## 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

## 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 = v_{x}t 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}

## Energy of a Wave

### As discussed in Higher Physics, the intensity of a wave is defined as "the amount of energy that passes though unit area perpendicular to the wave direction in unit time".

### As also discussed previously, the amplitude and the energy inherent to a wave are related to each other. They are not related proportionally, however, as the amplitude of a wave varies in a sinusoidal manner with time whereas the energy of the wave varies as sine squared.

### The graph below shows how the energy of a wave and the amplitude of a wave vary with time:-

### Since the variation in amplitude is simple harmonic, the energy variation with time is proportional to sine squared or more commonly :-

### "The intensity I (or energy E) of a wave is proportional to the square of its amplitude"

### Non-examinable derivation of above :

### A SHM wave is a displacement that is resisted by a restoring force. The larger the displacement x, the larger the force F = kx needed to create it. Because work W is related to force multiplied by distance (Fx) and energy is put into the wave by the work done to create it, the energy in a wave is related to amplitude. A wave’s energy is therefore directly proportional to its amplitude squared because W ∝ Fx = kx^{2} .

## 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.