# 9.1 Introduction to Waves

• It transfers energy.

• Usually involves a periodic, repetitive movement.

• Does not result in a net movement of the medium or particles in the medium (mechanical wave).

• There are some basic descriptors of a wave.

• Wavelength is the distance between two successive identical parts of the wave.

• Amplitude is the maximum displacement from the neutral position.

• This represents the energy of the wave. Greater amplitude carries greater energy.

• Displacement is the position of a particular point in the medium as it moves as the wave passes.

• Maximum displacement is the amplitude of the wave

• Frequency (ƒ) is the number of repetitions per second in Hz, and Period (T) is the time for one wavelength to pass a point.

• The velocity (v) of the wave is the speed at which a specific part of the wave passes a point. The speed of a light wave is c.

# 9.2 Types of Waves

• Transverse Waves

• Waves in which the medium moves at right angles to the direction of the wave.

• The high point of a transverse wave is a crest. The low part is a trough.

• Examples of transverse waves:

• Water waves (ripples of gravity waves, not sound through water)

• Light waves

• S-wave earthquake waves

• Stringed instruments

• Torsion wave

• Longitudinal Waves:

• A longitudinal wave has the movement of the particles in the medium in the same dimension as the direction of movement of the wave.

• Parts of longitudinal waves:

• Compression: where the particles are close together.

• Rarefaction: where the particles are spread apart.

• Examples of longitudinal waves:

• Sound waves

• P-type earthquake waves

• Compression wave

### Mechanical waves:

• A wave which needs a medium to propagate itself.

• Sound waves, waves in a slinky, and water waves are all examples of this.

### Matter Waves:

• Any moving object can be described as a wave.

• When a stone is dropped into a pond, the water is disturbed from its equilibrium position as the wave passes; it returns to its equilibrium position after the wave has passed.

### Electromagnetic Waves:

• These waves are disturbance that does not need any object medium for propagation and can easily travel through the vacuum.

• They are produced due to various magnetic and electric fields.

• The periodic changes that take place in magnetic and electric fields and therefore known as electromagnetic waves.

# 9.3 Properties of Waves

• The prime properties of waves are as follows:

• Amplitude – Wave is an energy transport phenomenon.

• Amplitude is the height of the wave, usually measured in metres.

• It is directly related to the amount of energy carried by a wave.

• Wavelength – The distance between identical points in the adjacent cycles of crests of a wave is called a wavelength.

• It is also measured in metres.

• Period – The period of a wave is the time for a particle on a medium to make one complete vibrational cycle.

• As the period is time, hence is measured in units of time such as seconds or minutes.

• Frequency – The frequency of a wave is the number of waves passing a point in a certain time.

• The unit of frequency is hertz (Hz) which is equal to one wave per second.

• The period is the reciprocal of the frequency and vice versa.

• Speed – The speed of an object means how fast an object moves and is usually expressed as the distance travelled per time of travel.

• The speed of a wave refers to the distance travelled by a given point on the wave (crest) in a given interval of time.

• The speed of a wave is thus measured in metres/second i.e. m/s.

# 9.4 Simple Harmonic Motion

• Definition:

• Simple Harmonic Motion (SHM) is described by Newton's Second Law through the following equations:

• x = x_0(cos(ωt))

• v = -ωx_0(sin(ωt))

• a = -ω^2(x_0)(cos(ωt))

• Here,

• x_0 is the amplitude (maximum displacement),

• x is the displacement,

• v is the velocity, a is the acceleration, and

• ω is the angular frequency related to the period (T) through ω= 2π/T.

• Energy Changes:

• In SHM, there's an exchange between kinetic energy (KE) and potential energy (PE) throughout the motion, while the total energy (KE + PE) remains constant.

# 9.5 Difference between Periodic, Oscillation and Simple Harmonic Motion

• Periodic Motion

• A motion repeats itself after an equal interval of time. For example, uniform circular motion.

• There is no equilibrium position.

• There is no restoring force.

• There is no stable equilibrium position.

• Oscillation Motion

• To and fro motion of a particle about a mean position is called an oscillatory motion in which a particle moves on either side of the equilibrium (or) mean position is an oscillatory motion.

• It is a kind of periodic motion bounded between two extreme points.

• For example, the oscillation of a simple pendulum, spring-mass system.

• The object will keep on moving between two extreme points about a fixed point is called the mean position (or) equilibrium position along any path (the path is not a constraint).

• A restoring force will be directed towards the equilibrium position (or) mean position.

• In an oscillatory motion, the net force on the particle is zero at the mean position.

• The mean position is a stable equilibrium position.

• Simple Harmonic Motion or SHM

• It is a special case of oscillation, along with a straight line between the two extreme points (the path of SHM is a constraint).

• The path of the object needs to be a straight line.

• A restoring force will be directed towards the equilibrium position (or) mean position.

• The mean position in Simple Harmonic Motion is a stable equilibrium.

Summary:

• At maximum displacement, PE is at its maximum while KE is zero.

• At zero displacement, KE is at its maximum while PE is zero.

• At minimum displacement, PE is at its maximum while KE is zero.

• Total energy remains constant throughout the motion.

# 9.6 Single-Slit Diffraction

• Nature of Single-Slit Diffraction:

• Distinct diffraction patterns emerge when light passes through a single slit comparable in size to the wavelength of the light.

• Representation of Diffraction Pattern:

• This pattern is represented by plotting light intensity against the angle of diffraction.

• Angle of Diffraction for First Minimum θ:

• sinθ = λ/a

• Here, λ is the wavelength, and a is the size/length of the slit.

• sinθ_m = m(λ/D)

• Where m is the order of the maximum, D is the distance from the slit to the screen.

# 9.7 Interference

• Young’s Double-Slit Experiment:

• In this experiment, interference patterns are observed when light passes through two slits, creating regions of constructive and destructive interference.

• Modulation of Double-Slit Pattern by Single-Slit Diffraction:

• A true double-slit pattern shows closely spaced dark and light areas, superimposed over the single-slit pattern.

• The single-slit profile modulates the double-slit pattern.

• Multiple Slit and Diffraction Grating Interference Patterns:

• Multiple Slit Interference Patterns:

• θ = m(λ/a)

• Diffraction Grating Interference Patterns:

• dsinθ = mλ

• Where d is the distance between gratings, m is the order of the maximum, and λ is the wavelength.

# 9.8 Resolution

• Diffracting Aperture Size:

• The resolution of an image passing through a diffracting aperture improves with a larger aperture diameter.

• Resolution of Two-Source Systems:

• The Rayleigh criterion determines whether two points are just resolved. The minimum angular separation θ for two points to be just resolved is given by θ = 1.22(λ/a)

• Importance of Resolution in Technology:

• Resolution is crucial in technologies like CDs, DVDs, electron microscopes, and radio telescopes for optimal performance.

# 9.9 Doppler Effect

• Doppler Effect Equations for Sound Waves:

• Four Doppler effect equations cater to different cases based on the movement of the source and/or observer.

• Doppler Equation for Electromagnetic Waves:

• change in f = (v/c)f

• This equation is applicable when the velocity of the observer is much smaller than the speed of light.

• Application in Speed Detectors:

• The Doppler effect is utilized in speed detectors where the frequency shift indicates whether a car is approaching or moving away, enabling the calculation of the car's speed.