Speed of Sound

Introduction

Sound travels through a medium in the form of waves. These waves move from the source of sound to the listener.

The speed of sound tells us how fast sound waves propagate through a medium.

In terms of compressions and rarefactions, speed of sound means how fast these density disturbances travel through the medium.

Speed of sound = Speed at which sound wave travels through a medium

Definition of Speed of Sound

The speed of sound is defined as the distance travelled by a point on a sound wave in unit time.

A point on a wave may be a crest, trough, compression, or rarefaction.

Speed = Distance travelled by sound wave per unit time

Formula of Speed of Sound

Speed is given by the general formula:

\(v=\frac{\text{distance}}{\text{time}}\)

For a sound wave, one wavelength is the distance travelled by the wave in one time period.

Therefore,

\(v=\frac{\lambda}{T}\)

where,

  • \(v\) = Speed of sound
  • \(\lambda\) = Wavelength
  • \(T\) = Time period

Relation Between Speed, Wavelength and Frequency

We know that frequency and time period are related as:

\(\nu=\frac{1}{T}\)

Using this relation in the formula \(v=\frac{\lambda}{T}\), we get:

\(v=\lambda\nu\)

Thus,

Speed of sound = Wavelength × Frequency

SI Unit of Speed of Sound

The SI unit of speed is metre per second.

SI unit of speed of sound = \(m\,s^{-1}\)

It is also written as m/s.

Figure

Speed of a Sound Wave Distance Density Average density Crest Crest λ One wavelength Direction of propagation v = λν Speed = Wavelength × Frequency

Explanation of the Figure

The figure shows a graphical representation of a sound wave. The distance between two consecutive crests is one wavelength, represented by \(\lambda\).

If this distance is covered by the sound wave in one time period, then the speed of sound is given by:

\(v=\frac{\lambda}{T}\)

Since \(\nu=\frac{1}{T}\), the speed of sound can also be written as:

\(v=\lambda\nu\)

Speed of Sound in Different Media

The speed of sound depends on the medium through which it travels.

Sound travels fastest in solids, slower in liquids, and slowest in gases.

Solids > Liquids > Gases

Why is Sound Faster in Solids?

Sound travels by the vibration of particles of a medium. In solids, particles are closely packed. Therefore, they can transfer vibrations quickly from one particle to another.

In liquids, particles are less closely packed than solids, so sound travels slower than in solids.

In gases, particles are far apart, so the transfer of disturbance is slower. Therefore, sound travels slowest in gases.

Approximate Speed Comparison

Medium Speed of Sound Reason
Solids Fastest Particles are very closely packed
Liquids Slower than solids Particles are less closely packed than solids
Gases Slowest Particles are far apart

Speed of Sound in Air, Water and Steel

The speed of sound in air is much less than its speed in water and solids.

Sound travels about 4 to 5 times faster in water than in air.

Sound usually travels about 15 to 20 times faster in solids than in air.

Medium Approximate Speed
Air About \(340\,m\,s^{-1}\)
Water About \(1500\,m\,s^{-1}\)
Steel About \(5000\,m\,s^{-1}\)

Speed of Sound in Air

The speed of sound in air depends on temperature and humidity.

  • When temperature increases, the speed of sound in air increases.
  • When humidity increases, the speed of sound in air also increases.

For example, the speed of sound in dry air is about \(331\,m\,s^{-1}\) at \(0^\circ C\), and nearly \(344\,m\,s^{-1}\) at \(22^\circ C\).

Higher temperature or humidity → Higher speed of sound in air

Does Speed Depend on Frequency?

In most media, such as air, the speed of sound depends mainly on the medium and not on the source or the frequency.

If the frequency of the source changes, the wavelength of the sound wave changes, but the speed remains almost constant in the same medium under the same conditions.

In the same medium: Frequency changes → Wavelength changes, but speed remains constant

Relation Between Speed, Frequency and Wavelength

The relation \(v=\lambda\nu\) shows that speed of sound depends on wavelength and frequency.

If the speed of sound is constant in a medium, then frequency and wavelength are inversely related.

  • Higher frequency means shorter wavelength.
  • Lower frequency means longer wavelength.

Example 1

A sound wave has wavelength \(2\,m\) and frequency \(170\,Hz\). Find the speed of sound.

Given,

  • Wavelength, \(\lambda = 2\,m\)
  • Frequency, \(\nu = 170\,Hz\)

Using,

\(v=\lambda\nu\)

\(v=2\times170\)

\(v=340\,m\,s^{-1}\)

Therefore, the speed of sound is \(340\,m\,s^{-1}\).

Example 2

The speed of sound in air is \(344\,m\,s^{-1}\). If the frequency of a sound wave is \(86\,Hz\), find its wavelength.

Given,

  • Speed, \(v = 344\,m\,s^{-1}\)
  • Frequency, \(\nu = 86\,Hz\)

Using,

\(v=\lambda\nu\)

Therefore,

\(\lambda=\frac{v}{\nu}\)

\(\lambda=\frac{344}{86}\)

\(\lambda=4\,m\)

Therefore, the wavelength of the sound wave is \(4\,m\).

Example 3

A sound wave travels \(680\,m\) in \(2\,s\). Find its speed.

Given,

  • Distance = \(680\,m\)
  • Time = \(2\,s\)

Using,

\(v=\frac{\text{distance}}{\text{time}}\)

\(v=\frac{680}{2}\)

\(v=340\,m\,s^{-1}\)

Therefore, the speed of sound is \(340\,m\,s^{-1}\).

Speed of Sound and Echo

The speed of sound is useful in understanding echo. An echo is heard when sound reflects from a distant surface and returns to the listener after a short time interval.

To calculate distance using echo, we use:

Distance travelled by sound = Speed × Time

Since sound travels to the reflecting surface and comes back, the actual distance of the reflecting surface is half of the total distance travelled by sound.

Sound and Lightning

During a thunderstorm, we see lightning first and hear thunder later.

This happens because light travels much faster than sound. Sound takes more time to reach our ears.

This delay helps us understand that sound has a finite speed.

Examples from Daily Life

  • We hear thunder after seeing lightning because sound travels slower than light.
  • Sound reaches faster through railway tracks than through air.
  • A person can hear knocking through a steel fence before hearing the same sound through air.
  • Sound travels faster in water than in air.
  • The speed of sound increases in warmer air.

Important Terms

1. Speed of Sound

The distance travelled by a sound wave in unit time.

2. Wavelength

The distance between two consecutive crests or two consecutive troughs of a wave.

3. Frequency

The number of complete oscillations per unit time.

4. Time Period

The time taken to complete one oscillation.

5. Medium

The material through which sound propagates.

Important Points

  • The speed of sound tells how fast sound waves propagate through a medium.
  • Speed of sound is the distance travelled by a point on a sound wave in unit time.
  • The general formula of speed is \(v=\frac{\text{distance}}{\text{time}}\).
  • For a sound wave, \(v=\frac{\lambda}{T}\).
  • The relation between speed, wavelength and frequency is \(v=\lambda\nu\).
  • The SI unit of speed of sound is \(m\,s^{-1}\).
  • Sound travels fastest in solids, slower in liquids, and slowest in gases.
  • Speed of sound in air increases with temperature.
  • Speed of sound in air also increases with humidity.
  • In the same medium, speed of sound remains almost constant even if frequency changes.

Conclusion

The speed of sound describes how fast sound waves travel through a medium. It is defined as the distance travelled by a point on a sound wave in unit time. The speed of sound is related to wavelength and frequency by the formula \(v=\lambda\nu\). Sound travels fastest in solids, slower in liquids, and slowest in gases. In air, the speed of sound increases with temperature and humidity. In most media, the speed of sound depends on the medium, not on the frequency of the source.

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