Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 149

where the notations

c 11
r

c 66
r

c L ¼ and c S ¼ for ð L Þ for ð S Þ;

have been introduced. The solution of these equations is so very similar that only
the longitudinal case is considered. These are differential equations whose solutions
can be explicitly calculated. Consider the differential equation for the longitudinal
wave and let c L t. The second derivatives that appear in

the differential equation are obtained using the chain rule, thus

x ¼ x 1 +c L t and ¼ x 1

2 u 1
@

2 u 1
@x

2 u 1 2 u 1

@

2 u 1
@

2 u 1
@x

2 u 1 2 u 1

@Z

@

x 1 ¼ @

2 @

@x@ þ @

and @ c L @ 2 @

@x@Z þ @

2 þ

t 2 ¼

2

:

2 2

When these expressions are substituted into the differential equation for the
longitudinal wave it reduces to

2 u 1
@x@ ¼

@

0 ;

which has the solution

u 1 ð x 1 ; t Þ¼ u 1 ðx;Þ¼ p ðxÞþ q ðÞ¼ p ð x 1 þ c L t Þþ q ð x 1 c L t Þ:

c L t represents a translation of the coordinate
system in the + x 1 direction by the amount c L t . Since this translation is proportional
to the time, a point

The transformation x ¼ x 1 þ

c L t held constant means @ x 1

x ¼ x 1 þ

@t ¼

c L , thus x ¼ constant

moves in the x 1 direction with speed c L . A solution of the form u 1 ð x 1 ; t Þ¼ p ðxÞ

¼ p ð x 1 þ c L t Þ represents a wave traveling with velocity c L without changing its

shape. For example u 1 ( x 1 , t ) ¼ sin( x 1 þ c L t ) represents a sine wave traveling with

velocity c L . Similarly u 1 ð x 1 ; t Þ¼ q ðÞ¼ q ð x 1 c L t Þ represents a wave traveling

with velocity +

c L without changing its shape. Thus the solution u 1 ð

x 1 ; t Þ¼ u 1 ðx;Þ

¼ of the differential equation for the lon-

gitudinal wave is the sum of a wave traveling to the left with velocity

p ðxÞþ q ðÞ¼ p ð x 1 þ c L t Þþ q ð x 1 c L t Þ

c L and one

traveling to the right with velocity + c L . Since the two waves travel in opposite

directions, the shape of u 1 ( x 1 , t ) will in general change with time.
The initial-boundary value problem is composed of the differential equation for
the longitudinal wave and the initial conditions u 1 ð and @u 1

@

x 1 ; 0 Þ¼ f ð x 1 Þ ð x 1 ; 0 Þ¼

t

g . These initial conditions determine the form of the functions

p and q in the solution. From the solution u 1 ð

ð x 1 Þ for 0 < x 1 < 1

x 1 ; t Þ¼ p ð x 1 þ c L t Þþ q ð x 1 c L t Þ and

the chain rule it follows that

c L @

p
@x ð

c L @

q
@Z ð

p ð x 1 Þþ q ð x 1 Þ¼ f ð x 1 Þ; x 1 Þ x 1 Þ¼ g ð x 1 Þ for 0 < x 1 <1: