## Treatise on materials science and technology, Volume 3 |

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Page 10

Actually the expression

volume t through the surface S. Therefore, theyth component of the energy flux

flowing outward from the volume t through S, i.e., in the direction in which the

energy flow associated with the elastic wave propagates is

52) It should be noted here that this energy propagates with the group velocity

which is always greater in magnitude than the phase velocity except in special

cases where the ...

Actually the expression

**given**in**Eq**. (51) represents the energy flow into thevolume t through the surface S. Therefore, theyth component of the energy flux

flowing outward from the volume t through S, i.e., in the direction in which the

energy flow associated with the elastic wave propagates is

**given**as Ej = -ayut. (52) It should be noted here that this energy propagates with the group velocity

which is always greater in magnitude than the phase velocity except in special

cases where the ...

Page 92

Again in this case the only nonvanishing displacement gradients are ua , u^, vb ,

and wc and therefore we may use the reduced equation of motion

245) as in the previous case. If we substitute the expressions from Eqs. (249),

properly differentiated, into Eq. (245) we obtain p0 Vt, = (2ai + X) + a (5A + 1 On +

2/ + 4m) + 2p (A + 21), (253) where again we have neglected terms of order a2 or

greater, and where we have set VLa = co/A:. (254) The subscripts L and a denote

...

Again in this case the only nonvanishing displacement gradients are ua , u^, vb ,

and wc and therefore we may use the reduced equation of motion

**given by Eq**. (245) as in the previous case. If we substitute the expressions from Eqs. (249),

properly differentiated, into Eq. (245) we obtain p0 Vt, = (2ai + X) + a (5A + 1 On +

2/ + 4m) + 2p (A + 21), (253) where again we have neglected terms of order a2 or

greater, and where we have set VLa = co/A:. (254) The subscripts L and a denote

...

Page 105

Nevertheless, if we attempt to propagate a pure mode longitudinal wave only,

Eqs. (312) reduce to the form of

If we rearrange

form used by Breazeale and Ford (750), namely p0 ii = K2 (uaa + 3ua Ğaa) + K3

ua uaa (3 1 6) where for the [1 1 1] direction K2 = a = i(c11+2c12 + 4c44) (317)

K3 = )8-3a = £(c111+6c,12+12c144 + 24c166 + 2c,23+16c456), using Brugger's

notation ...

Nevertheless, if we attempt to propagate a pure mode longitudinal wave only,

Eqs. (312) reduce to the form of

**Eq**. (196)**given**by p0 u - auaa = Pua uaa . (315)If we rearrange

**Eq**. (315) and use Eqs. (285), we may rewrite**Eq**. (312) in theform used by Breazeale and Ford (750), namely p0 ii = K2 (uaa + 3ua Ğaa) + K3

ua uaa (3 1 6) where for the [1 1 1] direction K2 = a = i(c11+2c12 + 4c44) (317)

K3 = )8-3a = £(c111+6c,12+12c144 + 24c166 + 2c,23+16c456), using Brugger's

notation ...

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