IB – PHYSICS (AHL)– Wave Phenomena



Topic 11.1 Standing WavesA standing wave, also known as a stationary wave, is a wave that remains in a constant position.Two opposing waves combine to form a standing wave.The main difference between travelling wave and standing wave is that in the latter no energy or momentum is transferred. One characteristic of every standing wave pattern is that there are points along the medium which appear to be standing still. These points, sometimes described as points of no displacement, are referred to as nodes. There are other points along the medium which undergo vibrations between a large positive and and large negative displacement. These are the points which undergo the maximum displacement during each vibrational cycle of the standing wave. In a sense, these points are the opposite of nodes, and so they are called antinodes. A standing wave pattern always consists of an alternating pattern of nodes and antinodes.Points D,C and B indicate nodes and points A,E and F indicate antinodes.Characteristics of standing waves:In stationary waves, there are certain points called nodes where the particles are permanently at rest and certain other points called antinodes where the particles vibrate with maximum amplitude. The nodes and antinodes are formed alternately.All the particles of the medium except those at the nodes, vibrate simple harmonically with a time period equal to that of the component waves.The amplitude of vibration increases gradually from zero to maximum from a node to an antinode.The medium is split up into segments. The particles in a segment vibrate in phase. The particles in one segment are out of phase with the particles in the neighbouring segment by 180o.Velocity and acceleration of all the particles separated by a distance l, are the same at a given instant.In a given segment, the particles attain their maximum or minimum velocity and acceleration at the same instant.There is no net transport of energy in the pressions and rarefactions do not travel forward as in progressive waves. They appear and disappear alternately, at the same place.During each vibration, all the particles pass simultaneously through their mean positions twice, with maximum velocity which is different for different particles.HarmonicsStanding wave patterns are produced within the medium when it is vibrated at certain frequencies. Each frequency is associated with a different standing wave pattern. These frequencies and their associated wave patterns are referred to as harmonics. A careful study of the standing wave patterns reveal a clear mathematical relationship between the wavelength of the wave which produces the pattern and the length of the medium in which the pattern is displayed. Furthermore, there is predictability about this mathematical relationship that allows one to generalize and deduce a statement concerning this relationship. To illustrate, consider the first harmonic standing wave pattern for a vibrating rope as shown below.The pattern for the first harmonic reveals a single antinode in the middle of the rope. This antinode position along the rope vibrates up and down from a maximum upward displacement from rest to a maximum downward displacement as shown. The vibration of the rope in this manner creates the appearance of a loop within the string. A complete wave in a pattern could be described as starting at the rest position, rising upward to a peak displacement, returning back down to a rest position, then descending to a peak downward displacement and finally returning back to the rest position. The animation below depicts this familiar pattern. As shown in the animation, one complete wave in a standing wave pattern consists of two loops. Thus, one loop is equivalent to one-half of a wavelength.In comparing the standing wave pattern for the first harmonic with its single loop to the diagram of a complete wave, it is evident that there is only one-half of a wave stretching across the length of the string. That is, the length of the string is equal to one-half the length of a wave. Put in the form of an equation:?Now consider the string being vibrated with a frequency that establishes the standing wave pattern for the second harmonic.The second harmonic pattern consists of two anti-nodes. Thus, there are two loops within the length of the string. Since each loop is equivalent to one-half a wavelength, the length of the string is equal to two-halves of a wavelength. Put in the form of an equation:?The same reasoning pattern can be applied to the case of the string being vibrated with a frequency that establishes the standing wave pattern for the third harmonic.The third harmonic pattern consists of three anti-nodes. Thus, there are three loops within the length of the string. Since each loop is equivalent to one-half a wavelength, the length of the string is equal to three-halves of a wavelength. Put in the form of an equation:?When inspecting the standing wave patterns and the length-wavelength relationships for the first three harmonics, a clear pattern emerges. The number of antinodes in the pattern is equal to the harmonic number of that pattern. The first harmonic has one antinode; the second harmonic has two antinodes; and the third harmonic has three antinodes. Thus, it can be generalized that the nth harmonic has n antinodes where n is an integer representing the harmonic number. Furthermore, one notices that there are n halves wavelengths present within the length of the string. Put in the form of an equation:?Observe that each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one. The table below summarizes the features of the standing wave patterns for the first several harmonics. HarmonicNo of NodesNo of AntinodesPattern1st212nd323rd434th545th656th76nthn + 1n--Comparison between Travelling and Standing Waves? ??Travelling waves?Standing waves?The disturbance produced in the medium travels onward, it being handed over from one particle to the next. Each particle executes the same type of vibration as the preceding one, though not at the same time.?There is no onward motion of the disturbance as no particle transfers its motion to the next. Each particle has its own characteristic vibration.?The amplitude of each partide is the same but the phase changes continuously,?The amplitudes of the different particles are different, ranging from zero at the nodes to maximum at the antinodes. All the particles in a given segment vibrate in phase but in opposite phase relative to the particles in the adjacent segment.?No particle is parmanently at rest. Different particles attain the state of momentary rest at different instants,?The particles at the nodes are permanently at rest but other particles attain their position of momentary rest simultaneously.All the particles attain the same maximum velocity when they pass through their mean positions.All the particles attain their own maximum velocity at the same time when they pass through their mean positions.?In the case of a longitudinal progressive wave all the parts of the medium undergo similar variation of density one after the other. At every point there will be a density variation.?In the case of a longitudinal stationary wave the variation of density is different at different points being maximum at the nodes and zero at the antinodes.?There is a flow of energy across every plane in the direction of propagation.?Energy is not transported across any plane.The Doppler Effect:The Doppler effect (or Doppler shift), named after Austrian physicist Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the recession.Doppler Effect Equation in SoundKinds of source:Doppler effect is true for all kinds of waves that is sound waves and light waves.It may be noted that1). Doppler's effect give the change in frequency of vibrations of source and it does not give any information about the intensity of vibrations of source.2). For sound waves doppler effect depends on the relative motion of source and observer and also it depends on whether the source or the observer is moving.Doppler effect in sound is studied in four parts:Source moving toward stationary observerWhen the source of sound is moving toward you, the pitch you hear is higher than what was emitted from the source and the wavelength is shorter than emitted.The equation for the observed frequency of sound when the source is traveling toward you is:fo = fv/(v ? vt)wherefo is the observed frequencyf is the emitted frequencyv is the velocity of soundvt is the velocity of the source toward youv > vt (vt is less than v)WavelengthAlso, since the velocity of the wave equals the frequency times the wavelength (v = fλ or f = v/λ), the equation for the observed wavelength when the source is traveling toward you is:λo = λ(1 ? vt/v)whereλo is the observed wavelengthλ is the emitted wavelength (Greek symbol lambda)v > vt (vt less than but not equal to v)Velocityf you know the resulting frequency, you can find the speed of the source moving toward you:vt = v(f/fo + 1)Source moving away from a stationary observerWhen the source of sound is moving away from you, the pitch you hear is lower, the frequency is slower and the wavelength is longer than what was emitted from the source.Note that the equations are the same as when the source is moving toward you, except that the "?" sign is replaced by a "+" sign to indicate the change in direction of the source. In the case of velocity, the "+" sign is replaced by a "?" sign.FrequencyThe equation for the observed frequency of a waveform when the source is traveling away from you is:fo = fv/(v + va)where va is the velocity of the source away from you.WavelengthThe equation for the observed wavelength when the source is traveling away from you is:λo = λ(1 + va/v)VelocityThe equation for the velocity of the source, when it is traveling away from you is:va = v(f/fo ? 1)2.?Source at Rest and observer in motion:???????????????? Suppose S is a sound-source at rest and O is observer. The true frequency of source is n. if observer is also at rest, he will hear n waves per second. Therefore, true wavelength .But the source is moving with velocity of observer.The number of waves in distance is λ'.The total number of waves received by observer in one second.The apparent frequency equation is fo = fs(1+vc)3. Source and observer both in motion:???????????????? Suppose source and observer both are moving with velocities vs and v0 respectively in the direction in which waves are being received. There will be change in frequency.The apparent frequency equation is n' = ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download