Why a Flute and a Tanpura Playing One Note Still Sound Different

A swing goes back and forth, a heart beats again and again, a plucked string vibrates hundreds of times a second. Each motion repeats itself, and the simplest repeating motion follows a sine curve.

That simple motion, simple harmonic motion, turns out to be the building block for springs, pendulums, sound and even light.

This part covers periodic and oscillatory motion, sine and cosine functions, the SHM equation, and writing equations for real situations.

**A motion that repeats itself after equal intervals of time is periodic; if the body also moves back and forth about a mean position, it is oscillatory; the repeat time is the period $T$, and the frequency is $f=\frac{1}{T}$ in hertz.

- Periodic but not oscillatory: the Earth spinning on its axis, a ceiling fan's blades going round
- Periodic and oscillatory: a swing, a pendulum, a vibrating string

Worked example 1 — a pendulum.** A pendulum makes $30$ oscillations in $45$ s.

$$T=\frac{45}{30}=1.5\text{ s}f=\frac{1}{1.5}\approx 0.67\text{ Hz}$$

Worked example 2 — a heartbeat. Suppose a heart beats $75$ times a minute.

$$T=\frac{60}{75}=0.8\text{ s}f=1.25\text{ Hz}$$

Worked example 3 — a fast vibration. A string vibrating at $440$ Hz has $T=\frac{1}{440}\approx 2.3\times 10^{-3}$ s.

An everyday example. A child on a park swing moves back and forth about the lowest point, completing each swing in almost the same time.

The substance. Every oscillation is periodic, but not every periodic motion is an oscillation.

**The functions $\sin \omega t$ and $\cos \omega t$ repeat every $T=\frac{2\pi }{\omega }$, so they describe periodic motion; a combination $A\sin \omega t+B\cos \omega t$ is a single sine wave, and any periodic function can be built by adding sines and cosines of suitable periods.

Combining two terms.**

$$A\sin \omega t+B\cos \omega t=D\sin (\omega t+\varphi )D=\sqrt{A^2+B^2}\tan \varphi =\frac{B}{A}$$

Worked example 1.

$$3\sin \omega t+4\cos \omega t=5\sin (\omega t+\varphi )\tan \varphi =\frac{4}{3}\Rightarrow \varphi \approx 53.1°$$

Worked example 2 — which are periodic?

- $\sin \omega t+\cos 2\omega t$ — periodic with period $\frac{2\pi }{\omega }$, the longer of the two periods
- ${\sin }^2\omega t=\frac{1-\cos 2\omega t}{2}$ — periodic with period $\frac{\pi }{\omega }$
- $e^{-\omega t}$ — not periodic, since it never repeats

An everyday example. A tanpura and a flute playing the same note give the same basic period, but each adds different higher sine waves, so they sound different.

The substance. Only a single sine or cosine is simple harmonic — sums with different periods are periodic but not SHM.

**Simple harmonic motion is oscillation whose displacement varies sinusoidally with time, $x(t)=A\cos (\omega t+\varphi )$, where $A$ is the amplitude, $\omega$ the angular frequency, $(\omega t+\varphi )$ the phase and $\varphi$ the phase constant.

- Amplitude $A$ — the largest displacement from the mean position
- Angular frequency** $\omega =\frac{2\pi }{T}=2\pi f$, in rad/s
- Phase $(\omega t+\varphi )$ — fixes both position and direction of motion at time $t$
- **Phase constant $\varphi$** — the phase at $t=0$

Worked example. $x=0.05\cos \left(4\pi t+\frac{\pi }{3}\right)$ m.

$$A=0.05\text{ m}\omega =4\pi \text{ rad/s}T=\frac{2\pi }{4\pi }=0.5\text{ s}f=2\text{ Hz}\varphi =\frac{\pi }{3}$$

**At $t=0$:** $x=0.05\cos \frac{\pi }{3}=0.025$ m.

**At $t=0.125$ s:** phase $=\frac{\pi }{2}+\frac{\pi }{3}=\frac{5\pi }{6}$, so $x=0.05\times (-0.866)\approx -0.043$ m.

An everyday example. A toy bobbing on a spring rises and falls through the same heights in a steady rhythm — close to simple harmonic motion.

The substance. Two moments with the same position can have different phases, because the body may be moving in opposite directions.

**Read off the amplitude and period to get $A$ and $\omega =\frac{2\pi }{T}$, then choose the phase constant $\varphi$ from where the body starts and which way it is moving at $t=0$.

Worked example 1 — starting at the positive extreme.** $A=4$ cm and $T=2$ s, so $\omega =\pi$ rad/s:

$$x=4\cos (\pi t)\text{ cm}$$

**Worked example 2 — starting at the mean position, moving towards $+x$.**

$$x=4\sin (\pi t)=4\cos \left(\pi t-\frac{\pi }{2}\right)\text{ cm}$$

**Worked example 3 — starting halfway, moving towards $+x$.** Put $x=4\cos (\pi t+\varphi )$ with $x(0)=2$ cm:

$$\cos \varphi =\frac{1}{2}\Rightarrow \varphi =\pm \frac{\pi }{3}$$

Moving towards $+x$ means the velocity $-A\omega \sin \varphi$ is positive, so $\sin \varphi <0$ and $\varphi =-\frac{\pi }{3}$:

$$x=4\cos \left(\pi t-\frac{\pi }{3}\right)\text{ cm}$$

It reaches the positive extreme when the phase becomes $0$, after $t=\frac{\pi /3}{\pi }=\frac{1}{3}$ s.

An everyday example. Starting a wall-clock pendulum by pulling it aside and letting go gives motion that begins at an extreme, just like example 1.

The substance. The sine and cosine forms describe the same motion — they differ only by a phase constant of $\frac{\pi }{2}$.