Mechanical Waves

• A mechanical wave is a disturbance that travels through some material or substance called the medium of the wave.

  • A transverse wave is one where the medium is displaced perpendicular to the direction of travel of the wave.
  • A longitudinal wave is one where the medium is displaced parallel to the direction of travel of the wave.
  • Waves can have both transverse and longitudinal components

• All mechanical waves have the following in common:

  • The disturbance propagates with a definite speed through the medium called the wave speed
  • The medium does not travel through space. Instead, its particles undergo motions that displace it from the equilibrium position.
  • Waves transport energy but not matter from one region to another.

Simple 1D Motion §

• Simple mechanical waves can be created using Simple Harmonic Motion by applying a periodic force on one end of the medium. We call such waves periodic waves

  • Any periodic wave can be represented as a combination of sinusoidal waves
  • When a sinusoidal wave passes through a medium, every particle in the medium undergoes simple harmonic motion with the same frequency.
  • The shape is a repeated pattern. The wavelength is the distance from one point to the corresponding point in the next pattern instance. It is computed as
    Here represents the phase velocity, also called the wave speed, the speed the disturbance propagates through the wave.

• The motion of a wave can be described using the following function. represents the distance from the start of the medium and represents time

  This can also be described using the angular wave number

  Or the wave number

  • If we have a negative , the wave travels in the negative direction.
  • The quantity is the phase. It determines what part of the sinusoidal cycle is occurring.
    • The phase of a wave does not change or vary with time.
  • The wave speed is calculated as
  • The kinematics of a sinusoidal wave are given by the following
    The second formula is the wave equation in one dimension.

• A wave’s amplitude is independent of its wavelength or frequency.

• The wave speed is dependent on two factors — a restoring force and an inertia resisting the return to equilibrium

  • For a string under tension and linear density (density per unit length) is ¹

  • The average power of a sinusoidal wave is given by

    The general equation for power is given by

    Where and are forces in the direction of the particle’s motion.

• A wave’s intensity is the time average rate at which energy is transported by the wave per unit area.

  It follows an inverse-square law where is the distance from the source

  This assumes waves spread out equally in all directions. This result follows from the Conservation of Energy. The power outputted (energy per time) must be constant.

Interacting Waves §

• Interference happens when two or more waves pass through the same region at the same time.

• When a wave travels and hits a boundary, the wave is reflected back which induces interference.

  • If the medium exerted an upward force on a fixed boundary, a downward reaction force is exerted on the medium and the pulse is sent “inverted”
  • On the other hand, if the medium exerted an upward force on a movable boundary, a pulse is sent from the boundary in an upward manner.

• If pulses overlap, the total displacement of the medium is the sum of the displacements at that point in the individual pulses.

  In general, the principle of superposition can be stated as follows

  Where is the wave function of the combined wave and the ’s are the individual wave functions.

  This does not hold for all waves, only waves whose mechanics allow for a motion that can be expressed as linearly (i.e. non-Hookean mediums do not exhibit this behavior)

• A standing wave is a wave pattern which does not appear to be moving in either direction of the stream. This happens when two waves interfere with each other.

  • Destructive Interference - happens when two waves cancel each other because at that point they have equal and opposite displacement.
  • Constructive Interference - happens when two waves interfere in such a way that the resulting displacement is larger.

• In a standing wave, the nodes are the points where the medium never moves, and the antinodes are the points where the amplitude of string motion is greatest.

  • Antinodes are found halfway between two nodes.
  • Nodes are apart.

• The wave function for a standing wave is given by

  Where is the amplitude of the standing wave. And we assume that the wave is fixed at .

• Standing waves do not transfer energy because at the points of destructive interference, the power from both sides cancel out. The net effect is that no net energy is transferred.

• Consider a string of length . By the nature of standing waves, the length of the string must be a multiple of half the wavelength

  Since the nodes of the wave must be half a wavelength apart. To support standing waves, needs to allow for this

• The fundamental frequency of a waveform is defined as the one with the largest wavelength and the lowest frequency. That is

  The Harmonic Series is defined as

• A normal mode of an oscillating system is a motion where all particles move sinusoidally with the same frequency

Generalizations §

• The Wave Equation is a generalization to the relation between a wave’s acceleration and its displacement. The denotes the Laplacian. We have

Links §

• University Physics with Modern Physics 15th Edition By Freedman - Ch. 15

• Periodic Motion

• Acoustics

Footnotes §

1. The intuition behind this is that, the total momentum increases with time due to the increasing mass being displaced. The mass displaced is equal to times the length displaced so far . The momentum of each particle on the string is proportional to . ↩