# Attenuation of Waves

### After reading this section you will be able to do the following:

- Describe what attenuation is and what causes it.
- Use the decibel unit with ratios and absolute quantities.

When a wave travels through a medium, its intensity diminishes with distance. In idealized materials, the wave amplitude is only reduced by the spreading of the wave. Natural materials, however, all produce an effect which further weakens the wave. This further weakening results from scattering and absorption. Scattering is the reflection of the wave in directions other than its original direction of propagation. Absorption is the conversion of the wave energy to other forms of energy. The combined effect of scattering and absorption is called **attenuation**.

The amplitude change of a decaying/attenuating plane wave can be expressed as:

$A=A_{0}e^{-\alpha z}$

In this expression **A _{0}** is the unattenuated amplitude of the propagating wave at some location. The amplitude

**A**is the reduced amplitude after the wave has traveled a distance

**z**from that initial location. The quantity $\alpha$ is the attenuation coefficient of the wave traveling in the z-direction. The dimensions of $\alpha$are nepers/length, where a neper is a dimensionless quantity. The term e is the exponential (or Napier's constant) which is equal to approximately 2.71828.

The units of the attenuation value in **Nepers per meter (Np/m)** can be converted to **decibels/length** by dividing by 0.1151. **Decibels** is a more common unit when relating the amplitudes of two signals. More information about decibels is provided below.

## The Decibel

The **decibel (dB)** is one tenth of a **Bel, **which is a unit of measure that was developed by engineers at Bell Telephone Laboratories and named for Alexander Graham Bell. The dB is a logarithmic unit that describes a ratio of two measurements. The basic equation that describes the difference in decibels between two measurements is:

$\Delta X(dB)=10log\frac{X_{2}}{X_{1}}$

where: delta X is the difference in some quantity expressed in decibels, X_{1} and X_{2} are two different measured values of X, and the log is to base 10.

### Why is the dB unit used?

Use of dB units allows ratios of various sizes to be described using easy to work with numbers. For example, consider the information in the table.

Ratio between Measurement 1 and 2 |
Equation |
dB |

1/2 |
dB = 10 log (1/2) |
-3 dB |

1 |
dB = 10 log (1) |
0 dB |

2 |
dB = 10 log (2) |
3 dB |

10 |
dB = 10 log (10) |
10 dB |

100 |
dB = 10 log (100) |
20 dB |

1,000 |
dB = 10 log (1000) |
30 dB |

10,000 |
dB = 10 log (10000) |
40 dB |

100,000 |
dB = 10 log (100000) |
50 dB |

1,000,000 |
dB = 10 log (1000000) |
60 dB |

10,000,000 |
dB = 10 log (10000000) |
70 dB |

100,000,000 |
dB = 10 log (100000000) |
80 dB |

1,000,000,000 |
dB = 10 log (1000000000) |
90 dB |

From this table it can be seen that ratios from one up to ten billion can be represented with a single or double digit number. Ease to work with numbers was particularly important in the days before the advent of the calculator or computer. The focus of this discussion is on using the dB in measuring sound levels, but it is also widely used when measuring power, pressure, voltage and a number of other things.

### "Absolute" Intensity Levels

Whenever the decibel unit is used, it always represents the ratio of two values. Therefore, in order to relate different intensities it is necessary to choose a standard reference level. The reference sound pressure (corresponding to a sound pressure level of 0 dB) commonly used is that at the threshold of human hearing, which is conventionally taken to be 2×10 −5 Newton per square meter, or 20 micropascals (20 μPa). To avoid confusion with other decibel measures, the term dB(SPL) is used. Similarly for EM waves the quantities dBW and dBm have been defined where the reference is 1 Watt and 1 milliwatt, respectively.

### Review:

- When a wave travels through a medium, its intensity diminishes with distance due to spreading, scattering, and absorption.
- The dB is a logarithmic unit that describes a ratio of two measurements.