Waves, Intensity & Doppler's Effect

You will need to know some basic terms which are related to this topic

Transverse waves

Waves which particles vibrate perpendicular to the direction of the wave energy

They contain points called crests and troughs

Crest and troughs are points in the transverse waves where particles have maximum displacement(amplitude).

Examples are electromagnetic waves and water waves

Longitudinal waves

Waves which particles vibrate parallel to the direction of the wave energy

So this wave contains rarefractions and compressions

Rarefractions are regions of the longitudinal wave where particles are far apart / low pressure

Compressions are regions of the longitudinal wave where particles are squashed together

Wavefronts

It is an imaginary line which is drawn through the crest of the wave which is perpendicular to the direction of the wave

It is used to represent the crests of the wave. And also the the distance between two wavefronts is the wavelength

Displacement of a wave

The distance of a particle in a wave from its equilibrium state in a specified direction

So Particles , depending on the type of wave can only move perpendicular or parallel, so its displacement of each particle is shown by an oscilliscope device

Amplitude

The maximum displacement of a particle from its equilibrium state

Rememebr amplitude has no direction and so amplitude is always positive

Speed of a wave

Is the distance travelled by the energy of the wave per unit time

Velocity = Frequency * Wavelength

We will come to this later

Frequency

The number of waves generated by the source per unit time

Remember that frequency of a wave depends on the source always and not the medium. In fact, frequency of wave remains constant in any medium unless the source is changed.

1 Hertz

When one wave is generated by the source per second

Period

The time taken for one wave to be produce by the source

It is in terms of seconds

Wavelength

It is the length between two similar points of two consecutive waves

Remember that this for the same wave or source

A wave is usually considered as one cycle only

Now let us derive some equations

Link between Frequency and period

So frequency is the number of waves per second where as period is the time taken for one wave

If we have a wave of frequency 25Hz, it means 25 waves are produced per second and so each wave takes 1/25 seconds. This gives us a new formula

Frequency = 1/Period

In fact, period is in seconds and so the base units of frequency is s^-1

Finding the velocity of a wave

To find the velocity of the wave we need to apple the same equation of velocity

v = d/t

The wavelength is the distance moved by one wave and the period is the time taken for this wave to be moved

The division of this gives us the speed

Velocity = Wavelength/Period

We know that Frequency = 1/Period. So instead of dividing the wavelength with period we can multiply it by just frequency

Velocity = Frequency * Wavelength

The above proof must be memorised and explained as above

Graphs of Waves

This shows the graph of a displacement time graph

There are few things you have to remember

• Amplitude

• Period

The graph below shows a displacement-distance graph ( how the displacement of particles changes with the distance moved by the wave ). It's the same as the above graph. However, remember this:

• Wavelength

Instead of period, the distance between two similar points of two consecutive waves is the wavelength. This because, the x-axis measure distance and not time. So always remember to check what quantity does the x-axis measure

Phase

The angle subtended at the center of the circle where the path of the oscillating particle is represented by the circumference of the circle

If you look at the graphs of waves we will see the path of the particle is stretched across time and this actually represents the circumference

So let us see what is phase of a wave

Let's take an transverse as an example. We know as time goes on, the particle vibrates up down down and up again in a cycle. If we stretch this across time we get the displacement-time graph of a wave. And it looks like a sine graph

When a particle is at a particular position at a specific time, we state the position in terms of radians or degrees, this is called phase.

So usually when we define a base point as 0° or 0rad then after one quater of the wave period has been passed, the position is at the peak and is now 90° and when half is complete it is 180°. Think of a sin graph where we plot the sine value against the angle, it loops every 360°. That sine curve is actually how the particle vibrates but, only perpendicular.

So if we have a circle, how do we calculate the angle of a specified arc length

If the arc length of the circle is 2m and the total circumference of the circle is 8m, what is the angle of the sector?

(2m/8m)* 2π = π/2 or 90°

Likewise, we apply it for waves

We can see that we get a ratio between the arc length and circumference likewise, if we need to find thephase difference, we find the difference between two point of the same wave or between two similar points of two different waves

Phase difference

When two points of the same wave or two similar points of two different waves a compared the difference in the angles subtended at the centers of the circle is called the phase difference

There are two ways we can calculate phase difference

If we get a graph of Displacement-time graph

Phase difference = (Difference in time/Period) * 2π

If we get a graph of Displacement-distance graph

Phase difference = (Difference in length/wavelength) * 2π

This can also be multiplied by 360° to get degrees. However, questions usually ask in radians

You need to remember an important point! The definition of phase difference states that it is used to compare two similar points of two waves. So say one crest of a wave could be measured with a crest far away, this can have a phase difference of more than 360° or 2π . Thus, when we usually calculate phase difference we get a value between 0° to 360° but, we can calculate SIMILAR POINTS of the two waves which are 2 , 3 or more waves away. So the phase difference could increase by Smallest phase difference + n360° where n is a positive integer

Confusing? Let's see the below example

We know , When we measure the difference of similar points between two CONSECUTIVE waves. Which for example is 2m, we can calculate the phase difference as 90°, if the wavelength is equal to 8m. This is because, the difference would be 1/4 of the wave. But you actually can compare not just consecutive points but, any SIMILAR points, so actually if we go to the next similar point it will be 360° away. So 90+360 is 450° and it could be another 360° away and so on

Phase difference = 90° or 450° or 810°..

Phase difference = Smallest phase difference + n360°

where n is a positive integer

This is actually an understanding - you have to know a phase difference of 30° is same as 390° and 750° and so on. It increases by 360°. Think of a sin graph, the sin graph has infinite solutions and each solution repeats every 360°

Coherent waves

For the above example, the waves must be coherent, which means that they have a constant phase difference at any given time

So if you compare two similar points and another two similar points you must have the same phase difference. This is not always the case and more will be talked about this in the superposition chapter

Intensity

The power delivered by the wave to per unit area

Intensity = Power / Area

we can write it in terms of base units: kgs^-3

SI units: Wm^-2

Think of a laser being focused on to your eyes. It's very small but, has a very high intensity!

So the power is actually the work done by the particles as they oscilate per unit time

Link between intensity and amplitude

We know amplitude is the maximum displacement of the wave. When the displacement increases more work is done per unit time, but when amplitude doubles, the intensity increase by 4. Why?

The intensity is directly proportional to the amplitude Squared

Intensity ∝ Amplitude²

Intensity/Amplitude² = Constant

The constant arises from direct proportionality

So why? A way I could explain is that when the amplitude increases, the one half of the wave vibrates up more(doubles) and the other half vibrate down more(doubles). So when you increase the amplitude, not only the top peak increases but, also the bottom peak increases doing squared the work per unit time

Fast Method of solving Intensity questions

Usually MCQs and questions will ask what the amplitude is when the intensity doubles or what is the new intensity when the amplitude halves. We will see that down.

Let's see an example:

The intensity of a wave is 10Wm^-2. The amplitude is 5m

What is the intensity, when the amplitude is 10m?

Now there are two ways of doing this. You can either find the constant by using the principle of proportionality.

Using the constant:

Intensity = K * Amplitude²

Intensity/amplitude² = Constant

10/25 = 0.4

So we can use the same constant for the next set of data

Intensity = 0.4*100

Intensity = 40Wm^-2

There is another and simple way of doing this and this requires a bit of logic

We know the amplitude has doubled and we know:

Intensity ∝ amplitude²

So when amplitude has doubled, the intensity would be the square of the double which is 4 times greater

(2 * amplitude )² ∝ Intensity * 4

So intensity is 4 times greater

Intensity = 4 * 10

Intensity = 40Wm^-2

Now say if the intensity was doubled, by how much will the amplitude increase

Intensity ∝ Amplitude²

√Intensity ∝ Amplitude

So when the intensity doubles, the amplitude only increases by √2

Intensity-Time/Distance graphs

The intensity-time or intensity-distance graph is different from a normal wave graph.

Intensity is always positive, in order words the negative part of the wave is upwards/reflected

And so it looks like this

Remember this as they might ask you to calculate the period/Frequency of the wave using this information only

Doppler's effect

When objects which generates a frequency moves, the observed frequency is different from the frequency of the source

So we can find the frequency observed using this formula

Frequency observed = Frequency(source)*Wave velocity / (Wave velocity + Velocity(source))

At different situations we get - minus

Frequency observed = Frequency(source)*Wave velocity / (Wave velocity - Velocity(source))

I know this equations looks hard but, here's an easier form

So remember this point, when an object is moving towards us, we use minus

If it is moving away, we use plus

Think of it as relative speeds

So this helps us to predict the frequency heard, when an source is moving relative to us

The questions will sometimes ask us to calculate the frequency of the source, when the observed frequency is known. So which equation should you use?

They will mention if this is the maximum frequency or the lowest frequency heard. If the highest frequency is heard, this means it's moving towards us so we use minus

If it's the lowest, it means it's moving away from us so we use plus!

Why? Think of it like this! When an object is moving towards us, the waves will be more squashed like below, so the wavelength is shorter and so frequency increases

Whereas, if it moves away, the waves are spread out when they reach us, so lower frequency

Remember that one side is always squashed and the other side is stretched

Doppler's effect practical uses:

We usually think of sound waves in Doppler's effect. However, Doppler's effect works in all types of waves, even light!

So it is used to find:

• Blue shift and red shift in stars

So it can predict the movement of stars

One of my friends once said:

What's the point knowing an asteroid is coming towards earth by using Doppler's effect?

He has a point, we will be dead anyways!

Electromagnetic waves

You need to know the characteristics and uses of the electromagnetic waves

They have these properties:

1. Travels at 3 * 10ms^-1

2. Doesn't require a medium to travel so these are called electromagnetic waves. Waves which require a medium to travel are called mechanical waves

3. These are transverse waves

You need to know all types and their wavelengths by heart

Type	Wavelength in vacuum	Uses
Radio waves	106m to 10-1m	Radio signals & FM
Microwaves	10-1m to 10-3m	Satelites/space or phone waves or Radar
Infrared	10-4m to 10-7m	Grills & remote signals
Light	7 * 10-7m to 4 * 10-7m	Vision
Ultraviolet	3 * 10-7m to 10-8m	Sterilisation & tanning
X-ray	10-8m to 10-13m	Bone image & treating cancer(can cause also if too much exposure) & Also for Secuirity checks
Gamma Rays	10-10m to 10-16m	Sterilisation & killing bacteria & treating cancer

There are some points you need to remember:

1. If we need to calculate the frequency just divide speed of light by the wavelength

There is a reason why this is the wavelength in vacuum as the frequency remains the constant but, the wavelength and the speed changes from different mediums

2. The main difference between x -rays and gamma rays are the sources

X - rays are formed by slowing down fast moving electrons whereas, gamma rays are from radioactive sources

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