Grade 12 Full notes on Wave and Optics

Overview

A wave is a continuous transfer of disturbance from one part of a medium to another through successive vibrations of particles of the medium about their mean positions. This note provides us an information on types of wave.

Types of wave

A wave is a continuous transfer of disturbance from one part of a medium to another through successive vibrations of particles of the medium about their mean positions. In wave motion, energy and momentum are carried from one region to another region of the medium. If there is no transfer energy, it is not a wave but an oscillation as there is no transfer of energy.

Types of wave

1. Mechanical wave
2. Non-mechanical wave

Mechanical Wave

If the wave requires a material medium to carry energy from one point to another point then it is called a mechanical wave. For a mechanical wave, a medium has elastic nature.

Non-mechanical Wave

If the wave does not require material medium to carry energy from one point is called a non-mechanical wave. Example: X-ray, microwave, electromagnetic wave.

Crest

It is the maximum displacement of the vibrating particle above equilibrium line of mean position.

TroughIt is the maximum displacement of the vibrating particle below the equilibrium line of the mean position. The velocity of a wave is maximum at equilibrium line and minimum at an extreme point.

Wavelength

It is the distance travelled by the wave to complete one oscillation of vibrating particle on the medium. In other words, a distance between two nearest crests or trough is called wavelength. It is denoted by ‘λ’. It’s unit is a meter. Hence, wave velocity is equal to the product of wavelength and frequency.

Wave velocity

The distance travelled by a wave in one second is called wave velocity, denoted by v.

wave velocity, v∴v∴v=λf=distance travelled by waveλtime taken, t=λT=λf=λf$$\begin{array}{rl}\text{wave velocity, v}& =\frac{\text{distance travelled by wave}\lambda }{\text{time taken, t}}\\ & =\frac{\lambda }{T}=\lambda f\\ \therefore v& =\lambda f\\ \therefore v=\lambda f\end{array}$$

Phase

The angular displacement of the wave at any time‘t’ is called phase of the wave. Phase indicates where the wave reaches and in which direction is the wave.

Characteristics of Wave Motion

1. Wave motion is the disturbance travelling through a medium.
2. When a disturbance is produced in a medium, the disturbed particles vibrate about their mean positions.
3. Particles handover their energy to their neighbors through the disturbance but their net displacement over one period is zero.
4. The energy transfer in the medium takes place with a constant, v = λf which depends on the nature of the medium.
5. As the disturbance reaches to a particle, it starts to vibrate. As the disturbance is communicated to the next neighbor a little later, so there is a phase difference in the vibratory motion of the consecutive particles.
6. The wave motion is possible in a medium which possess the property of elasticity and inertia.
7. Particle velocity is a function of time, so it is different in different points of a displacement. But wave velocity is constant in a medium.
8. Vibrating particles of the medium possess both kinetic and potential energies.

Types of Wave

1. Transverse Wave
2. Longitudinal Wave

Transverse Wave

If the vibrating particle of the medium has oscillated perpendicularly to the direction of propagation of the wave is called a transverse wave. Example: wave in a surface of liquid and wave in solid.

Propagation of Transverse Wave

To understand the propagation of a transverse wave, suppose nine particles of a medium numbered as 1 to 9 lying at an equal distance at the mean position shown in the figure. When a wave travels from to right, the particles vibrate up and down about their mean positions and the disturbance travel from the 1st particles to a 9th particle.

1. At t= 0, all the particles are at their mean positions.
2. After t = T/8 seconds, the particle 1 travels vertically a certain distance upward and the disturbance reaches to particle 2.
3. After t= 2T/8 seconds, particle 1 has reached to the maximum position and the disturbance has reached to particle 3.
4. After t= 3T/8 seconds, particle 1 has completed 3/8th of its vibration and the disturbance has reached to particle 4. The positions of particles 2 and 3 are also shown in a figure.
5. In this way after t/2 seconds, particle 1 has come back to its mean position and the particles 2, 3 and 4 are at the positions shown in the figure in a figure. The disturbance has reached to particle 5.

After T seconds, the particle 1, 5 and 9 are at their mean positions and the wave has reached to particle 9. Particle 1 and 9 are in the same phase. The wave travelled a distance between particles 1 and 9 in the time in which the particle 1 has completed oscillation.

The top point on the wave at the maximum distance from the mean position is called crest, and the point at the maximum distance below the mean position is called trough. Thus in a transverse wave, crests and trough are alternately produced.

Properties of Transverse Waves

1. The particles of the medium vibrate simple harmonically perpendicular to the direction of propagation of the wave.
2. All particles vibrate with the same amplitude, frequency and period.
3. There is a gradual phase difference between the successive particles.
4. All the particles vibrating in phase will be at a distance equal to nλ, where n = 1, 2, 3 …. Etc. it means minimum distance between two particles vibrating in phase is equal to one wavelength.
5. The velocity of each particle is the maximum at the mean position and is zero at extreme positions.
6. At any instant, different particles have different displacements.
7. When the particles move above the mean position, it is in the region of a crest and when the particles move below the mean position, it is in the region of a trough.
8. Due to the repeated periodic motion of the particles, crests and troughs are produced alternately.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa,et al. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.

Overview

If the vibrating particles of the medium oscillate to the same direction of propagation of the wave is called a longitudinal wave. This note provides us an information on longitudinal wave and progressive wave.

Longitudinal Wave and Progressive Wave

Longitudinal Wave

If the vibrating particles of the medium oscillate to the same direction of propagation of the wave is called a longitudinal wave. Example: sound wave travelling in air. A longitudinal wave can travel in solid, liquid and gas.

Propagation of Longitudinal Wave

To understand the propagation of longitudinal waves in a medium considers nine particles named 1, 2, 3, 4, 5, 6, 7, 8, 9 of the medium lying at equal distances at their mean positions. The wave travels from left to right and the particles vibrate about their mean positions. After T/8 seconds, the particle 1 goes to the right and completes 1/8th of its vibration. The disturbance reaches to the particle 2. After T/4 seconds, the particle 1 has reached its extreme right position and completes 1/4th of its vibration. The disturbance reaches to the particle 3. The process continues. The waves reach to particle 9. Here 1 and 9 are again in the same phase. Here particles 1, 5 and 9 are at their mean positions. The particles 1 and 3 are close to the particle 2. This is the position of condensation. Similarly, particles 9 and 8 are close to the particle 7. This is also the position of condensation or compression. On the other hand, particles 4 and 6 are far away from the particle 5. This is the position of rarefaction. Hence in a longitudinal wave motion, condensations (compressions) and rarefactions are alternately formed.

Properties of Longitudinal Waves

1. The particles of the medium vibrate simple harmonically along the direction of propagation of the wave.
2. All particles have same amplitude, frequency and period.
3. There is a gradual change in phase between the successive particles.
4. The velocity of each particle is maximum at its mean position and zero at extreme points.
5. When the particles move in the same direction as the propagation of the wave, it is in the region of compression but when they move in the opposite direction to the direction of propagation of the wave, it is in the region of rarefaction.
6. When the particle is at the mean position, it is a region of maximum compressions or rarefaction.
7. All particles vibrating in phase will be at a distance equal to nλ, where n = 1, 2, 3, 4, 5, etc. It means minimum distance between two vibrating particles in phase is equal to one wavelength.
8. Due to repeated periodic motion of the particles, compressions and rarefactions are produced alternately.

Comparison between Transverse and Longitudinal Waves..

S.N.	Transverse Wave	Longitudinal Wave
1.	Particles of the medium vibrate perpendicular to the direction of wave propagation.	Particles of the medium vibrate in the same direction to the wave propagation.
2.	Alternate crest and trough.	Alternate compression and rarefaction.
3.	It can travel in solid and surface of a liquid medium.	It can travel through solid, liquid and gas medium.
4.	It can travel in a vacuum.	It cannot travel through a vacuum.
5.	There are pressure and density variation. Examples: waves in the surface of liquid, wave in solid.	There pressure and density become maximum at compression and minimum at rarefaction. Examples: sound wave travelling in air.

Progressive Wave

If the wave travels from one region to another region is called a progressive wave. Transverse wave and longitudinal wave are both progressive wave.

Consider a progressive wave (transverse wave ) is travelling on a medium. Ï´ be the point of wave starting. At any time ‘t’, the displacement of the wave at point ‘O’ is given by equation,

y=asinωt…(i)$$y=a\sin \omega t\dots (i)$$

Where, yaωt=displacement of wave=amplitude of wave=angular velocity of wave=time taken by wave$$\begin{array}{rl}\text{Where, y}& =\text{displacement of wave}\\ a& =\text{amplitude of wave}\\ \omega & =\text{angular velocity of wave}\\ t& =\text{time taken by wave}\end{array}$$

The disturbance travels later at point P than point O. So, particles at point P vibrates simple harmonically after certain time. Let particle ‘P’ is at distance x from the point o. the distance travelled by the wave in one complete oscillate is equal to λ. i.e λ = wavelength of the wave

For displacement,λPhase angle=2π1 Phase angle=2πλx phase angle=2πλ.x$$\begin{array}{r}\text{For displacement,}\lambda \text{Phase angle}=2\pi \\ \text{1 Phase angle}=\frac{2\pi }{\lambda }\\ \text{x phase angle}=\frac{2\pi }{\lambda }.x\end{array}$$

when the wave reaches at point P then the equation of wave isy=asin(ωt−ϕ)ϕ=phase angle=2πλ.x∴yor,y=asin(ωt−2πλ.x)…(ii)=asin(ωt–k.x)…(iii)$$\begin{array}{r}\text{when the wave reaches at point P then the equation of wave is}\\ y=a\sin (\omega t-\varphi )\\ \varphi =\text{phase angle}=\frac{2\pi }{\lambda }.x\\ \therefore y& =a\sin (\omega t-\frac{2\pi }{\lambda }.x)\dots (ii)\\ \text{or,}y& =a\sin (\omega t–k.x)\dots (iii)\end{array}$$

Where,2πλis the propagation constant, it is also called wave number.since,ωthen,y=asin(2πT.t−2πλ.x)ySince,ω=2πf=2πvλThen, from equation(ii),we can write,yy=2πT=asin2π(tT−xλ)…(iv)=asin(2π.vλ.t−2πλ).x=asin2πλ(vt–x)…(v)$$\begin{array}{r}\text{Where,}\frac{2\pi }{\lambda }\\ \text{is the propagation constant, it is also called wave number.}\\ \text{since,}\omega & =\frac{2\pi }{T}\\ \text{then,}y=a\sin \left(\frac{2\pi }{T}.t-\frac{2\pi }{\lambda }.x\right)\\ y& =a\sin 2\pi (\frac{t}{T}-\frac{x}{\lambda })\dots (iv)\\ \text{Since,}\omega =2\pi f=2\pi \frac{v}{\lambda }\text{Then, from equation}(ii),\text{we can write,}\\ y& =a\sin \left(2\pi .\frac{v}{\lambda }.t-\frac{2\pi }{\lambda }\right).x\\ y& =a\sin \frac{2\pi }{\lambda }(vt–x)\dots (v)\end{array}$$

Equations(ii),(iii),(iv)and(v)are the general equation for progressive wave.If the progressive wave travels from right to left direction.Then,y=asin2πλ(vt+x)…(vi)$$\begin{array}{r}\text{Equations}(ii),(iii),(iv)\text{and}(v)\\ \text{are the general equation for progressive wave.}\\ \text{If the progressive wave travels from right to left direction.}\\ \text{Then,}\\ y& =a\sin \frac{2\pi }{\lambda }(vt+x)\dots (vi)\end{array}$$

Differential Equation of Wave Motion

The equation of wave isyDifferentiating equation(i)with respect to t, we getdydtand again differentiating,d2ydt2When the equation(i)is differentiated with respect to x, we getdydxd2ydt2From equation(ii)and equation(iii),we haved2ydt2=v2d2ydx2…(iv)which is the differential wave equation.=asin2πλ(vt–x)…(i)=2πvλacos2πλ(vt–x)=−4π2v2λ2asin2πλ(vt–x)…(ii)=−2πλacos2πλ(vt–x)=−4π2λ2asin2πλ(vt–x)…(iii)$$\begin{array}{r}\text{The equation of wave is}\\ y& =a\sin \frac{2\pi }{\lambda }(vt–x)\dots (i)\\ \text{Differentiating equation}(i)\text{with respect to t, we get}\\ \frac{dy}{dt}& =\frac{2\pi v}{\lambda }a\cos \frac{2\pi }{\lambda }(vt–x)\\ \text{and again differentiating,}\\ \frac{d^{2}y}{d{t}^2}& =-\frac{4{\pi }^2{v}^2}{{\lambda }^2}a\sin \frac{2\pi }{\lambda }(vt–x)\dots (ii)\\ \text{When the equation}(i)\text{is differentiated with respect to x, we get}\\ \frac{dy}{dx}& =-\frac{2\pi }{\lambda }a\cos \frac{2\pi }{\lambda }(vt–x)\\ \frac{d^{2}y}{d{t}^2}& =-\frac{4{\pi }^2}{{\lambda }^2}a\sin \frac{2\pi }{\lambda }(vt–x)\dots (iii)\\ \text{From equation}(ii)\text{and equation}(iii),\text{we have}\\ \frac{d^{2}y}{d{t}^2}=v^2\frac{d^{2}y}{d{x}^2}\dots (iv)\\ \text{which is the differential wave equation.}\end{array}$$

Overview

According to superposition principle, if two or more waves are travelling in a medium at the same time, then the resultant displacement of waves is equal to the vector sum of individual displacement of the wave. This note provides us an information on principle of superposition and stationary waves.

Principle of Superposition and Stationary Waves

Principle of Superposition

If two or more waves are travelling in a medium at the same time, then the resultant displacement of waves is equal to the vector sum of individual displacement of the wave.

Let y⃗ 1,y⃗ 2,y⃗ 3……y⃗ n${\overrightarrow{y}}_1,{\overrightarrow{y}}_2,{\overrightarrow{y}}_3\dots \dots {\overrightarrow{y}}_n$ be the displacements of different waves travelling in medium at same time. Then, according to the principle of superposition

Resultant displacement,

y⃗ =y⃗ 1+y⃗ 2+y⃗ 3+…⋯+y⃗ n$$\overrightarrow{y}={\overrightarrow{y}}_1+{\overrightarrow{y}}_2+{\overrightarrow{y}}_3+\dots \cdots +{\overrightarrow{y}}_n$$

If two waves each of same amplitude ‘a’ superposition the resultant displacement is

yandy=a+a=2a(constructive interference)=a−a=0(destructive interference)$$\begin{array}{rl}y& =a+a=2a(\text{constructive interference})\\ \text{and}\\ y& =a-a=0(\text{destructive interference})\end{array}$$

The principle of superposition can be used to explain many wave phenomena. Some of them are as follows:

1. When two waves of same frequency moving in the same direction superpose, constructive interference of waves is produced.
2. When two waves of same frequency moving in the opposite direction superpose, stationary waves are produced.
3. When two waves of slightly different frequency moving in the same direction superpose, beats are produced.

Stationary Wave

If two waves having same amplitude and frequency are travelling in opposite direction on the medium at the same time then the resultant displacement of the wave becomes zero and the waves come in rest or stationary state. This type of wave is called stationary wave or standing wave. There are different point appear in a stationary wave.

Nodes

Two different waves meet at a point are called nodes. In node particles of the medium are in permanent rest.

Antinodes

The maximum displacement of the wave in either side from equilibrium position is called antinodes. The antinode particles are not in permanent rest.

Expression of Stationary Wave:

Consider two waves having displacement y1 and y2 travelling in opposite direction with same amplitude and frequency on the medium at same time. Then, the general progressive wave equation for first wave is given by

y⃗ 1=asin(ωt–kx)…(i)$$\begin{array}{rl}{\overrightarrow{y}}_1& =a\sin (\omega t–kx)\dots (i)\end{array}$$

and general progressive wave equation for second wave is given by,

y⃗ 2where,KAccording to principle of superposition. The resultant displacement of two wave=asin(ωt+kx)…(ii)=2πλ=wave number or propagation constant,$$\begin{array}{rl}{\overrightarrow{y}}_2& =a\sin (\omega t+kx)\dots (ii)\\ \text{where,}K& =\frac{2\pi }{\lambda }\\ & =\text{wave number or propagation constant,}\\ \text{According to principle of superposition. }\\ \text{The resultant displacement of two wave}\end{array}$$

y⃗ y⃗ y⃗ =y⃗ 1+y⃗ 2=asin(ωt–kx)+asin(ωt+kx)=a[sin(ωt–kx)+sin(ωt+kx)]=a[2sin(ωt–kx+ωt+kx)2]cos(ωt–kx−ωt–kx2)=a[2sinωt.cos(−kx)]=2asinωt.coskx=2acoskx.sinωt…(iii)$$\begin{array}{rl}\overrightarrow{y}& ={\overrightarrow{y}}_1+{\overrightarrow{y}}_2\\ \overrightarrow{y}& =a\sin (\omega t–kx)+a\sin (\omega t+kx)\\ & =a[\sin (\omega t–kx)+\sin (\omega t+kx)]\\ & =a[2\sin \frac{(\omega t–kx+\omega t+kx)}{2}]\cos (\frac{\omega t–kx-\omega t–kx}{2})\\ & =a[2\sin \omega t.\cos (-kx)]\\ & =2a\sin \omega t.\cos kx\\ \overrightarrow{y}& =2a\cos kx.\sin \omega t\dots (iii)\end{array}$$

Equation (iii) gives the expression for stationary wave where, 2acoskx$2a\cos kx$gives the amplitude of the stationary wave and sinωt$\sin \omega t$gives the nature of oscillation

If2acoskx=0kx2πxλor,x=(x+12)π=(x+12)π=(x+12)π2$$\begin{array}{r}\text{If}2a\cos kx=0\\ kx& =\left(x+\frac{1}{2}\right)\pi \\ \frac{2\pi x}{\lambda }& =\left(x+\frac{1}{2}\right)\pi \\ \text{or,}x& =\left(x+\frac{1}{2}\right)\frac{\pi }{2}\end{array}$$

Equation (iv) represents the position of node where displacement of wave is 0 and n is interger.

Ifcoskxkx2πλ.xor,x=±1=nπ=xπ=n.λ2…(v)$$\begin{array}{rl}\text{If}\cos kx& =\pm 1\\ kx& =n\pi \\ \frac{2\pi }{\lambda }.x& =x\pi \\ \text{or,}x& =\frac{n.\lambda }{2}\dots (v)\end{array}$$

Equation} (v) represents the position of antinodes of stationary wave where displacement is maximum. If sinωt=±1$\sin \omega t=\pm 1$ Then the wave oscillates in either side of mean position

Ifsinωt=0The wave just crosses the nodes$$\begin{array}{r}\text{If}\sin \omega t=0\\ \text{The wave just crosses the nodes}\end{array}$$

Properties of Stationary Wave

1. When two progressive waves of same amplitude and frequency travel in a medium in opposite direction to each other, a stationary wave is produced.
2. Nodes and antinodes are formed alternatively in the wave. A node is a position of zero displacement and maximum strain whereas an antinode is the position of maximum displacement and zero strain. However, if these waves travel in the same direction, they produce an interference pattern.
3. All the particles except at the nodes vibrate simple harmonically with the period equal to that of the wave.
4. The amplitude of vibration gradually increases from zero at node to maximum at antinode.
5. The medium splits into segments and all the particles of segment vibrate in phase. The particles in one segment have a phase difference of p with the particles in the neighboring segment.
6. The disturbance does not travel forward; there is no transfer of energy.
7. The disturbance between two adjacent nodes is λ/2 and that between two adjacent antinodes is also λ/2. The distance between a node and adjacent node is λ/4.
8. The velocity and acceleration of all the particles separated by a distance λ are the same at the given instant.

Difference between a Progressive and a Stationary Wave

S.N.	Progressive Wave	Stationary Wave
1.	The disturbance travels forward in a medium and is handed over from one particle to the next after some time.	The disturbance is at rest and does not move at all. So there is no transfer of disturbance to the neighboring particles.
2.	The amplitude of oscillation is same at all positions in the medium.	The amplitude of oscillation varies from zero at the node to a maximum at the antinode.
3.	No particle is permanently at rest.	The particles at nodes are permanently at rest.
4.	Energy is transmitted from particle to particle across every section of the medium.	Energy is not transmitted from particle to particle, i.e. no transfer of energy across every section of the medium.
5.	As the disturbance moves, every part of the medium suffers a change in density.	At antinodes, there is no change in density but at node there is maximum.
6.	At every point, there is variation in pressure.	Pressure variation is maximum at nodes and zero at antinodes.
7.	Regular phase difference exists between successive particles.	All the particles in between two successive nodes are in phase.
8.	The value of maximum velocity for all particles of the medium is same.	The value of maximum velocity for different particles is different and velocity of the particles at the node is always zero.

Overview

Sound waves are reflected from the surfaces like light waves, obeying laws of reflection; angle of incident is equal to the angle of reflection. This note provides us an information on reflection, refraction, diffraction and interference of sound waves.

Reflection, Refraction, Diffraction and Interference of Sound Waves

Reflection

Reflection

Sound waves are reflected from the surfaces like light waves, obeying laws of reflection; angle of incident is equal to the angle of reflection. This can be observed in a simple experiment in which a source of sound such as a transmitter sends sound at a certain angle to the normal to a wall and a higher sound is received from reflection at the same angle on another side of the normal.

The regular reflection occurs from a surface if a whole part of an incident wavefront is reflected uniformly. Since the sound wave in speech has a wavelength of various meters, they are reflected from rough surfaces such as cliffs, walls etc. Reverberation occurs after incident rays undergo multiple reflections.

Refraction

Refraction

Soundwave bends when the parts of wave fronts travel at different speeds. This occurs in uneven winds or when sound is travelling through an air of uneven temperatures. On a warm day, the air near the ground becomes warmer than the rest of the air and the speed of sound near the ground increases. So, sound waves then tend to bend away from the ground resulting in a sound that does not seem to travel well as shown in the figure. Due to the effect, sounds are easier to hear at night than during the day times as shown in the figure. Different speeds of sound produce refraction.

Diffraction

Diffraction

Diffraction is the phenomenon of spreading of waves around the corners of an obstacles or apertures. It is a wave phenomenon and occurs if the wavelength of the wave is of the same order as the dimension of a diffracting obstacle. Due to diffraction of sound through the doors and windows, a sound is heard inside the room from an outside source. This is due to the wavelength of a sound wave is nearly equal to the dimension of doors and windows. However, light having small wavelength can be diffracted only through a very sharp object such as the edge of a blade and diffracting grating. Diffraction of water waves in a ripple tank through a small aperture is shown in the figure. It is observed that smaller the width of the aperture, greater is the spreading of the waves.

Interference

interference

When two frequency and amplitude overlap, this produces interference of the waves. It is characteristics of all wave motion, whether the waves are sound, light or water waves. The interference pattern consists of a region where intensity is maximum at certain points and minimum at certain points. The positions of maximum intensity are called maxima and the superposition is called constructive interference while the positions of minimum intensity are called minima and the superposition is called destructive interference.

Undamped and Damped Oscillations

A system executing simple harmonic motion is called a harmonic oscillator. A harmonic oscillator produces sinusoidal oscillations. The sinusoidal oscillations can be of two types. They are damped and undamped oscillation.

Undamped Oscillations

The oscillations whose amplitude remains constant with time are called undamped oscillations. Such oscillations can occur if frictional forces on the oscillating systems are absent. For example, if the bob of a simple pendulum is displaced in a vacuum and then released, the bob executed simple harmonic motion with constant amplitude.

Damped Oscillation

The oscillations whose amplitude goes on decreasing with time are called damped oscillations. In a real oscillating system, forces like friction are always present that dissipate the energy of the oscillator. Unless energy somehow is added to the system, dissipation eventually brings the system to rest i.e. in equilibrium.

Free, Forced and Resonant Oscillations

Free Forced and Resonant Oscillations

When a body capable of oscillation is displaced from its equilibrium position and then left free, it begins to oscillate with a definite amplitude frequency. If the body is not resisted by any kind of friction, the motion continues. Such oscillation is called free oscillation. The frequency of vibration depends on the intrinsic properties (shape, elasticity etc.) of the body which is as the natural frequency. The force acting on the system is restoring force. For example, when a simple pendulum is displaced from its mean position and then left free, it executes free oscillations. The frequency of the simple pendulum is f=(12π)1/g−−−√$f=(\frac{1}{2}\pi )\sqrt{1/g}$.

Forced Oscillation

When a body is maintained in a state of oscillation by an external periodic force of frequency other than the natural frequency of the body, the oscillation is called the forced vibration. In such oscillation, the frequency of oscillation is equal to the frequency of the periodic force. The externally applied force on the body is called the driver and the body set into oscillation is called driven oscillation.

Resonant Oscillation

When a body is maintained in a state of oscillations by a periodic force having the same frequency as the natural frequency of the body, the oscillations are called resonant oscillations. The phenomenon of producing resonant oscillations is called resonance. Note that the resonance is a particular case of forced oscillations in which the two frequencies are equal.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.