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Maharashtra State BoardHSC Science (Computer Science) 12th Standard Board Exam
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Syllabus

Overview: Superposition of Waves

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Estimated time: 48 minutes
Maharashtra State Board: Class 12

Definition: Mechanical Wave

A mechanical wave is a disturbance produced in an elastic medium due to periodic vibrations of particles of the medium about their respective mean positions.

Maharashtra State Board: Class 12

Definition: Progressive Wave

A wave, in which the disturbance is produced in the medium travels in a given direction continuously, without any damping and obstruction, from one·particle to another, is a progressive wave or a travelling wave.

Maharashtra State Board: Class 12

Key Points: Properties of Progressive Waves

  • In a progressive wave, each particle of the medium vibrates about its mean position in simple harmonic motion.
  • All particles have the same amplitude, frequency, and time period.
  • The phase of vibration changes from particle to particle.
  • Energy is transferred through the medium, but no matter is transferred.
  • Particles have maximum velocity at the mean position and zero velocity at extreme positions.
  • Wave velocity depends on the properties of the medium.
  • Progressive waves are of two types:
    Transverse (vibrations perpendicular to the direction of propagation)
    Longitudinal (vibrations parallel to the direction of propagation).
Maharashtra State Board: Class 12

Definition: Reflection of Waves

When a progressive wave, travelling through a medium, reaches an interface separating two media, a certain part of the wave energy comes back in the same medium. The wave changes its direction of travel. This is called reflection of a wave from the interface.

Maharashtra State Board: Class 12

Definition: Principle of Superposition

When two or more waves, travelling through medium, pass through  common point, each wave produces its own displacement at that point, independent of the presence of the other wave. The resultant displacement at that point is equal to the vector sum of the displacements due to the individual wove at that point.

Maharashtra State Board: Class 12

Key Points: Superposition of Waves

  • When two waves meet, the resultant displacement is the sum of their individual displacements.
  • After crossing, waves continue with their original shape and speed.
  • In-phase waves (ϕ = 0) produce maximum amplitude (constructive interference).
  • Out-of-phase waves (ϕ = π) produce a minimum or zero amplitude (destructive interference).
  • Intensity depends on amplitude², so it is maximum in constructive and minimum in destructive interference.
Maharashtra State Board: Class 12

Definition: Stationary Wave

When two identical waves travelling along the same path in opposite directions interfere with each other, resultant wave is called stationary wave.

Maharashtra State Board: Class 12

Key Points: Properties of Stationary Waves

  • Stationary waves are formed by the superposition of two identical waves travelling in opposite directions.
  • In a stationary wave, some points remain at rest (nodes), and some vibrate with maximum amplitude (antinodes).
  • The distance between two consecutive nodes or two consecutive antinodes is \[\frac {λ}{2}\], and between a node and adjacent antinode is \[\frac {λ}{4}\].
  • All particles between two adjacent nodes vibrate in the same phase, but adjacent loops vibrate in opposite phase.
  • In stationary waves, energy is not transferred; the wave is localised, and the wave velocity is zero.
Maharashtra State Board: Class 12

Key Points: Progressive and Standing Waves

  • A progressive wave travels through the medium, whereas a stationary wave does not.
  • In progressive waves, all particles have the same amplitude; in stationary waves, the amplitude varies.
  • In progressive waves, particles cross the mean position at different times; in stationary waves, they cross together.
  • In progressive waves, all particles move; in stationary waves, nodes remain at rest.
  • Progressive waves transfer energy; stationary waves do not transfer energy.
Maharashtra State Board: Class 12

Definition: Natural Frequency

The frequency at which a body vibrates when disturbed and left free to vibrate on its own.

Maharashtra State Board: Class 12

Definition: Free Vibrations

Vibrations in which a body vibrates at its natural frequency without any external periodic force.

Maharashtra State Board: Class 12

Definition: Forced Vibrations

Vibrations in which a body vibrates under the influence of an external periodic force, with the frequency of the external force.

Maharashtra State Board: Class 12

Definition: Resonance

The phenomenon in which the amplitude of forced vibrations becomes maximum when the driving frequency equals the natural frequency.

Maharashtra State Board: Class 12

Definition: Fundamental Frequency

The lowest natural frequency of a vibrating system is called the fundamental frequency or first harmonic.

Maharashtra State Board: Class 12

Definition: Fundamental Mode

The mode of vibration corresponding to the fundamental frequency is called the fundamental mode or fundamental tone.

Maharashtra State Board: Class 12

Key Points: Vibrations of Air Columns and End Correction

  • End correction is the extra length added because the antinode forms slightly outside the open end.
    e ≈ 0.3d
  • For a closed pipe: corrected length L = l + e; only odd harmonics are present.
  • For an open pipe: corrected length L = l + 2e; all harmonics are present.
  • Fundamental frequency:
    Closed pipe → n = \[\frac {v}{4L}\]
    Open pipe → n = \[\frac {v}{2L}\]
  • For the same length, the fundamental frequency of an open pipe is double that of a closed pipe.
Maharashtra State Board: Class 12

Formula: Fundamental Frequency of a Vibrating String

n = \[\frac{1}{2l}\sqrt{\frac{T}{\mu}}\]

where:

  • l = length of string
  • T = tension
  • μ = mass per unit length (linear density)
Maharashtra State Board: Class 12

Key Points: Vibrations of a Stretched String

  • A stretched string fixed at both ends forms stationary waves with nodes at the ends.
  • The fundamental mode has one loop, with wavelength λ = 2l.
  • Frequencies of harmonics are integral multiples of the fundamental:
    np = pn
  • Frequency increases with tension and decreases with length and linear density.
Maharashtra State Board: Class 12

Law: Law of Length

The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string, if tension and mass per unit length are constant.
n ∝ \[\frac {1}{l}\], if T and m are constant.

Maharashtra State Board: Class 12

Law: Law of Tension

The fundamental frequency of vibrations of a string is directly proportional to the square root of tension, if vibrating length and mass per unit length are constant.
n ∝ \[\sqrt {T}\] , if l and m are constant.

Maharashtra State Board: Class 12

Law: Low of Linear Density

The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density), if the tension and vibrating length of the string are constant.

n ∝ \[\frac{1}{\sqrt{m}}\], if T and l are constant.

Maharashtra State Board: Class 12

Definition: Sonometer

A sonometer is an apparatus used to study the vibrations of a stretched string and to verify the laws of a vibrating string.

Maharashtra State Board: Class 12

Definition: Sound Box

The hollow wooden box used to amplify the sound produced by the vibrating string is called the sound box.

Maharashtra State Board: Class 12

Key Points: Verification of Laws of Vibrating String

  • First law (Law of length):
    For constant tension and linear density, frequency is inversely proportional to length:
    n ∝ \[\frac {1}{l}\]
  • Second law (Law of tension):
    For constant length and linear density, frequency is directly proportional to the square root of tension:
    n ∝ \[\sqrt {T}\]
  • Third law (Law of linear density):
    For constant length and tension, frequency is inversely proportional to the square root of mass per unit length:
    n ∝ \[\frac {1}{\sqrt {m}}\]These three laws are verified using a sonometer.
Maharashtra State Board: Class 12

Definition: Beats

Beats are the periodic variation in intensity of sound produced due to the superposition of two sound waves having nearly equal (slightly different) frequencies.

Maharashtra State Board: Class 12

Formula: Frequency of Beats

N = n1 - n2

Maharashtra State Board: Class 12

Definition: Time Period of Beats

The interval between two maximum sound intensities is the time period of beats.

Maharashtra State Board: Class 12

Key Points; Application of Beats

  • Beats are used to tune musical instruments; when beat frequency becomes zero, the instruments are in unison (same frequency).
  • Beats are used in Doppler RADAR and SONAR to determine the speed of moving objects like aeroplanes, vehicles, and submarines.
  • Beats help in finding an unknown frequency by adjusting a known frequency source until beat frequency becomes zero.
Maharashtra State Board: Class 12

Key Points: Sound and Musical Instruments

  • Sound has three main characteristics: loudness, pitch, and quality (timbre).
  • Loudness depends on intensity and amplitude; sound level is measured in decibels (dB).
    β = 10log⁡10\[\frac {I}{I_0}\]
  • Pitch depends on frequency; a higher frequency means a higher pitch.
  • Quality (timbre) depends on the number and relative amplitudes of overtones; it helps distinguish between sounds of the same pitch and loudness.
  • A musical sound is produced by regular, periodic vibrations; irregular vibrations produce noise.
  • Musical instruments work by producing stationary waves in strings, air columns, or membranes.
  • Musical instruments are classified into stringed, wind, and percussion instruments.
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