Harmonic wave function

The simplest wave is the harmonic wave: its profile is a cosine function, the same smooth, periodic curve as simple harmonic motion, but “frozen” in space. Every point of the medium performs a harmonic oscillation about its own equilibrium position; neighbouring points oscillate slightly out of phase with one another, and this growing phase shift along space is what gives the wave its sinuous shape and makes it appear to move.

Harmonic wave: $A$ = amplitude, $\lambda$ = wavelength, $v$ = propagation speed.

The curve is described by a wave function $\phi(x,t)$ that depends both on position $x$ and on time $t$: fix an instant, and it gives the “snapshot” of the wave profile; fix a point, and it gives the oscillation in time of that point of the medium.

Key formula — Harmonic wave function

$\ev{\phi(x,t) = A\cos(kx - \omega t + \varphi_0)}$

The argument of the cosine, the phase $kx - \omega t + \varphi_0$, contains all the information. Let’s look at the meaning of each quantity.

The quantities of the harmonic wave

• $A$ = amplitude: maximum displacement from the equilibrium position.
• $\lambda$ = wavelength: spatial distance after which the profile repeats identically.
• $T$ = period: time after which the oscillation of a point repeats identically.
• $f$ = frequency: number of oscillations per second, $f = 1/T$.
• $k = 2\pi/\lambda$ = wave number: how many radians of phase “fit” into every metre of space.
• $\omega = 2\pi/T = 2\pi f$ = angular frequency: how many radians of phase elapse every second.
• $\varphi_0$ = initial phase: fixes the value of the wave at $x=0$, $t=0$.

The two factors of $2\pi$ are not a whim: they serve to turn a “geometric” repetition (a wavelength, a period) into the $2\pi$ radians the cosine needs to complete a full cycle. So, advancing by one wavelength $\lambda$, the spatial phase $kx$ grows by exactly $2\pi$, and the passing of one period $T$ makes the temporal phase $\omega t$ grow by $2\pi$: in both cases the cosine returns to its starting value, as is fitting for a periodic phenomenon.

Links

Topics: Onde Concepts: Onda armonica · Periodo e frequenza · Moto armonico semplice

Related exercises: Problema — Periodo e frequenza massa-molla · Problema — Frequenza propria di un grattacielo · Problema — Ranking di molle e masse