Showing posts with label Waves. Show all posts
Showing posts with label Waves. Show all posts

Waves Complete Lesson

Wave motion is a periodic disturbance through which energy is transferred from one place to other. In general if this is need of a medium, then it is called mechanical wave. No particle of the medium has permanent displacement in wave motion. Each particle of the medium vibrates about its mean position and transfers the energy from one particle to other and this keeps on happening until the energy reached its destination. For this wave propagation to happen, the medium shall have elastic property. 

Sound in air travels from one place to other in the form of a mechanical wave and it means it cannot be propagated on the moon since there is no medium there.

We can understand that each particle of the medium during wave motion executes oscillatory motion and hence rules of oscillations can be applied here also.

In wave motion, we study about wave propagation, velocity of the wave in a given medium, open pipes and closed pipes. We also deals with the concept of beats. It means when when two waves of slightly different frequencies travel in the same direction are superimposed together, there is alternate big and small sounds called waxing and waning.

We also deal with Doppler effect of sound here. It is the study of apparent change in the frequency of sound due to relative motion of source and observer.

Here are the concepts of all the above explained in detail.

Wave Motion an introduction 





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Ray Optics Complete Lesson

Light is a form of energy. It exhibits a wide variety of properties. If the size of the object is much larger than the wavelength of the light, light appears like travelling in straight lines. It exhibits certain properties under these conditions and that properties are studied and rename called Ray optics.

In Ray optics we study about reflection, refraction, dispersion and the deviation. 

Reflection is the phenomena of light bouncing back into the same medium after striking a boundary that is separating the two media.

Refraction is the phenomenon of light due to which light travels into the other medium after striking a boundary that is separating the two media.

Dispersion is the phenomenon of splitting up of a white light into multiple colors when it is passed through a prism. 

Deviation is the phenomena of changing its path when the light is passing through a different media.

In this chapter we are also going to study regarding mirrors, lenses, prisms, critical angle, total internal reflection, microscopes and telescopes.


This post is a list of all the topics in Ray optics which includes problems and solutions.

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Doppler Effect and Its Applications

The apparent change in the frequency due to the relative motion of source and observer is called Doppler Effect. We can experience the change in the frequency only when there is a relative motion. The original frequency of the source is not actually changing. Due to the relative motion it is appearing like changing and that’s why it is called as apparent change in frequency.

We can derive the equation for the apparent frequency in different possible cases. When the observer is in the motion he will receive more number of the waves than when he is in the state of rest. It is simply because waves are not only crossing him and he is also crossing the waves.



When the observer is crossing the stationary source there will be difference in the frequencies. If the observer is approaching the apparent frequency increases and when the observer is receding the apparent frequency decreases. The difference between these frequencies can be heard like beats to the observer. We can calculate the number of the beats as shown below.



There will be apparent change in the frequency even when the observer is the state of rest and the source is moving towards the Observer. Here is the source is approaching the observer, its wavelength towards the observer decreases and hence frequency increases. We can derive the equation for the apparent frequency in this case as shown below.



When the observer is in the state of the rest and source is approaching him, apparent frequency increases. When the source is moving away from the stationary observer, apparent frequency decreases. The difference between these two frequencies can be heard like the beats to the observer. We can derive the equation for the number of the beats as shown below.



When a source is revolving around the stationary observer, he is not going to have any Doppler effect experience. It is simply because there is no relative motion between the source and observer. No component of the velocity of the source is acting towards the observer and hence we cannot find any change in the frequency.



If the source is moving in the circular path and observer is far away from the Centre of the circular path, he can hear apparent frequency with different possible frequencies. When the source is moving away from the observer, apparent frequency decreases and vice versa. We can write the equation is as shown below.



When a source is moving by making an angle  to the direction of the observer, still there will be apparent change in the frequency due to the component of the velocity of the source towards Observer. We can write the equation for it as shown below.



Problem and solution



Problem and solution



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Wave Motion an introduction 
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Beats and Its Applications

The phenomenon in which to sound notes of slightly different frequency travelling together in the same direction are superimposed and produce alternate waxing and waning s called beats.

When the two waves are met in the same phase, they produces a maximum resultant intensity and it is called waxing. When the two waves are met in the opposite phase, they producers minimum resultant intensity and it is called waning.

We have derived a mathematical equation for beat frequency here.It is number of beats per one second.The time taken for completion of one beat that is one waxing and one waning is called time period of beat.A mathematical equation is derived for both of them as shown below.


The time interval between two maximum intensities as well as the two minimum intensities is always fixed. This is called Beat time period. The reciprocal of this time period is called beat frequency. We can derive the equation for them as shown below.


The diagrammatic view of the phenomena is as shown below.We can see one two waves are met in same phase,their resultant intensity is maximum and it is called waxing.As the time progresses,the phase difference increases and minimum intensity is produced and it is called waning.

The interval between two waxings and wanings is regular and systamatic.

Every ordinary human being needs a time interval of 0.1 second between the two successes sounds to understand the sound properly. This is called persistence of hearing. Hence difference between the frequencies of two sources shall not be greater than 10 to hear the beats.

Problem and solution

A tuning fork A has a frequency 5% more than the standard fork K and another tuning fork B has a frequency 3% less than the standard fork K. When this two tuning forks are vibrated together calculate the number of the beats generated?

Number of the beats generated is equal to difference between the frequencies.We can solve the problem as shown below.


Problem and solution

 Solving problems in the concepts of beat is very simple.Just follow the concept given and comment if any clarification is required.



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Wave Motion an introduction 
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Stretched String Problems and Solutions

Problem and solution

A sonometer wire has a length of 114 cm between the two fixed ends. Where shall we place two movable bridges to divide the wire into three segments whose fundamental frequency surrender ratio of  1:3:4 ?

When the tension and linear density of the wire is kept constant, frequency of the wire is inversely proportional to its length. Taking this law into consideration the problem is solved as shown below.



Problem and solution

A wire with density and length given and extension under a load is given in the below problem.We need to calculate the frequency of the wire under fundamental mode using the formula for the frequency of a stretched string.



This problem  is based on law of tension.When frequency is changed its tension will change as shown below.












Frequency of the tuning fork is directly proportional to thickness of the fork, velocity of the wave and inversely proportional to Squire of its length.

Speed of a longitudinal wave in a medium

The velocity of a wave in a medium can be expressed as the ratio of Squire root of  modulus of elasticity of the medium to the density of the medium. It is assumed that the propagation of the sound happens in a isothermal way. Anyway practically it is found that the temperature of the particles of the medium is not going to remain constant during the propagation of the wave. It is rather in adiabatic process when the heat energy of the system remains constant but the temperature increases.


It can be further proved that velocity of sound is independent of pressure.When ever pressure changes its volume also changes which generates same change in density and hence the ratio of pressure to density remain constant.



we can further compare this velocity with RMS velocity of a gas as shown below.Both of them depend on the absolute temperature similarly.



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Wave Motion an introduction 
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Frequency of Stationary wave in a streched String

Speed of the transverse wave in a string

When a string is attached tightly between the two points there will be tension generated in the string. Linear density of the string can be defined as the mass per unit length of the string. It can be proved that velocity of the string is directly proportional to Squire root of the tension and inversely proportional to Squire root of linear density.We can express the equation different formats as per the requirement as shown.




If Young's modulus of the wire is given with can express the tension in terms of Young’s modulus as shown below.



Problem and solution

We need the find the velocity of the wave in a stretched string using the regular formula as shown below.



Standing waves

Two waves of same amplitude, frequency and velocity moving in opposite directions are superimposed then stationary waves are formed.

The superposition of the waves can be done basing on the vector laws of addition. In the stationary waves there are some points that the displacement is minimum and the points are called nodes. There are some other points where the displacement is maximum and that points are called anti-nodes. The interval between two successive who nodes as well as the anti-nodes is always fixed as shown below.



Depending on the point of disturbance stationary waves can be formed under different modes of vibration. At the point of disturbance the displacement is going to be maximum and there is a formation of anti-node. Depending on the point of disturbance, a string can vibrate under different modes of vibration.



Laws of stretches strings

The frequency of a stretched string is inversely proportional to its length when it’s tension and linear density are kept constant. This is called law of lengths.

The frequency of stretches string is directly proportional to Squire root of the tension when its length and linear densities are constants. This is called law of tensions.

The frequency of a stretched string is inversely proportional to Squire root of the linear density when the length and tension are kept constant. This law is called as law of linear densities. 

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Wave Motion an introduction 
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Progressive Wave Representation and Problems

Equation of a progressive wave

To represent the displacement of a particle in wave motion we have a mathematical equation.

When the particle is advancing from the origin towards the positive x-axis, any particle who is at a certain distance from the origin will receive the wave lately than the origin by a specified time.

Taking that into consideration we can write a mathematical equation as shown below.


When the wave is moving along the negative x-axis, the particles will receive the vibration not with the time lag but with the time addition. The corresponding representation of the wave in the different possible formats is as shown below.



Relation between phase difference and path difference

This can be obtained basing on the definitions of the basic terms itself. We know that when the particles are separated by a distance equal to wavelength they are going to have a phase difference of 360°. By calculating the phase difference for unit separation we can get the relation as shown.


Intensity of the sound energy is defined as the energy emitted by a body per unit surface area per unit time. When the number of the waves are acting simultaneously on the same point, we can get the resultant of them using the vector laws of addition.

Reflection of waves

When waves travel from one medium to other, a part of it returns to the other medium. When the wave strikes the obstacle, it reflects back in the opposite direction. The string through which the wave is travelling applies a force on a wall as an action. The wall applies the reaction the string in opposite direction which is in satisfaction with the Newton third law.

When the wave travels from wherever medium to denser medium velocity decreases and the pulses inverted upon reflection.



Problem and solution

The audible range of frequencies for a human being leaves from 20 to 20,000 Hz. Express them in terms of the wavelength?

We can solve the problem using the simple relation between where velocity and wavelength. We shall also remind ourselves that when the wave changes its medium its frequency is going to remain constant. Frequency is a characteristic property of the source and it is independent of the medium.


Problem and solution

 We are going to solve the following problem basing on the very definition of frequency.It is the number of vibrations made per one second.Then time taken for one vibration is called time period.



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Wave Motion an introduction

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