Wave Profile of Harmonic Wave

Theorem

Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.

Then the wave profile of $\phi$ can be expressed as:

$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$

Proof

By definition, a harmonic wave is a wave whose wave profile can be expressed as a sine curve.

By definition, a sine curve can be expressed in the form:

$\map \phi x = a \map \sin {\omega x + \epsilon}$

where $a$, $\omega$ and $\epsilon$ are arbitrary constants.

We select $\epsilon$ so as to set:

$\epsilon = \dfrac \pi 2$

We then have:

\(\ds \map \phi x\)

\(=\)

\(\ds a \map \sin {\omega x + \dfrac \pi 2}\)

\(\ds \)

\(=\)

\(\ds a \cos \omega x\)

Hence at time $t = 0$ the equation expressing a wave in the form of a sine curve is:

$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$

Hence the result by definition of wave profile.

$\blacksquare$

Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$