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School-college Physics Notes: SOUND 1. Sound wave characteristics

SOUND  1. The characteristic properties of sound waves - longitudinal waves - calculations using speed and wave formula

Doc Brown's Physics exam study revision notes

1. The characteristic properties of sound waves

Longitudinal waves and calculations using the wave equation

Sound waves are caused by some object or material vibrating e.g. your vocal chords vibrating, plucked guitar strings, something grating on a surface, the diaphragm (cone) moving back and forth in a loudspeaker etc.

The vibrations from a sound source are carried along by any available medium - gas (e.g. air), liquid (e.g. water) or solid (e.g. wall).

Sound cannot travel in a vacuum of empty space - there is no material to vibrate.

You can do a simple experiment with an electric ringing bell paced in a large bell jar attached to a pump.

As you pump out the air, the sound of the bell gets fainter and fainter until you can't hear it as the medium is removed from around it. However, you can still see the striker of the bell vibrating.

BUT, what exactly happens to the medium and enables it to convey the energy of a sound wave?

The above diagram gives an idea of a longitudinal wave of sound where the oscillations are in the direction the wave moves.

The oscillations in the same direction as the wave progression, can be considered as vibrations or disturbances in the medium through which the sound wave is travelling.

Reminder: Contrast this oscillation with transverse waves like water waves or electromagnetic radiation where the oscillations are at 90o to the direction of wave movement.

These oscillations in longitudinal sound waves show areas of compression and rarefaction.

A compression is where the particles of the medium are compressed to a maximum pressure and a rarefaction is where the particles of the medium are spaced out the most to the minimum pressure.

When the particles get squeezed closer together or spaced apart more than 'normal' they will want to return to their rest position.

This they do, driving the wave in a forward direction. So the wave is a continuous series of compressions and decompressions (rarefactions) in which the particles pushed together and then spaced apart again.

In the diagram above, where the lines are close together, imagine the particles e.g. air molecules or atoms in a metal crystal, are also packed together - the opposite is true where the vertical lines are far apart.

In the diagram above for longitudinal sound waves, wave B has twice the frequency and half the wavelength of wave A.

Reminder of the general wave equation applied to sound:

speed of sound wave (m/s) = frequency of sound (Hz) x wavelength of sound wave (m)

in symbolic 'shorthand'    v = f x λ, rearrangements:  f = v ÷ λ   and   λ = v ÷ f

Sound waves that we hear travel at the same speed whatever the frequency, so, with reference to the diagram, if the speed stays constant and you halve the wavelength, you must then double frequency for the equation to be valid.

Where the vertical lines are close together you can imagine the particles in a material being compressed closer to one another (compression) and where the lines are spaced well apart, so are the particles of the medium (the rarefaction).

The frequency of a sound wave (or any wave) equals the number of compressions passing a point per second, and is perceived as the pitch eg of a musical note.

The frequency of sound is the number of vibrations per second (unit hertz, Hz).

The amplitude is the maximum compression with respect to the 'rest line' and is perceived as loudness.

The 'rest line' is effectively the point of no disturbance, zero amplitude - neither compression or rarefaction.

You can think of a sound wave as a pressure wave - a continuous variation of high and low pressure regions of the wave.

The diagram above illustrate the simulation of sound waves by push and pulling on a slinky spring to create pulses of energy being transmitted along the slinky spring (the 'medium'). Its a good simulation of the compression and rarefaction behaviour of a longitudinal wave.

NOTE: Not all frequencies of sound can be transmitted through an object or material.

The nature of the material can affect which frequencies can be transmitted.

The shape, size as well as the structure of an object affects the frequencies that can pass through.

When you hit a solid object that is 'sonorous' e.g. a metal block that rings when structure, it will vibrate-resonate most strongly with certain 'natural' frequencies - you will hear one particular note most loudly e.g. as with a tuning fork in music.

The frequency of sound doesn't change as it passes from one medium to another.

However, the speed does change and therefore the wavelength must change too.

v = f x λ, rearrangements:  f = v ÷ λ   and   λ = v ÷ f

If the frequency f stays constant, then increase in speed v must be matched by an increase in the wavelength λ to keep the ratio speed / wavelength constant.

Unlike electromagnetic light waves, sound cannot travel through empty space (vacuum) because you need a material substance (gas, liquid or solid) which can be compressed and decompressed to transmit the wave vibration.

The more dense a material, the faster the sound wave travels (its the opposite for light in transparent materials).

Therefore in general for sound: speed in solids > speed in liquids > speed in gases

This is borne out by the data of sound speeds quoted below:

Typically at room temperature, the speed of sound waves in various materials at ~20-25oC

air 343 m/s (0.34 km/s),

that's why if a thunderstorm is 1 km away, you hear the thunderclap about 3 seconds after the flash of lightning - the speed of light is so much greater than the speed of sound that the flash is virtually instantaneous,

the speed of sound increases in air with temperature, an empirical formula (from experiment, no theoretical basis) I found on the internet is: speed of sound in air in m/s = 331 + 0.6T (where T = 0 to 100oC).

water 1493 m/s (1.49 km/s),  sea water 1533 (1.53 km/s), high velocity useful in sonar scanning of sea bed

kerosene 1324 m/s (1.32 km/s) (liquid hydrocarbon)

ordinary glass 4540 m/s, pyrex glass 5640 m/s (5.64 m/s), much more dense than water

iron 5130 m/s (5.1 km/s), steel 5790 m/s (5.8 km/s)

rocks 2000 to 7000 m/s (2-7 km/s), the speed tends to be greater in the more dense igneous rocks compared to less dense sedimentary rocks.

for comparison:

longitudinal earthquake P-waves typically travel at 2 to 7 km/s depending on whether the wave is in the Earth's crust, mantle or core and the speed will also depend on density and temperature

Some sound wave calculations

speed of sound wave (m/s) = frequency of sound (Hz) x wavelength of sound wave (m)

v = λ x f, rearrangements:  f = v ÷ λ   and   λ = v ÷ f

Q1 A typical audible high frequency sound might be 2 kHz.

Speed of sound = 340 m/s

(a) Calculate the wavelength of this sound wave.

(b) The musical note 'middle C' has a frequency of ~262 Hz

(c) How is it that you can hear sounds from room to other rooms in the house?

Q2 A musical note has a wavelength of 1.13 m.

If the speed of sound in air is 340 m/s calculate the frequency of the note.

Q3 Two people, 20 m apart, conduct an experiment using iron railings.

One taps the railings and the other places their ear on the iron railings.

(a) Why does the listener hear two sounds?

(b) Which sound arrives first and why?

Keywords, phrases and learning objectives for sound waves

Be able to draw and interpret diagrams to explain the characteristic properties of longitudinal sound waves.

Be able to do calculations using speed and wave formulae with the correct matching units.

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Worked out ANSWERS to sound wave calculations

Q1 A typical audible high frequency sound might be 2 kHz.

Speed of sound = 340 m/s

(a) Calculate the wavelength of this sound wave.

λ = v ÷ f = 340 / 2000 = 0.17 m

(b) The musical note 'middle C' has a frequency of ~262 Hz

Speed of sound ~340 m/s

Therefore λ = v ÷ f = 262 / 340 = 0.77 m

(c) How is it that you can hear sounds from room to other rooms in the house?

The second wavelength is similar to the width of a doorway.

Sound will travel throughout a house by both reflection and refraction!

Q2 A musical note has a wavelength of 1.13 m.

If the speed of sound in air is 340 m/s calculate the frequency of the note.

f = v ÷ λ = 340 / 1.13 = 301 Hz (3 s.f.)

Q3 Two people, 20 m apart, conduct an experiment using iron railings.

One taps the railings and the other places their ear on the iron railings.

(a) Why does the listener hear two sounds?

The listener hears the sound coming through the iron railings themselves and through the air.

(b) Which sound arrives first and why?

The sound coming through the internal vibration of the iron railings arrives first because iron is much more dense than air and sound travels faster the more dense the medium.

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