the equation of motion in the
y-direction is

T{^e + ^ c b c ^ - T d = p c t ^
or

£. ^ (9.1-1)
dx T dt^
Because the slope of the string is d = dy/dx, the preceding equation reduces to
d^y
(9.1-2)
dx^ c" dt^
where c = ^T /p can be shown to be the velocity of wave propagation along the
string.
The general solution of Eq. (9.1-2) can be expressed in the form
y = F^{ct - x) + F^{ct + x) (9.1-3)
where F^ and F2 are arbitrary functions. Regardless of the type of function F, the
argument {ct ± x) upon differentiation leads to the equation

d^F _L ^ (9.1-4)
dx^ dt^
and hence the differential equation is satisfied.
Considering the component y = F^{ct —x), its value is determined by the

argument {ct —x) and hence by a range of values of t and x. For example, if

c = 10, the equation for y = Fj(lOO) is satisfied by i = 0, x = -100; t = 1,

jc = —90; / = 2, X = —80; and so forth. Therefore, the wave profile moves in the
positive x-direction with speed c. In a similar manner, we can show that F2{ct + x)
represents a wave moving toward the negative x-direction with speed c. We
therefore refer to c as the velocity of wave propagation.
One method of solving partial differential equations is that of separation of
variables. In this method, the solution is assumed in the form
y { x , t ) = Y( x ) G{ t ) ( 9.1-5)