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.