Progressive Waves

Wave motion is a means of moving energy from one place to another. Vibrating or oscillating objects act as source of waves. Waves that move through a material (or vacuum) are called progressive waves. A progressive wave transfers energy and momentum from one place to another.

Waves are formed when particles vibrate about a mean position. Waves can be observed in different situations. Waves are formed on the surface of water when an object is dropped into the water or when a wind blows across the surface of water. Waves can also be observed when a string on a table is shaken from side to side.

Terminology to describe waves

The following terms are used to describe waves.

Displacement of a vibrating particle is its distance from its equilibrium position.

Amplitude is the maximum displacement of the vibrating particle from its equilibrium position.

The length of a complete oscillation is known as the wavelength of the wave. The symbol for wavelength is λ. It is also the distance from crest to crest or trough to trough of a wave.

Period of oscillation is the time taken for one oscillation or time take for wave to move one whole wavelength past a fixed point. The symbol for period is T.

The frequency of a wave is the number of cycles of vibration of a particle per second, or the number of complete waves passing a point per second. The symbol for frequency is f. The unit of frequency is hertz (Hz).

Frequency and period are related by the equation f = \displaystyle\frac{1}{T}

Wave speed is the distance travelled by the wave per unit time.

The wave equation

The speed of a particle is given by the equation

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

Hence wave speed = \displaystyle\frac{\text{distance travelled by the wave}}{\text{time}}

In time T, the period of oscillation, the wave travels one wavelength λ. Hence

v = \displaystyle\frac{\lambda}{T}

We know from above f = \displaystyle\frac{1}{T}

Hence v = fλ

The speed of a wave depends only on the properties of the medium and not how it is produced.

Intensity of radiation in a wave

Progressive waves transfer energy. When the wave travels out from a source the radiant power spreads out, reducing the intensity. Intensity is defined as the energy transmitted per unit time per unit area at right angles to the wave velocity. Energy transmitted per unit time is the power transmitted.

Hence Intensity I = \displaystyle\frac{Power}{Area}

where I is the intensity of the wave at a surface, P is the radiant power passing through the surface and A is the cross-sectional area of the surface.

The unit of intensity is Wm-2.

The radiant power from a point source spreads out in a sphere. So the intensity of a wave depends on the area over which the power is spread out. As we move away from the center of the source, the surface area of the sphere (A=4πr2) increases. From above equation, greater the area, lower the intensity.

I = \displaystyle\frac{Power}{Area} = \displaystyle\frac{P}{A} = \displaystyle\frac{P}{ 4\pi r^2}

We can see from the above equation that the intensity has an inverse square relationship with the distance from the source. If the distance doubles, the intensity decreases by a factor of 4.

The intensity of wave is proportional to the square of the amplitude of the wave.

I ∝ xo2

Thus if amplitude of the wave is halved, its intensity is reduced by a factor of 4.