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Summary

  • Mechanical waves can exist in material media and are governed by Newton's laws.
  • Transverse waves are waves in which the particles of the medium oscillate perpendicular to the direction of wave propagation.
  • Longitudinal waves are waves in which the particles of the medium oscillate along the direction of wave propagation.
  • Progressive wave is a wave that moves from one point of medium to another.
  • The displacement in a sinusiodal wave propagating in the positive x direction is given by
  • Where a is the amplitude of the waves, k is the angular wave number, ω is the angular frequency , is the phase, and is the phase constant or phase angle.
  • Wavelength λ of a progressive wave is the distance between two consecutive points of the same phase at a given time. In a stationary wave, it is twice the distance between two consecutive nodes or anti-nodes.
  • Period T of oscillation of a wave is defined as the time any element of the medium takes to move through one complete oscillation. It is related to the angular frequency ω through the relation
     
  • Frequency of a wave is defined as i/T and is related to angular frequency by
     
  • Speed of a progressive wave is given by
  • The speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with tension T and linear mass density μ is
  • Sound waves are longitudinal mechanical waves that can travel through solid,liquids, or gases. The speed v of sound wave in a fluid having bulk modulus B and density ρ is
  • The speed of longitudinal wave in a metallic bar is
  • For gases, since , the speed of sound is
  • When two or more waves traverse the same medium, the displacement of any element of the medium is the algebraic sum of the displacements due to each wave. This is known as the principle of superposition of waves
  • Two sinusoidal waves on the same string exhibit interference, adding or cancelling according to the principle of superposition. If the two are travelling in the same direction and have the same amplitude a and frequency but differ in phase by a phase constant Φ, the result is a single wave with same frequency ω:
  • If Φ=0 or an integral multiple of , the waves are exactly in phase and the interference is constructive: if Φ=∏, they are exactly out of phase and the interference is destructive.
  • A travelling wave, at a rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary takes place without any phase change.
  • For an incident wave
  • The reflected wave at a rigid boundary is
  • For reflection at an open boundary
  • The interference of two identical waves moving in opposite directions produce standing waves. For a string with fixed ends, the standing wave is given by
  • Standing waves are characterised by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The seperation between two consecutive nodes or antinodes is λ/2.
  • A stretched string of length L fixed at both the ends vibrates with frequencies given by
  • The set frequencies given by the above relation are called normal modes of oscillation of the system. The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscillation mode with n=2 and so on.
  • A pipe of length L with one end closed and other end open (such as air columns) vibrates with frequencies given by
  • The set of frequencies represented by the above relation are the normal modes of oscillation of such a system. The lowest frequency given by v/4L is the fundamental mode or the first harmonic.
  • A string of length L fixed at both ends or an air column closed at one end open at the other end, vibrates with frequencies called its normal modes. Each of these frequencies is a resonant frequency of the system.
  • Beats arise when two waves having slightly different frequencies, and and comparable amplitudes, are superposed. The beat frequency is
  • The doppler effect is a change in the observed frequency of a wave when the source and the observer O moves relative to the medium. For sound the observed frequency is given in terms of the source frequency by
Here v is the speed of the sound through the medium, v0 is the velocity of observer relative to the medium, and vs is the source velocity relative to the medium. In using this formula, velocities in the direction OS should be treated as positive and those opposite to it should be taken to be negative.




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