Three Chapter 7

• Wave - a Transfer of energy in the form of a disturbance.
  • On a rope - the disturbance is a side ways displacement of the rope which is not normally there.
  • For sound - the disturbance is the slight crowding together or separating of air molecules.
  • For light - the disturbance is a fluctuation in the electromagnetic field where none exists.
• The speed of a wave is determined by the properties of the medium and not by the frequency or wavelength.
  • For a rope, wave speed is determined by the tension in the rope.
    • high tension means high speed.
  • As frequency changes, wavelength changes and the speed stays constant.
• Amplitude has no effect on speed, frequency or wavelength.
• Energy of a wave depends upon:
  • Amplitude - the larger the amplitude, the larger the energy.
  • Frequency - the higher the frequency, the more times the source applies a force through a distance in order to produce the wave, and therefore, the higher the energy.

• Plane polarized - the displacement of the particles in a medium of a transverse wave in one plane only.
  • Only transverse waves can be polarized.
• Test for polarization - A wave is polarized if there is some aspect of the wave that changes as the medium the wave must pass through is rotated.

• A continuous disturbance of a medium at regular intervals will produce a wave in the medium
  • E.G. A hand holding a rope is swung up and down in a regular manner
• The frequency of the wave will be the same as the frequency of the disturbance of the medium (source)
• The source alone determines frequency.
• We defined the period of a wave as the time it takes a wave to travel one wavelength. This means:

• The above equation is the universal wave equation, and it applies to all waves.
• The speed of the wave (V) is a constant as long as the medium's characteristics remain constant.
• If the frequency (f) increases (because the source frequency increases), then the wavelength (l) decreases so that frequency times wavelength is a constant (V).

• Waves travel with uniform motion unless the medium changes.
• It is the characteristics of the medium that determines wave speed.
• It is the source that determines the frequency of a wave, therefore, since V = fl, if V changes, then l must change.
  • If V increases, then l increases.
  • If V decreases, then l decreases.
• When a wave moves from a less dense to a more dense medium, then V decreases and l decreases. On the other hand, if the wave moves from a more dense to a less dense medium, then V increases and l increases.
• If wave speed remains constant but the frequency of the source changes, then if f increases then l decreases. Similarly, if f decreases, then l increases.
• Speed remains constant if the medium remains unchanged.
• Fixed End Reflection - reflection of a wave off of a rigid obstacle.
  • e.g. a rope attached to a wall.
  • The wave is reflected in an inverted state with no frequency change (i.e. a crest reflects as a trough and a trough reflects as a crest).
• Free End Reflection - reflection of a wave off of the end of the medium which is not fixed.
  • e.g. a rope not attached to anything (a whip)
  • The wave is reflected without inversion and there is no frequency change (a crest reflects as a crest and a trough as a trough).
• Partial Reflection - When a wave moves from one medium to the next, there will be part of the wave reflected and part transmitted into the second medium.
• Consider a wave moving from fast to slow medium (i.e. light to heavy or less dense to more dense).
  • The reflected wave is inverted (the heavy medium acts like a rigid barrier).
  • The transmitted wave in not inverted.
  • The reflected and transmitted waves have smaller amplitude (They share total energy).
  • The transmitted wave moves slower.
  • The reflected wave moves at the same speed.
  • The wavelength of the transmitted wave decreases.
• Now consider a wave moving from a slow to fast medium (i.e. heavy to light or more dense to less dense).
  • The reflected wave is not inverted (The boundary acts like a free end).
  • The transmitted wave is not inverted.
  • The reflected and transmitted waves have smaller amplitude (They share the total energy).
  • The transmitted wave moves faster.
  • The reflected wave moves at the same speed.
  • The wavelength of the transmitted wave increases.
• In both cases, wavelength changes because speed changes. Frequency remains constant because the source is the same regardless of the medium the wave is in.

• Waves when they meet, pass through each other, unaffected by the other.
• Wave Interference - Occurs when two or more waves act simultaneously on the same particles of a medium.
• Principle of Superposition - The amplitude of the medium at the points of interference of waves is the sum of the displacements that would be present for each of the waves acting independently.

• Constructive Interference - Occurs when two or more waves interfere to produce a resulting displacement greater than the displacements that would be caused by either wave by itself.
• Destructive Interference - Occurs when two or more waves interfere to produce a resulting displacement less than the displacements that would be caused by either wave by itself.
• The principle of superposition is used regardless of the number of waves present.

• Interfering waves often have different wavelengths and frequencies.
• Standing wave Interference pattern - the interference pattern in the medium when two waves of the same wavelength, frequency and amplitude moving in opposite directions interfere.
  • The pattern is what you see.
  • This pattern remains relatively stationary, therefore, we call it standing.
• Node or Nodal Points - points that remain at rest throughout the interference of pulses or waves.
  • These occur when two waves of equal amplitude interfere, only one is of negative and the other is of positive amplitude.
  • The distance between nodes is one half of the wavelength of the interfering waves.
• Loops or Anti nodes - regions midway between nodes where double crests and double troughs occur.
  • Occurs with standing waves

• A single source may produce standing waves if the wave is reflected back to the source off of a fixed end.
• The distance between nodes can be changed by changing the frequency.
• For a given medium, only certain wavelengths will produce standing waves that will be maintained. This is because the reflecting end must be a nodal point.
  • Only certain allowed wavelengths will fit into the length of the medium.

• Resonant Frequencies - (natural frequencies) - The frequencies at which standing waves can exist in a medium of a given length whose ends are fixed
  • Some objects have only 1 resonant frequency. Examples would be a tuning fork and a pendulum.
• Fundamental - lowest resonant frequency (fundamental frequency).
• Resonant frequencies are whole number multiples of the fundamental.
• A stretched string which is plucked has many wavelengths that travel its length and reflect off of the ends.
  • Because they interfere with themselves, most reflections cancel the incident wave and therefore, the waves die out quickly.
  • The frequencies that equal a resonant frequency, will not cancel themselves, and therefore will persist.
• For a string attached at both ends
  • the ends must be a node
  • Resonant frequencies allow the ends to be nodes. This means that the length of the medium relates to the wavelength of the wave by the equation

• In the above equation, L is the length of the medium, l is the wavelength of the wave, and n is a whole number (1, 2, 3, ... ) equal to the number of loops.
• Fundamental mode - The mode of vibration where the medium vibrates as one segment.
  • The number of loops = 1.
  • produces the fundamental frequency.
• Overtones - The modes of vibration that have the medium vibrate in more than one segment.
  • The number of loops is > 1
• Harmonics - The various resonant frequencies of standing waves.
  • The number of loops equals 1, 2, 3, 4, ... etc.
  • Are whole number multiples of the fundamental frequency (f_o)
  • The fundamental is the first harmonic
  • Overtones are second, third, fourth, etc. harmonics.
• Most stringed instruments vibrate with a complex mixture of overtones superimposed on the fundamental
• Tuning forks produce fundamentals with no overtones.

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November 6, 2013