Archive for March, 2012
When more than one wave travels through the same location in the same medium at the same time, the total displacement of the medium is governed by the principle of superposition. The principle of superposition simply states that the total displacement is the sum of all the individual displacements of the waves. The combined effect of the interaction of the multiple waves is known as wave interference.
Question: The diagram shows two pulses approaching each other in a uniform medium. Diagram the superposition of the two pulses.
When two or more pulses with displacements in the same direction interact, the effect is known as constructive interference. The resulting displacement is greater than the original individual pulses. Once the pulses have passed by each other, they continue along their original path in their original shape, as if they had never met. An animation of two pulses interfering constructively is shown at the site below: Courtesy Penn State Schuylkill
Notice the top pulse travels to the right with a positive displacement, while the middle pulse travels to the left with a positive displacement. When the two pulses meet (shown at bottom), the interfere constructively before continuing on their path as if they had never met.
When two or more pulses with displacements in opposite directions interact, the effect is known as destructive interference. The resulting displacements negate each other. Once the pulses have passed by each other, they continue along their original path in their original shape, as if they had never met. An animation of two pulses interfering destructively is shown at the site below: Courtesy Penn State Schuylkill
Notice the top pulse travels to the right with a positive displacement, while the middle pulse travels to the left with a negative displacement. When the two pulses meet (shown at bottom), the interfere destructively before continuing on their path as if they had never met.
Question: Two wave sources operating in phase in the same medium produce the circular wave patterns shown in the diagram. The solid lines represent wave crests and the dashed lines represent wave troughs. Which point is at a position of maximum destructive interference?
Answer: Point B is at a position of maximum destructive interference, since point B represents the intersection of a crest and a trough.
When waves of the same frequency and amplitude traveling in opposite directions meet, a standing wave is produced. A standing wave is a wave in which certain points (nodes) appear to be standing still and other points (anti-nodes) vibrate with maximum amplitude above and below the axis.
Looking at the standing wave produced on the right, we can see a total of five nodes in the wave, and four anti-nodes. For any standing wave pattern, you will always have one more node than anti-node.
Standing waves can be observed in a variety of patterns and configurations, and are responsible for the functioning of most musical instruments. Guitar strings, for example, demonstrate a standing wave pattern. By fretting the strings, you adjust the wavelength of the string, and therefore the frequency of the standing wave pattern, creating a different pitch. Similar functionality is seen in instruments ranging from pianos and drums to flutes, harps, trombones, xylophones, and even pipe organs!
Question: The diagram shows a standing wave in a string clamped at each end. What is the total number of nodes and antinodes in the standing wave?
Answer: Five nodes and four anti-nodes.
Due to their very nature, waves exhibit a number of behaviors that may not be obvious upon first inspection, including the Doppler Effect, reflection, refraction, and diffraction. Understanding these behaviors brings us closer to understanding the universe, while also providing a number of useful applications including, but not limited to, radar, sonography, digital televisions, mirrors, telescopes, glasses, contact lenses, atomic research, and even holography!
The shift in a wave’s observed frequency due to relative motion between the source of the wave and the observer is known as the Doppler Effect. In essence, when the source and/or observer are moving toward each other, the observer perceives a shift to a higher frequency, and when the source and/or observer are moving away from each other, the observer perceives a lower frequency.
This can be observed when a vehicle travels past you. As you hear the vehicle approach, you can observe a higher frequency noise, and as the vehicle passes by you and then moves away, you observe a lower frequency noise.
The Doppler Effect results from waves having a fixed speed in a given medium. As waves are emitted, a moving source or observer encounters the wave fronts at a different frequency than they waves are emitted, resulting in a perceived shift in frequency.
Question: A car’s horn is producing a sound wave having a constant frequency of 350 hertz. If the car moves toward a stationary observer at constant speed, the frequency of the car’s horn detected by this observer may be:
- 320 Hz
- 330 Hz
- 350 Hz
- 380 Hz
Answer: If source is moving toward stationary observer, observed frequency must be higher than source frequency, therefore the correct answer is (4) 380 Hz.
An exciting application of the Doppler Effect involves the analysis of radiation from distant stars and galaxies in the universe. Based on the basic elements that compose stars, we know what frequencies of radiation to look for. However, when analyzing these objects, we observe frequencies shifted toward the red end of the electromagnetic spectrum (lower frequencies), known as the Red Shift. This indicates that these celestial objects must be moving away from us. The more distant the object, the greater the red shift. Putting this together, we can conclude that more distant celestial objects are moving away from us faster, and therefore, the universe as we know it must be expanding!
Defining & Describing Waves
A pulse is a single disturbance which carries energy through a medium or through space. Imagine you and your friend holding opposite ends of a slinky. If you quickly move your arm up and down, a single pulse will travel down the slinky toward your friend.
If, instead, you generate several pulses at regular time intervals, you now have a wave carrying energy down the slinky. A wave, therefore is a repeated disturbance which carries energy. The mass of the slinky doesn’t move from end of the slinky to the other, but the energy it carries does.
When a pulse or wave reaches a hard boundary, it reflects off the boundary, and is inverted. If a pulse or wave reaches a soft, or flexible, boundary, it still reflects off the boundary, but does not invert.
Waves can be classified in several different waves. One type of wave, known as a mechanical wave, requires a medium (or material) through which to travel. Examples of mechanical waves include water waves, sound waves, slinky waves, and even seismic waves. Electromagnetic waves, on the other hand, do not require a medium in order to travel. Electromagnetic waves (or EM waves) are considered part of the Electromagnetic Spectrum. Examples of EM waves include light, radio waves, microwaves, and even X-rays.
Further, waves can be classified based upon their direction of vibration. Waves in which the "particles" of the wave vibrate in the same direction as the wave direction are known as longitudinal, or compressional, waves. Examples of longitudinal waves include sound waves and seismic P waves. Waves in which the particles of the wave vibrate perpendicular to the wave’s direction of motion are known as transverse waves. Examples of transverse waves include seismic S waves, electromagnetic waves, and even stadium waves (the "human" waves you see at a baseball or football game!).
Waves have a number of characterisics which define their behavior. Looking at a transverse wave, we can identify specific locations on the wave. The highest points on the wave are known as crests. The lowest points on the wave are known as troughs. The amplitude of the wave, corresponding to the energy of the wave, is the distance from the baseline to a crest or the baseline to a trough.
The length of the wave, or wavelength, noted with the Greek letter lambda (), is the distance between corresponding points on consecutive waves (i.e. crest to crest or trough to trough). Points on the same wave with the same displacement from equilibrium moving in the same direction (such as a crest to a crest or a trough to a trough) are said to be in phase (phase difference is 0° or 360°). Points with opposite displacements from equilibrium (such as a crest to a trough) are said to be 180° out of phase.
Question: The diagram represents a periodic wave. Which point on the wave is in phase with point P?
Answer: Point C is in phase with point P, since point C represents a point with the same displacement from equilibrium moving in the same direction. Points that are in phase are located one or more whole wavelengths apart on a wave.
Question: Two waves having the same frequency and amplitude are traveling in the same medium. Maximum constructive interference occurs at points where the phase difference between the two superposed waves is:
In similar fashion, longitudinal, or compressional, waves also have amplitude and wavelength. In the case of longitudinal waves, however, instead of crests and troughs, the longitudinal waves have areas of high density (compressions) and areas of low density (rarefactions), as shown in the representation of the particles of a sound wave. The wavelength, then, of a compressional wave is the distance between compressions, or the distance between rarefactions. Once again, the amplitude corresponds to the energy of the wave.
Question: A longitudinal wave moves to the right through a uniform medium, as shown below. Points A, B, C, D, and E represent the positions of particles of the medium. What is the direction of the motion of the particles at position C as the wave moves to the right?
Answer: The particles move to the left and right at position C, as the particles in a longitudinal wave vibrate parallel to the wave velocity:
Question: Between which two points on the wave could you measure a complete wavelength?
Answer: You could measure a complete wavelength between points A and C, since A and C represent the same point on successive waves.