SPH3U Resonance, Harmonics, Intensity and Hearing: Notes and Practice

Send a wave down a string fixed at both ends and it reflects, then interferes with the wave still arriving. At most frequencies the result is a mess. At certain special frequencies the two line up perfectly and produce a standing wave — a pattern that appears to stay still, with points that never move and points that swing wildly.

Nodes and antinodes

A node is a point of zero displacement, where the two waves always cancel. An antinode is a point of maximum displacement, where they always reinforce. Adjacent nodes are half a wavelength apart, which is the fact every length calculation in this topic rests on.

Resonance

Resonance is what happens when a system is driven at one of its natural frequencies: the energy accumulates cycle after cycle and the amplitude grows dramatically. A child on a swing pushed at exactly the right moment each time goes higher and higher; pushed at random they barely move at all.

A wine glass shattered by a sustained note at its natural frequency.
A radio tuner, which resonates at one chosen frequency and ignores every other station.
Bridges and buildings, which engineers deliberately design so that their natural frequencies avoid wind and footfall rhythms.
Every musical instrument, which works by resonating a string or an air column at a chosen set of frequencies.

Standing waves in strings and open pipes

Fig 1Each harmonic fits a whole number of half-wavelengths between the two ends.

A string fixed at both ends must have a node at each end. The longest wave that fits is one where the string holds exactly half a wavelength. Fit two halves and you get the second harmonic, three halves the third, and so on. The same arithmetic applies to a pipe open at both ends, which must have an antinode at each end.

l_{n} = \frac{nλ}{2} f_{n} = n f_{0}

For strings and open pipes, with n = 1, 2, 3, … — every whole number is allowed.

Closed pipes are different

Fig 2A closed end forces a node; an open end forces an antinode. That is what removes the even harmonics.

A pipe closed at one end must have a node at the closed end and an antinode at the open end. The shortest pattern that manages both is a quarter of a wavelength, not a half. The consequence is striking: only the odd harmonics exist at all.

l_{n} = \frac{( 2 n - 1 ) λ}{4} f_{n} = (2 n - 1) f_{0}

For closed pipes. The (2n − 1) factor generates 1, 3, 5, 7 … — odd numbers only.

AString or open pipe

Same boundary condition at both ends
$l_{n} = \frac{nλ}{2}$
Harmonics: $f_{0}$ , $2 f_{0}$ , $3 f_{0}$ , $4 f_{0}$ , …
All harmonics present
Fundamental fits half a wavelength

BClosed pipe

Node at the closed end, antinode at the open end
$l_{n} = \frac{( 2 n - 1 ) λ}{4}$
Harmonics: $f_{0}$ , $3 f_{0}$ , $5 f_{0}$ , $7 f_{0}$ , …
Odd harmonics only
Fundamental fits a quarter wavelength

A closed pipe sounds an octave lower

Because the fundamental fits only a quarter wavelength instead of a half, a closed pipe of a given length produces a note one octave below an open pipe of the same length. Organ builders exploit this constantly — a stopped pipe gives a deep note from half the physical length, which is why stopped ranks are used where height is limited.

Intensity: how sound spreads out

Fig 3The same energy spread over a sphere that grows as the square of the radius.

Sound from a point source spreads over an expanding sphere. Double the distance and the same energy covers four times the area, so the intensity — power per unit area — falls to a quarter. This is the inverse square law.

\frac{I _{2}}{I _{1}} = \frac{r _{1}^{2}}{r _{2}^{2}} I \propto \frac{1}{d ^{2}}

Note the ratio is inverted: the subscripts on the radii are the other way round from those on the intensities.

Watch which way round the ratio goes

Because intensity decreases as distance increases,

I_{2} / I_{1}

equals

r_{1}^{2} / r_{2}^{2}

and not

r_{2}^{2} / r_{1}^{2}

. Getting it backwards turns a quarter into a factor of four. The check is always physical: if you moved further away, the intensity must come out smaller.

The decibel scale

Human hearing spans an enormous range of intensities — the loudest tolerable sound is around a trillion times more intense than the quietest audible one. A linear scale is useless over that range, so sound is measured logarithmically in decibels.

d B = 10 lo g (\frac{I _{2}}{I _{1}})

A logarithm base 10, comparing two intensities.

Fig 4Each 10 dB step is ten times the intensity, and sounds roughly twice as loud.

Change in intensity	Change in dB	How it sounds
$\times 2$	$+ 3 dB$	Just noticeably louder
$\times 10$	$+ 10 dB$	Roughly twice as loud
$\times 100$	$+ 20 dB$	About four times as loud
$\times 1000$	$+ 30 dB$	About eight times as loud

Sound	Level	Safe exposure
Rustling leaves	$20 dB$	Unlimited
Normal conversation	$60 dB$	Unlimited
Busy traffic	$80 dB$	Unlimited
Power tools, loud headphones	$100 dB$	About 15 minutes
Rock concert, chainsaw	$110 dB$	Under 2 minutes
Jet engine close up	$140 dB$	Immediate damage

Hearing damage is permanent and cumulative

Loud sound destroys the hair cells in the cochlea, and they do not grow back. Damage depends on both level and duration, which is why safe exposure time drops sharply as the level rises. Ringing ears after a concert is a sign that damage has already occurred — the standard advice is to keep headphones below 60% of maximum and take breaks.

Resonance, Harmonics, Intensity and Hearing

Key things to remember

Resonance

Standing waves in strings and open pipes

Closed pipes are different

Intensity: how sound spreads out

The decibel scale

Try it yourself

Worked examples

Where marks get lost

Flashcards 11

Practice