4.2 COMPLEX NUMBER NOTATION

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In working with the wave equation and its solution, it is convenient to use

the complex number notation, because we are often interested in simple

harmonic waves (sinusoidal waves). Even if the wave is not simple harmonic,

the waveform may be expanded in a Fourier series, which involves a

series of sinusoidal terms. In addition, the complex notation provides information

about both the magnitude of a quantity and its phase angle in

compact form.

A complex number may be written in cartesian form:

z ј x ю jy ј Re ю jIm (4-21)

where x ј Re ј ‘‘real’’ and y ј Im ј ‘‘imaginary’’ part of the complex

quantity. We have used the symbol, j ј

ffiffiffiffiffiffiffi

p_1, because the symbol i is

often used to represent electric current. The complex quantity may also be

written in polar form:

z ј jzj e j           (4-22)

where jzj is the magnitude of the quantity and    is the phase angle. We may

convert from one form to the other by using the following relations:

jzj ј рRe2 ю Im2Ю1=2 (4-23)

tan        ј Im=Re (4-24)

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Copyright © 2003 Marcel Dekker, Inc.

Re ј jzj cos        (4-25)

Im ј jzj sin         (4-26)

Mathematical manipulations (derivatives, integrations, etc.) are somewhat

simpler to handle in terms of exponentials, because we do not need to

utilize the trigonometric identities. If we have a sinusoidal expression,

yрtЮ ј Y cos !t ј Y cosр2_ftЮ (4-27)

then, we may also write:

yрtЮ ј Y ej!t ј Yрcos !t ю j sin!tЮ ј Reр yЮ ю j Imр yЮ (4-28)

Equations (4-27) and (4-28) are identical if we adopt the convention that

only the ‘‘real’’ part of the complex expression is representative of the ‘‘real’’

physical quantity. This procedure is illustrated by an example in the following

section.

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