A sound wave traveling in one-dimension (along x):

In more general, sound travels in three dimensions, spreading out and losing intensity as it spreads.

If it is a periodic sound wave (i.e. musical tone), one still has vsound = fλ.

Notes that sounds that are pleasant together (i.e. make pleasant chords) are at frequencies that are at “small integer” ratios.

If f2: f1 = 2:1, we say that they are an octave apart. [In the scale do-re-mi-fa-sol-la-ti-do, the upper and lower “do’s” are an octave apart.]

If f2: f1 = 3:2, we say that they are at an interval of a “fifth”.[This corresponds to “do” and “sol” on the scale.]

If f2: f1 = 4:3, we say that they are at an interval of a “fourth”.[This corresponds to “do” and “fa” on the scale.]

If the wave (in 1D) reflects so the original wave meets an identical wave traveling in the opposite direction, the result is a standing wave.

x / l-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

l P /

l P0

-1.0

-0.5

0.0

0.5

1.0

Instead of traveling to the right or left, the pressure at each point oscillates at the wave frequency, with different amplitudes at each point: standing wave.

This illustrates a standing wave that is 4 wavelengths long: the wave reflects back and forth after hitting obstacles that are separated by 4λ. In this case, there are nodes (points that have zero oscillation) at the ends (and in this case 7 points in between). The points that have maximum amplitude oscillations are called antinodes.

antinodes

nodes

wave at t = 0, t = T, t = 2T, …

wave at t = T/2, t=3T/2, =5T/2, …

t = 0, T, 2T, ...

t = T/8, 7T/8, 9T/8, ...

l = l/4, 3l/4, 5l/4, ...

t = 3T/8, 5T/8,11T/8

t = T/2, 3T/2, 5T/2, ...

Here is a standing wave that is ½ wavelength long, with nodes at the ends.

x x

It’s also possible to have standing waves with antinodes at the ends; the length is still = λ/2.

Here is a standing wave with a node at one end and antinode at the other end. Now the length = λ/4.

x

t = 0, T, 2T, ...

t = T/8, 7T/8, 9T/8, ...

l = l/4, 3l/4, 5l/4, ...

t = 3T/8, 5T/8,11T/8

t = T/2, 3T/2, 5T/2, ...

If you pluck (or bow) a string, it will vibrate in a standing wave (e.g. traveling waves bouncing back and forth) with L = λ/2, with L = length of the string (i.e. for a guitar or violin, the distance between the bridge and your finger or the fret line).

Therefore, the frequency at which it will vibrate is f = vstring/λ = vstring/2L, where vstring is the speed of the wave in the string, which depends on the tension (T) in the string and its mass/length (ρ): vstring = √T/ρ. Therefore you can change the frequency by changing your finger position (how you play different notes) and by tightening or loosening the string (tuning the instrument).The vibrating string transfers some of its vibrational energy to the bridge which transfers it to the sound board (and body) of the instrument. This gets the air vibrating at that frequency and → sound.

λ = 2L: f = v/2L = ffun

λ = L: f = v/L = 2ffun

λ = 2L/3: f = 3v/2L = 3ffun

λ = L/2: f = 2v/L = 4ffun

Of course, other standing waves with nodes on both ends are possible. The lowest frequency wave is called the fundamental (ffun) and the others are called overtones. For the waves on a string, the overtones are all at integer multiples of the fundamental: fn = nffun . In this case, the overtones are called harmonics.

In general, all of these are excited (and contribute to the sound), depending on the design of the instrument and how it is plucked or bowed.

The quality of a sound of an instrument (including the voice) is called the timbre, and it is a function of relative intensity of the overtones. It is what makes different instruments sound different.

Note that the 2nd harmonic is one octave above the fundamental.The 2nd harmonic is at one octave above the fifth; i.e. if the fundamental is “do”, the 3rd harmonic is at the second “sol” .

The 3rd harmonic is 2 octaves above the fundamental.

Etcetera: the overtones sound nice together.

Fourier’s Theorem: Any waveform with period T = 1/f can be constructed from sums of sines f and its harmonics. (The nth harmonic of f is the wave with frequency = nf, where n = integer.)

x(t) = ∑[xn sin(2πnft + φn)]

f t = t / T0.0 0.5 1.0 1.5 2.0

x(t)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5sin(2lf0t)

0.5 cos(4lf0t)

0.33 sin(6lfot)

sum

f t = t /T0.0 0.5 1.0 1.5 2.0

x(t)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5sin(2lf0t)

0.5 sin(4lf0t)

0.33 sin(6lf0t)

sum

For example, even triangle-waves and square-waves can be constructed as sums of sine-waves of the fundamental and its harmonics.

f t = t / T0.0 0.5 1.0 1.5 2.0

x / x

0

-1.0

-0.5

0.0

0.5

1.0

sine

square-wave

triangle-wave

In an organ pipe, air rushes in and out of the pipe, again creating standing waves. Since the open ends of the pipe are at atmospheric pressure, they correspond to pressure nodes; for the fundamental, the length of the pipe is again L = λ/2. There will also be harmonic overtones, with λ = 2L/n

Unlike a string instrument, now the wave is carried by the air, so the relevant speed is vsound. Therefore, the frequency of the fundamental ffun=vsound/2L and the frequency of the overtones are fn = nffun.

Some organ pipes are closed at one end and open at the other. While the open end is a pressure node, the closed end is a pressure antinode. Therefore, the fundamental wavelength λfun = 4L, so ffun = vsound/4L, half the value (i.e. one octave lower) than an open/open pipe of the same length. [Also: an open/closed pipe will only have odd harmonics (homework problem).]

fundamental

Similarly, in wind instruments, a standing wave is established in the air column inside the instrument. One end of the column is closed (e.g. by mouthpiece, reed, or cap), and is at a pressure antinode while the open end is a pressure antinode, so for the fundamental L = λ/4 and ffun = vsound/4L. The length is set by sliding the tube (trombone), opening/closing valves (trumpet, etc.), or (effectively) by opening holes (woodwinds).Vibrations are established by the lips (brass instruments), the “whistle mouthpiece” (flute) or reeds (other woodwinds).

These wind instruments also have (odd) harmonic overtones, and the relative size of these determine the timbre of the instrument. These sizes are determined by the shape of the instrument, especially the taper in the tube and the shape of the bell. An untapered flute, with no bell, has relatively weak harmonics, i.e. the sound is almost completely fundamental. A trumpet and flugel horn have the same length but different tapers, giving rise to the more “mellow” sound of the flugel horn (less intense high-frequency harmonics) vs. the “brassier” sound of the trumpet.

A drum-head is two dimensional, as compared to the one-dimension for wind and string instruments. One consequence of this is that the vibrations couple (i.e. lose energy) to the air more rapidly, so the drum sound can be loud but short lived. Another consequence is that the overtones are not at integer ratios with the fundamental (and last longer), so they do not sound like musical tones.

The shape of the timpani’s kettle changes the frequencies of the overtones to make them more harmonic (i.e. in ratios of integers), leading to its more “musical” sound

Sound frequency: f = vsound / λ. This is the frequency (pitch) you hear if you and the source of the sound are not moving.

However, if the source of the sound and the detector (you) are approaching each other with speed Δv, your relative motion means that the successive wave crests reach your ear more rapidly, and f = (vsound + Δv) / λ, i.e. the pitch increases. For example, if a train whistle’s note is at 500 Hz and the train is approaching you at 33 m/s (~ 74 miles/hour), you will hear the sound at 550 Hz (> 1 note higher).

If the source of the sound and the detector (you) are traveling away from each other with speed Δv, the frequency you hear decreases: f = (vsound - Δv) / λ. For example, if a train whistle’s note is at 500 Hz and the train is traveling away from you at 33 m/s, you will hear the sound at 450 Hz (> 1 note lower).

These shifts are called Doppler shifts and are why train whistles have a higher pitch when the train is arriving and a lower pitch when it is leaving.

[A policeman’s “radar” gun measures the Doppler shift of radio waves he reflects off your car to determine its speed. For these, instead of vsound, the radio waves travel at the speed of light (c) so

fradar = (c ± Δv) / λ. ]

Exercises:10.A tuning peg slips on your violin, lessening the tension in one of the strings. What effect does this change

have on the string’s pitch?11.The strings that play the lowest notes on a piano are made of thick steel wire wrapped with a spiral of heavy

copper wire. The copper wire doesn’t contribute to the tension in the string, so what is its purpose?12.Why are the highest pitched strings on most instruments, including guitars, violins, and pianos, the most likely

strings to break?13.Some wind chimes consist of sets of metal rods that emit tones when they’re struck by wind-driven clappers.

These rods vibrate like the baseball bat in Fig. 3.2.7. Why do the longer rods emit lower pitched tones than theshorter rods?

14.Why would replacing the air in an organ pipe with helium raise its pitch?15.A flute and a piccolo are both effectively pipes that are open at both ends, with holes in their sides to allow

them to produce more tones. The piccolo is very nearly a half-size version of the flute. How does this factexplain why the piccolo’s tones are one octave above those of a flute?

Problem: Show that only odd harmonics of the fundamental exist for a standing wave in a column which has a node at one end and an antinode at the other end.