Travelling Waves

Waves Recap

"Waves transfer energy from one point to another with no net transfer of mass"

This means that the particles of the medium do not travel with the wave, apart from being disturbed from their equilibrium position as the wave passes.

As can be seen from the previous section, the formula to describe a wave's displacement is given by :-

y = 0 at t = 0 y = A Sin wt

y = A at t = 0 y = A Cos wt

Link to Wave Generator applet

The Travelling Wave Equation

The diagram below shows a sinusoidal wave ( y = 0 at t = 0 ) :-

In the above diagram, two points on the wave have been marked :-

i ) - displacement on x axis = 0

ii ) - displacement on x axis = x

Given the above points and by applying s = vxt it can be shown that :-

t = x/v

Where :-

t = time taken ( s )

x = displacement from origin ( m )

V = velocity of the wave ( ms-1 )

Since wave motion is repetitive after each Wavelength, the formula describing the wave can be derived as follows :-

At point i ) x = 0, t = x/v

At point ii ) x = x, t = t

The motion is repetitive, therefore motion of particles at i ) and ii ) are identical, allowing the replacement of t within the calculation by :-

t = t - x/v

The general formula can then be derived as follows :

y = a sin ωt

Substitute in t = t - x/v

y = a sin ω(t - x/v)

Substitute in ω = 2 π f and v = f λ

y = a sin 2 π (ft - x/λ)

Note - As sine is involved, ensure that the calculation is performed in radians.

Also, the above derivation applies to moving to the right along the x axis, for a wave moving to the left (-x) direction :-

y = a sin 2 π (ft + x/λ)

The above formula allows the calculation of the Displacement of the medium at any given point or time as long as the Wavelength and Frequency of the wave is known.

Example 1 -

A periodic wave travelling in the x-direction is described by the following equation :-

y = 0.2 sin 2 π ( 4π t + 0.1x)

Calculate :-

1. The Amplitude of the wave

2. The Frequency of the wave

3. The Wavelength of the wave

4. The Speed of the wave

In order to complete this question, the above formula must be rearranged to appear in the form of the general equation above :-

y = 0.2 sin 2 π ( 4π t + 0.1x )

y = 0.2 sin 2 π [ 2t + (0.1x / 2π) ]

By comparing the two equations together, the following answers can be found :-

a = 0.2 m

f = 2 Hz

λ = 2π /0.1 = 63m

v = 2 x 63 = 126 ms-1

Phase Difference

In the above derivation, the concept of repetition of waves was discussed. In order to understand how the motion of two separate points in a wave act in relation with each other, an understanding of Phase Difference is required.

Consider the two waves below :-

Both waves are :-

1. Sinusoidal

2. Identical Wavelength

3. Identical Frequency

4. Identical Amplitude

By comparing how the motion of the medium acts at different parts of the wave, Phase Difference can be explained.

Points 0 and 3 are separated by one Wavelength, therefore perform the identical motion, and are said to have a Phase Difference of 2π radians . These points are in Phase.

Points 0 and 2 are separated by a half wavelength, therefore perform the opposite motion to each other, and are said to have a Phase Difference of π radians. These points are exactly out of Phase.

All important points on a wave are summarised below :-

Also, by comparing the above table, it can be shown that :-

As the Wavelength is unchanging, the above ratio is equal to a constant value.

This implies that if Phase Difference (ϕ) between two points on the wave is equal to x, then :-

By rearranging, the general form of this formula can be found :-

Where :-

ϕ = Phase Difference ( radians )

x = separation of points ( m )

λ = wavelength of the wave ( m )

Example 2 -

A travelling wave has a wavelength of 60mm. A point P is 75mm from the wave's origin and a point Q is 130mm from the wave's origin.

Calculate :-

1. The Phase Difference between P and Q

2. Identify which following Phrase best describes this Phase Difference :-

a. Almost completely out of Phase

b. Almost in Phase

c. Exactly out of Phase

d. Approximately 1/4 cycle out of Phase

Answer :-

Separation of points = 130mm - 75mm = 55mm = 0.055m

Phase Difference = 2π ( 0.055 / 0.060 ) = 5.76 radians

P and Q are separated by 55mm, which is close to one whole wavelength ( 60mm ), meaning that they are almost in Phase.

Phase Change and Reflection

Interference patterns can be seen when a wave is reflected from a surface.

This is because that during reflection, a phase change occurs.

As can be seen from the above diagram, the reflected pulse has a phase difference of π compared to the incident pulse.

There is a similar phase change within light, when incident on a glass surface.

However, there is a slight additional point...

If :-

Moving to a higher optical density (eg. air to glass), a phase change of π occurs.

Moving to a lower optical density (eg. glass to air), no phase change occurs.