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**WAVE MOTION**

**Waves in Pipes**

closed pipes |

Pipes produce standing waves similar to stretched strings. However it must be emphasized that:

in **stretched strings** the waves are **transverse**

in **pipes** the waves are** longitudinal**

**Transverse** - particles of the wave vibrate at right angles to the direction of travel of the wave.

**Longitudinal** - particles of the wave vibrate in the same line as the direction of travel.

__Closed Pipes__

In the diagrams, **P** is the site of a **node**, while **Q** is at an **antinode**.

**Nodes** are always formed at the **closed end** of a pipe, where the air cannot move.

**Antinodes** are always formed at the **open end** of pipes.

As with stretched strings, the distance between node and antinode is 1/4 of a wavelength.

The diagram above represents the **Fundamental Frequency**, where **n = 1**.

This is the **1st harmonic**.

The diagram above represents the **3rd harmonic**, sometimes called the **First Overtone**.

The diagram above represents the **5th harmonic**, sometimes called the **Second Overtone**.

Looking at the different wavelengths in terms of the length of the pipe * L *,

we can then make wavelength the subject of each equation.

Using the wave equation and making the frequency ** f** the subject:

Substituting the different values of wavelength to obtain different expressions for frequency:

Looking at the form of these equations it is observed that each is a multiple of * f_{x}* (the Fundamental Frequency).

where * n* is 1, 3, 5, ... (odd)

__Open Pipes__

The diagram above represents the **Fundamental Frequency**, where **n = 1**.

This is the **1st harmonic**.

The diagram above represents the** 2nd harmonic**, sometimes called the **First Overtone**.

The diagram above represents the **3rd harmonic**, sometimes called the **Second Overtone**.

Looking at the different wavelengths in terms of the length of the pipe * L *,

We can then make wavelength the subject of each equation.

Using the wave equation and making the frequency ** f** the subject:

We can now substitute the different values of wavelength to obtain different expressions for frequency:

Looking at the form of these equations it is observed that each is a multiple of * f_{x}* (the Fundamental Frequency).

where * n* is 1, 2, 3, 4, 5, ... (odd + even)

__A comparison of 'closed' and 'open' pipes __

1. Comparing expressions for the Fundamental Frequency (n=1) for closed and open pipes respectively,

For a pipe of the same length * L*, the open pipe frequency is twice that of the closed pipe frequency.

2. For a given length of pipe, an open pipe gives more harmonics (odd & even) than a closed pipe (odd only). This results in a richer note from the open pipe.

__End correction __

The 'end correction' (* c*) is a length that must be added on to the the length (

**L**_{o}) of a pipe to take account of antinodes extending beyond the open end of the pipe.

End correction for a **closed pipe** :

The effective length (* L_{E}*) is given by:

where * r* is the radius of the pipe

End correction for an **open pipe** :

The effective length (* L_{E}*) is given by:

where * r* is the radius of the pipe

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