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An Introductory Physics Course with Peter Eyland

Lecture 5 (Resonance)

In this lecture the following are introduced:

•Resonance

•Standing waves

•Harmonics, Overtones and Partials

•Musical instruments with strings

•Musical instruments with pipes

Resonance

When a pulse sets a system vibrating, it will do so at the system's natural frequency. This frequency depends on the system's construction.

When a periodic vibration sets a system vibrating then the system is forced to vibrate at that frequency.
When the frequency of the forcing vibration is the same as the natural frequency there is a big increase in the amplitude of the vibrations.
This big increase in amplitude is called *resonance*.

The Tecoma Narrows bridge (while not strictly "resonance" because there is no periodic driving force) provides a graphic image.

Standing Waves

A wave transfers the energy of a vibrating source through the neighbouring medium.
When waves are reflected back on themselves with the right phase, a *resonant stationary pattern* can appear.
The pattern is called a *standing* wave because the energy from the source stays in place.
When the standing wave generates a secondary *moving* wave (as musical instruments do when they disturb the air around them)
then the energy from the source will then move away through the neighbouring medium.

Transverse Standing Waves

If a stretched wire is vibrated at right angles to its length then transverse (shear) waves will be propagated along it. If the frequency is right then "loops" are seen to form as shown. "Nodes" are points where there is no lateral movement. "Antinodes" are points of maximum lateral vibration. The picture on the right shows five nodes (one at each end) and four antinodes.

*Fundamental frequency for a fixed length wire*

The *Fundamental* or *First Harmonic* is the lowest frequency that forms a single "loop". See the left picture above.
The "loop" will be half a wavelength long so that the wave goes out and returns in one period.
The vibration frequency needed is given by .
Where v is the velocity of the wave, λ is the wavelength and L is the vibrating length.

*Harmonics and Overtones*

With musical instruments, standing waves can be formed at multiples of the fundamental frequency.

The 1^{st} Harmonic = ν_{0}

The 2^{nd} Harmonic = 1^{st} Overtone = 2ν_{0}

The 3^{rd} Harmonic = 2^{nd} Overtone = 3ν_{0}

The 4^{th} Harmonic = 3^{rd} Overtone = 4ν_{0}

The distance between adjacent nodes is always half a wavelength.

Stringed Musical Instruments

A stringed instrument will produce the fundamental plus a number of harmonics with varying intensities. The harmonic content (that is, the number and strength of each harmonic excited) determines the wave shape so it determines the timbre of the sound (the perceived quality). This enables us to tell a guitar from a mandolin and a violin etc. Some harmonics will be enhanced and some suppressed by the design. The actual harmonics that do form are called the Partials.

If an instrument cuts out all the *even* harmonics then:

The 1^{st} Harmonic = Fundamental= 1^{st} Partial = ν_{0}

The 3^{rd} Harmonic = 2^{nd} Overtone = 2^{nd} Partial = 3ν_{0}

The 5^{th} Harmonic = 4^{th} Overtone = 3^{rd} Partial = 5ν_{0}

The 7^{th} Harmonic = 6^{th} Overtone = 4^{th} Partial = 7ν_{0}

For other information, see A/Prof. Joe Wolfe's page about waves in strings, reflections, standing waves and harmonics.

Pipe Instruments

Pipes fall into two main classes depending on whether or not they are closed at one end.

*Open pipes*

These are open to the atmosphere at the origin of the sound waves and also the other end.

Sound waves are *pressure waves* so pressure nodes appear symmetrically at each end where they open to the atmosphere.
The *displacement wave* representation (here called "Amplitude wave") is out of phase with the pressure waves so maximum displacements occur
at the pressure nodes i.e at the ends.
Although they are longitudinal waves, displacement waves are drawn as transverse waves for convenience.
Displacement wave forms drawn with this convention look like this

It should be noted that the pressure nodes (displacement antinodes) actually occur a little bit outside the ends of the pipe. This is called the "end effect" and should be taken into account. The distance from the end to the node outside is usually designated as "ε"

so L+2ε = λ_{0}/2 for the fundamental

and L+2ε = λ_{1} for the 1^{st} harmonic

A flute is *effectively* an open pipe as shown by these pressure wave forms.

*Closed Pipes*

Closed pipes are not symmetrical and if they have a constant diameter they form only the odd harmonics.

The pressure node/displacement antinode is at the open end. The pressure antinode/displacement node is at the closed end. The wave forms are drawn like this

Taking end effects into account L + ε = λ_{0}/4

Taking end effects into account L + ε = 3λ_{3}/4

A flute is an open pipe so it can form all the harmonics. So is the recorder and some organ pipes. But most other wind instruments are closed pipes. An clarinet is a closed, cylindrical pipe and only supports odd harmonics (at least in its bottom register). An oboe and a saxophone are also closed pipes but because they are cone-shaped rather than cylindrical, they can form all the harmonics. This is explained in more detail in pipes and harmonics

*Measuring the speed of sound in a closed pipe*

An "air column" uses water to adjust the column length. A vibrating tuning fork will provide a known frequency.

The distance between the first resonance position and the second resonance will give exactly one half-wavelength.
Knowing frequency and wavelength the speed can be found.

Example:

A flute has a fundamental frequency of G sharp i.e. 415.3 Hz. The velocity of sound in air is 340 m.s^{-1}. Find the effective length of the flute.

Example:

A (closed) organ pipe is 193 mm long. The velocity of sound in air is 340 m.s^{-1}. Find the frequency that the pipe emits under excitation.

*Summarising:*

Resonance occurs when the frequency of a forcing vibration is the same as the natural frequency of a system
and there is a big increase in the amplitude of the vibrations.

A standing wave is a resonant condition with nodes and antinodes where the energy stays in place.

The distance between adjacent nodes is always a half a wavelength.

The lowest standing wave frequency is called the fundamental or first harmonic or first partial.

Harmonic frequencies are multiples of the fundamental, and the second harmonic is the first overtone etc.

The harmonic content determines the timbre of a sound.

The actual harmonics that do form are called partials.

Strings are symmetrical and may form all the harmonics (depending on how they are played).

Open pipes are symmetrical and may form all the harmonics.

Closed pipes (with constant diameter) are not symmetrical and cut out the even harmonics.

Pressure nodes occur just outside the open ends of a pipe.

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