 # Wave properties

A wave is a disturbance moving through a medium. The medium can be air, liquid, or solid. The wave travels through the medium and carries energy with it.

#### Wavelength

The wavelength of a wave is the distance between two crests (highest point) of consecutive waves. Its unit is metre (m).

#### Amplitude

The amplitude of a wave is a measure of the intensity of the energy the wave carries. For example, a note played on a musical instrument may be played louder (greater amplitude) without changing the pitch (frequency), wavelength, or velocity (sound in air is a constant 340 m/s). The velocity of a wave is equal to the product of its wavelength and frequency.

#### Frequency

The frequency of a wave is equal to the wave's velocity divided by its wavelength. As the wavelength increases, the frequency decreases. Frequency is a measure of the number of waves (crest to crest) passing a point in space per second. It has the unit of hertz (Hz), which means 'per second'.

v = f . λ

## Wavelength

### Transverse Waves

The wavelength of a transverse wave is the distance between two consecutive crests of the wave. Its unit is metre (m). A transverse wave may be plotted against time. The period of the wave is the time interval between two successive wave crests passing a point.

Wavelengths of electromagnetic radiation can be any length, from nanometres to thousands or even millions of kilometres long.

As the wavelength increases, the time it takes for the next wave to pass a point is longer, so the period increases.

Since the period is longer, there are fewer waves passing per second, so the frequency will decrease.

### Longitudinal Waves A longitudinal wave is a wave of compression passing through a medium.

The wavelength of a longitudinal wave is the distance between two consecutive areas of compression. Its unit is metre (m).

As the wavelength of sound grows longer, the frequency of the sound will decrease. This we hear as a fall in pitch from a higher note to a lower note. That is why as a guitar string is made shorter the sound gets higher in pitch.

## Frequency

The frequency of a wave is how many wave crests (for transverse waves) or compression zones (longitudinal waves) pass a fixed point in a second.

As the frequency of sound increases, the sound gets higher in pitch. That is why a piano has the longest strings on the left-hand side, to play the lower notes, and the strings get gradually shorter as the hands move up the keyboard.

As the frequency of light increases, the light changes colour, and very high frequency becomes x-ray or gamma radiation.

Higher frequency has higher energy. That is why radio waves (low frequency) are harmless, but ultraviolet and x-rays (higher frequency) can be dangerous.

Since the frequency is the number of waves passing a point per second, the period is the inverse of the frequency. For example, if two people pass the door per second, the interval between the people is half a second. If four pass per second, the time between them, or period, is 1/4 s.

Therefore, \$f = 1/T\$, where f is the frequency, and T is the period.

The period is measured in seconds, so the frequency unit is \$1/s\$, or 'waves per second'. In the S.I. unit system, this has a special name: hertz (Hz).

### The Maths

The frequency and the wavelength are inversely proportional. This means that if one increases, the other decreases.

This can be written: \$f ∝ 1/λ\$

The frequency is also the inverse of the period: \$f = 1/T\$, where f is the frequency, and T is the period.

The period, T, is measured in seconds, so the frequency unit is \$1/s\$, or 'waves per second'. In the S.I. unit system, this has a special name: hertz (Hz).

If we know the speed of a wave, v (\$m/s\$), then we can calculate the wavelength, λ (m), from the frequency, f (Hz), or the frequency from the wavelength, using the formula:

\$\$v = f⋅λ\$\$ The velocity of a wave is equal to the product of its wavelength and frequency.

## Amplitude

The amplitude of a wave is a measure of the intensity of the energy the wave carries. For example, a note played on a musical instrument may be played louder (greater amplitude) without changing the pitch (frequency), wavelength, or velocity (sound in air is a constant 340 m/s).

The amplitude has no affect on the wavelength, frequency, period, or speed of a wave.

## Doppler Effect Christian Doppler, 1803-53, Austrian physicist who termed the Doppler Effect

Christian Andreas Doppler, 29 Nov 1803 - 17 March 1853, Austrian physicist.

Doppler gave his name to the phenomenon of changing perceived frequency of a wave-emitting source as it moves at velocity towards a receiver. The change in frequency compared to the static frequency as perceived by the listener or observer is proportional to the velocity.

##### Change in frequency

The Doppler Effect is used to determine the velocity of distant galaxies and the relative speeds of binary stars. Objects moving at high speed away from the observer display a red shift (overall decrease in their spectra frequency range), while moving towards the observer causes an increase in frequency, or a blue shift. A listener will receive waves from a static sound source at a frequency proportional to the rate of their propagation

A static source sends waves at speed c towards a receiver. The formula v = f⋅λ gives the frequency \$f = c/{λ}\$. A listener hears an increase in freqency from a source moving towards him, because the distance between waves is shortened
A moving source on the other hand, sends waves at speed c, but the distance from the source to the receiver decreases at each wave emission. Hence the wavelength λ has decreased to: \$λ_0 = {c - v_s}/{f_s}\$, where \$f_s\$ is the frequency of emission.

The frequency as perceived by the receiver is \$f_0 = {f_s}/{1 - {v_s}/c}\$.

For a source moving away from the receiver, the frequency is \$f_0 = {f_s}/{1 + {v_s}/c}\$.

##### Light

The Doppler Effect is used to measure the velocity of receding galaxies. The technique was invented by Edwin Hubble in 1929, and led to the theory that the universe was expanding, rather than remaining fixed in size, as had previously been assumed.

If the speed of the observer or the emitter is comparable to the speed of light, relativity has a significant influence on the change in frequency.

However, at lower speeds, the relationship is:

\$\$Δf = v/c⋅f\$\$

where c is the speed of light in a vacuum, \$3.0 x 10^8 m/s\$.

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