5. Phase and Phase Velocity

Consider a harmonic wave function $\psi (x,t)=A\sin (kx-\omega t)$, where A,k and ω are real constants. A is the amplitude of the wave, $k={}^{2\pi }/{}_{\lambda }$  is the propagation wave number that denotes the number of wavelengths at a distance 2π, and $\omega ={}^{2\pi }/{}_{\tau }$  is the angular temporal frequency that gives the number of period in duration 2π. From Equation (4.5), the wavelength λ is given by:

$\lambda =\frac{2\pi }{k}$                                                                  (5.1)

Since $\omega =\frac{2\pi }{\tau }$, the temporal period τ can be expressed in terms of ω as follows:

$\tau =\frac{2\pi }{\omega }$                                                                  (5.2)

The entire argument of the harmonic wave function $\psi (x,t)$ is the phase φ of the wave where:

$\varphi (x,t)=(kx-\omega t)$                                                        (5.3)

While holding x constant, the partial derivative of φ with respect to time t is given by:

$\left| {{\left( \frac{\partial \varphi }{\partial t} \right)}_{x}} \right|\equiv \omega $                                                       (5.4)

Equation 5.4 gives the rate of change of phase at any fixed location, which is the angular frequency ω of the wave.

Equivalently, the partial derivative of φ with respect to x while holding t constant is by definition the rate of change of phase with distance, which is by definition the propagation number k of the wave:

$\left| {{\left( \frac{\partial \varphi }{\partial x} \right)}_{t}} \right|\equiv k$                                                       (5.5)

Since the angular frequency ω and propagation number k, are positive constants, the absolute value are set between the partial derivatives of phase in Equation (5.4) and (5.5).

Combining both expressions of the rate of change of phase in Equation (5.4) and (5.5), we obtain the propagation speed at constant phase, i.e. the phase velocity of the wave:

${{\left( \frac{\partial x}{\partial t} \right)}_{\varphi }}=\frac{-{{\left( {}^{\partial \varphi }/{}_{\partial t} \right)}_{x}}}{{{\left( {}^{\partial \varphi }/{}_{\partial x} \right)}_{t}}}$                                                  (5.6)

For harmonic wave function $\psi (x,t)=A\sin (\omega t-kx)$, the phase velocity is given by:

${{\left( \frac{\partial x}{\partial t} \right)}_{\varphi }}=\pm \frac{\omega }{k}=\pm v$                                                    (5.7)

The phase velocity in Equation (5.7) is accompanied by a positive sign when the wave moves in the ‘+’ x-direction and a negative sign for the wave moves in the ‘–‘ x-direction.

The harmonic wave function $\psi (x,t)=A\sin (kx-\omega t)$ is the solution of one-dimensional wave equation given by Equation (3.5). By taking the second derivative of the harmonic wave function with respect to x, we obtain:

$\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{x}^{2}}}=-A{{k}^{2}}\sin (kx-\omega t)$                                             (5.8)

And by working out the term on the right hand side of Equation (3.5), it can be proven that the harmonic function $\psi (x,t)=A\sin (kx-\omega t)$  meets the one-dimensional wave Equation (3.5):

$\frac{1}{{{v}_{phase}}^{2}}\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{t}^{2}}}=\frac{1}{{}^{{{\omega }^{2}}}/{}_{{{k}^{2}}}}\frac{\partial }{\partial t}(-A\omega \cos (kx-\omega t))=-A{{k}^{2}}\sin (kx-\omega t)=\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{x}^{2}}}$           (5.9)