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Lesson 3.7
Lesson 3.5 |
Lesson 3.7 Phases & Beats Overview This lesson deals with
MINI LAB
CHOICE OF ACTIVITIES
Wave Interference Waves that are moving through the same medium can interfere with each other in ways that can be desirable or undesirable. For example, surfers can be bounced into the air when the wave they are on meets a wave that has been reflected from the shore or a pier. On the other hand, massive waves like those at Jaws on Maui or Mavericks in California can occur as a result of constructive interference in a wave front. Destructive interference can adversely affect acoustics in an auditorium. Sound waves that are reflected from walls and objects can cancel the same sounds coming directly from the performers and produce "dead" spots in the acoustics of the auditorium. If loudspeakers in a stereo system are incorrectly connected, this can result in poor performance due to cancellation of certain sounds by speakers moving in opposite directions. Noise reducing earphones detect sounds and produce equal and opposite frequencies to cancel noises. As a wave moves through a
material, the potential energy of each particle in the
path of the wave varies periodically. The kinetic energy
of each particle increases as the potential energy
decreases and visa versa. When two waves that are moving
through the same medium meet, particles are imparted with
the combined energies of both waves. The energy state of
one wave is superimposed on the second wave at any point
at which their paths intersect. The interference that
occurs will be either constructive or destructive and
will depend upon the phase of each wave at the point at
which the waves meet. Phases in Cycles The position in the cycle of a rotating body can be described in terms of its angle of rotation from an initial state. If a wheel rotates through 45º, the wheel has rotated through 25% of its cycle. If the wheel rotates at a constant rate of 2 rotations per second, the period of its cycle is 0.5 seconds. If an identical wheel (Wheel B) is placed next to the first wheel (Wheel A) and both wheels start rotating from the same state with a cycle of 0.5 seconds, the two wheels will rotate in phase. At any point in time, the wheels will be identically oriented. Any two corresponding points in the two wheels will have moved through the same angle of rotation. Rotation Out Of Phase However, if Wheel B starts
from the same point and at the same speed of rotation as
the first wheel (Wheel A) but 0.25 seconds after Wheel A
has started rotating, the wheels will rotate out of phase.
At any point in time, there will be a difference of 180º
between the positions of corresponding points on the two
wheels. Wheel B will lagging Wheel A by 180º or 50% of
its cycle. Oscillations in Phase Any two oscillators can oscillate in phase or out of phase. If two identical pendulums start oscillating from the same position at the same time, they will oscillate in phase. If they start from different positions or at different times within their cycle, they will oscillate out of phase. For each of the pendulums, a plot of the distance of the pendulum from its normal position at rest will be similar in shape to a sine wave. Phases in Waves When waves move through a material, the particles of the material oscillate around their normal position at rest. In transverse waves, the particles oscillate at right angles to the direction of the wave. In longitudinal waves, the particles oscillate in the direction of the wave. In simple waves each particle in the path of the wave oscillates. The movement of each particle is similar to simple harmonic motion. The distance of a particular particle from its normal position at rest varies periodically. Any two particles that are at a distance equivalent to one wavelength apart in the path of the wave will oscillate in phase. The same applies to any two particles that are separated by a whole number of wavelengths in the path of the wave. Two waves are said to be oscillating in phase if they have the same period and particles in corresponding positions oscillate in phase. Constructive Interference As a wave moves through a material, the potential energy of each particle in the path of the wave varies periodically. The kinetic energy of each particle increases as the potential energy decreases and visa versa. When two waves that are moving through the same medium meet, particles are imparted with the combined energies of both waves. The energy state of one wave is superimposed on the second wave at any point at which their paths intersect. For example, a particle of liquid in a surface wave that is 10 mm above its normal position at rest (in Wave A) would be elevated by a further 5 mm if its position in Wave B, in the absence of Wave A, would be 5 mm above its normal position at rest at the same time. The principle of superposition applies to all waves moving through the same medium. When the paths of two or more waves cross, interactions between the waves results in interference. The principle of superposition comes into effect. The principle of superposition states that when the superposition of two or more waves occurs at a point, the resultant displacement is equal to the sum of the displacements of the individual waves. Displacements can be either positive or negative in accordance with the sign convention used. For example, with waves moving across the surface of a liquid, the crest will have a positive value and the trough, a negative value. The simplest example of wave interference is when two waves meet while moving in opposite directions. The four diagrams below show the combined effects of the two waves at four different times as the paths of the two waves cross. It can be seen that at any time, the combined amplitude of the wave is equal to the sum of the amplitudes of the two waves at that point. Constructive and Destructive Interference Constructive interference
occurs when the combined amplitude of the two waves is
greater than the amplitude of each of the waves at that
point. Destructive interference occurs when the combined
amplitude represents the reduction of the amplitude of
one wave by the other. Beats Piano tuners listen for beats that help them to compare the frequencies of notes in different strings. When sound waves with similar frequencies or multiples of frequencies interfere, lower frequency beats can occur. The beat frequency is related to the difference between the frequencies creating the beat. By eliminating these beats, tuners are able to ensure the correct relationship between the frequencies of different strings. When waves with different frequencies move through the same medium, the degree of interference will vary as a result of the difference in frequency. A high degree of constructive interference will occur when overlapping frequencies are similar. Overlapping waves will cancel each other to a large extent when there is a large difference between the phases of the waves. The degree of interference will vary periodically and will produce a beat that has a frequency represented by the difference between the two frequencies. When beating occurs, the combined amplitude of the two waves or oscillations moves periodically from a maximum to a minimum value with a frequency equal to the difference between the two interfering frequencies. Standing Waves Standing waves can occur when two waves with the same frequency move in opposite directions in the same medium. Standing waves commonly occur in musical instruments. In a stringed instrument, waves move back and forth due to reflection between the two ends of the string. The result is that certain parts of the string oscillate with varying degrees of amplitude. Nodes occur where destructive interference between waves moving in opposite directions is complete and constant. Antinodes are points at which the maximum amount of constructive interference occurs periodically.
Review Questions
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