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Work Associated with Unsteady Pressure
The unsteady pressure can be decomposed into contributions from the cosine and sine components of the complex mode shape. For a forward traveling wave
Preal — pcr + Psi Pimag — pci psr
Wup — К J (etc (Pci – Psr) + as (Pcr + Psi)) • ndA
A
Wup — к f (etcPci) • ndA + к J (asPsi) • ndA — к f (etcPsr) • ndA + к J (asPcr) • ndA
A A A A
Wup (forward traveling wave) — Wcc + Wss — Wcs + Wsc
As can be seen Wcc and Wss (diagonal terms) represent the work due to the unsteady pressure acting on the mode shape producing that unsteady pressure. The Wcs and Wsc terms (off-diagonal) represent the interaction terms. That is, the work caused by the unsteady pressure due to the sine mode acting upon the mode shape due to the cosine mode, and vice versa.
