When a sound wave hits a wall, it is partially absorbed and partially
reflected. A person far enough from the wall will hear the sound twice. This
is an **echo**. In a small room the sound is
also heard more than once, but the time differences are so small that the sound
just seems to loom. This is known as **reverberation**.

Music is the sound that is produced by instruments or voices. To play most
musical instruments you have to create standing waves on a string or in a tube or pipe. The perceived
**pitch** of the sound is related
to the frequency of the wave. The higher the frequency, the higher is the
pitch.

**Wind instruments** produce sounds by means
of vibrating air columns. To play a wind instrument you push the air in a tube
with your mouth or a reed. The air in the tube starts to vibrate with the same
frequency as your lips or the reed. Resonance increases the amplitude of the
vibrations, which can form standing waves in the tube. The length of the air
column determines the resonant frequencies. The mouth or the reed produces a
mixture if different frequencies, but the resonating air column amplifies only
the natural frequencies. The shorter the tube the higher is the pitch. Many
instruments have holes, whose opening and closing controls the effective pitch.

We can create a standing wave in a tube, which is open on both ends, and in a
tube, which is open on one end and closed on the other end. **Open and closed ends** reflect waves differently. The closed end of a tube is an antinode in the
pressure (or a node in the longitudinal displacement). The open end of a tube
is approximately a node in the pressure (or an antinode in the longitudinal
displacement).

The
longest standing wave in a tube of length L with two open ends** **
has displacement antinodes (pressure nodes) at both ends.

It is called the **fundamental**
or **first harmonic**.

The next longest standing wave in a tube of length L
with two open ends is the **second harmonic****.**It also has
displacement antinodes at each end.

An integer number of half wavelength have to fit into the tube of length L.

L = nλ/2, λ = 2L/n, f = v/λ = nv/(2L).

For a tube with two open ends all frequencies f_{n }= nv/(2L) = nf_{1}, with n equal to an integer, are natural frequencies.

The longest standing wave in a tube of length L with one open end and one
closed end has a displacement antinode at the open end and a displacement node
at the closed end. This is the **fundamental**.

The next longest standing wave in a tube of length L
with one open end and one closed end is the **third harmonic**.

An odd-integer number of quarter wavelength have to fit into the tube of length L.

L = nλ/4, λ = 4L/n, f = v/λ = nv/(4L), n = odd.

For a tube with one open end and one closed end all frequencies f_{n }= nv/(4L) = nf_{1}, with n equal to an odd integer are natural frequencies,
i.e. only odd harmonics of the fundamental are natural frequencies.

Link: The Physics Classroom: Sound Waves and Music (Lessons 4 and 5)

What length should a pipe open at both ends have to produce a fundamental frequency of 110 Hz on a day when the speed of sound is 343 m/s?

Solution:

- Reasoning:

One half wavelength has to fit into the tube of length L. - Details of the calculation:

L = λ/2, λ = v/f = 2L, L = v/(2f) = (343/220) m = 1.55 m.

Is the wavelength of the fundamental standing wave in a tube open at both ends greater than, equal to, or less than the wavelength of the fundamental standing wave in a tube of the same length with one open end and one closed end?

Solution:

- Reasoning:

In a tube with two open ends f_{1 }= v/2L, λ = v/f = 2L.

In a tube with one open end and one closed end f_{1 }= v/4L, λ = v/f = 4L.

The wavelength of the fundamental standing wave in a tube open at both ends is less than the wavelength of the fundamental standing wave in a tube with one open end and one closed end.

Why does a clarinet play a lower note than a flute when both instruments are about the same length?

Solution:

- Reasoning:

Let the instruments play the lowest note.

A flute is a tube with two open ends.

Wavelength of the fundamental standing wave: λ = 2L

A clarinet is open at one end and closed at the other end by the player's lips and reed.

Wavelength of the fundamental standing wave: λ = 4L.

Link: Visualizing standing sound waves (Youtube)

The quality of a sound depends on the relative
intensities of the waves with the natural frequencies. It depends
on the spectrum of the sound.
The sound quality of a musical instrument is called its
timbre. A sinusoidal sound wave of
frequency f is a pure tone. A note played by a musical instrument
is not a pure tone. Its wave function is not sinusoidal, i.e. it
is not of the form ∆P(x,t) = ∆P_{max}sin(kx
- ωt + φ). The wave function is a sum
of sinusoidal wave functions with frequencies nf, (n = 1, 2, 3, ...,)
with different amplitudes, which decrease as n increases. The
harmonic waves with different frequencies which sum to the final wave
are called a **Fourier series**.
Breaking up the original sound wave into its sinusoidal components is
called **Fourier analysis**.

Fourier analysis implies that any wave pattern can be produced by superimposing the appropriate sinusoidal waves. Fast computer algorithms exist that calculate the required amplitudes. Fourier synthesis the basis for electronic music.

Link: Fourier series simulation Explore! It can be fun.