ABSTRACT
Tω πν π ω= = but a different phase constant, since ( )xε is a linear function of position. Because the difference between the phase constants at any two given points does not vary with time, it is convenient to define the wavelength λ as
22 vvT k πλ π ω= = = (16.10)
and this gives the distance between two successive points whose oscillations are in phase. From Eqs.(16.10) it follows that k is the number of waves in a 2π distance, known as the wave number. Expanding the cosine in Eq.(16.4) gives a linear combination of two waves in quadrature
( ) ( ) ( )vtxkAvtxkAtx ±+±=Ψ sincos, 21 (16.11) where the information about the amplitude and the phase constant of the wave is expressed by means of two new constants
i.e. , (16.12)
sin tan AA A A
ϕ ϕ= − = − Equation (16.11) can be generalized in the form
(16.13) ( ) ( ϕ+ω±=ϕ+ω±∑ =
which states that any linear combination of harmonic waves, having the same frequency ω and wave number k but arbitrary amplitudes and phase constants jA jϕ reduces to a harmonic wave of definite amplitude A given by
A A A A )lϕ ϕ = > =
= + −∑ ∑∑ (16.14) and phase constant ϕ of the form
sin tan
cos
A
A
jϕϕ ϕ =
The formulation of wave theory is essentially simplified by means of the complex linear combination of two waves in quadrature, having the same amplitude and frequency, which reads ( ) ( ) ( ) ( ) ( ), cos sin Ai kx t i kx tx t A kx t iA kx t Ae eω ϕω ϕ ω ϕ ± + ±Ψ = ± + + ± + = = ω (16.16) We shall call A the complex amplitude of the wave A iAe ϕ= (16.17) of modulus equal to the amplitude A and argument equal to the phase constant .ϕ The complex representation (16.16) can be easily differentiated and integrated, hence providing a convenient way of handling harmonic waves. The real part of the complex function (16.16) has the standard form (16.5) of a harmonic wave
( ) ( )Re A Rei kx t i kx tie Ae eω ωϕ± ±⎡ ⎤ ⎡ ⎤⎦ ( ) ( )Re cosi kx tA e A kx tω ϕ=⎣ ⎦ ⎣ ω ϕ± +⎡ ⎤= = ±⎣ ⎦ +
which leads to the following rule:
A wave may be represented in an equation by a complex function, in which case the real part of the complex function is to be taken, provided the operations are restricted to addition, subtraction, multiplication or division by a real quantity and differentiation or integration with respect to a real variable.
