Course PHYSICS260 Assignment 4 Due at 11 00pm on Wednesday February 27 2008 A Simple Introduction to Interference Description Interference is discussed for pulses on strings and then for sinusoidal waves Learning Goal To understand the basic principles underlying interference One of the most important properties of waves is the principle of superposition The principle of superposition for waves states that when two waves occupy the same point their effect on the medium adds algebraically So if two waves would individually have the effect 1 on a specific point in the medium then when they are both at that point the effect on the medium is 2 If a third wave with effect 2 happens also to be at that point then the total effect on the medium is zero This idea of waves adding their effects or canceling each other s effects is the source of interference First consider two wave pulses on a string approaching each other Assume that each moves with speed meter per second The figure shows the string at time The effect of each wave pulse on the string which is the medium for these wave pulses is to displace it up or down The pulses have the same shape except for their orientation Assume that each pulse displaces the string a maximum of and that the scale on the x axis is in meters Part A meters At time what will be the displacement at point Express your answer in meters to two significant figures ANSWER Part B Choose the picture that most closely represents what the rope will actually look like at time ANSWER A B C D The same process of superposition is at work when we talk about continuous waves instead of wave pulses Consider a sinusoidal wave as in the figure Part C How far to the left would the original sinusoidal wave have to be shifted to give a wave that would completely cancel the original The variable in the picture denotes the wavelength of the wave Express your answer in terms of ANSWER Part D In talking about interference particularly with light you will most likely speak in terms of phase differences as well as wavelength differences In the mathematical description of a sine wave the phase corresponds to the argument of the sine function For example in the function the value of at a particular point is the phase of the wave at that point Recall that in radians a full cycle or a full circle corresponds to radians How many radians would the shift of half a wavelength from the previous part correspond to Express your answer in terms of ANSWER phase difference radians Part E The phase difference of radians that you found in the previous part provides a criterion for destructive interference What phase difference corresponds to completely constructive interference i e the original wave and the shifted wave coincide at all points Express your answer as a number in the interval ANSWER phase difference radians Part F Since sinusoidal waves are cyclical a particular phase difference between two waves is identical to that phase difference plus a cycle For example if two waves have a phase difference of the interference effects would be the same as if the two waves had a phase difference of The complete criterion for constructive interference between two waves is therefore written as follows Write the full criterion for destructive interference between two waves Express your answer in terms of and ANSWER phase difference The phase for a plane wave is a somewhat complicated expression that depends on both position and time For most interference problems you will work at a specific time and with coherent light sources so that only geometric considerations are relevant Consider two light rays propagating from point A to point B in the figure which are apart One ray follows a straight path and the other travels at a angle to that path and then reflects off a plane surface to point B Both rays have wavelength Part G Find the phase difference between these two rays at point B Part G 1 Find the difference in distance Find the difference in length between the direct path and the reflected path You can use the fact that triangle ABC is an equilateral triangle Express your answer in terms of ANSWER path length difference Now that you have the difference in path length convert that to radians Recall that every cycle of radians is equivalent to one wavelength Express your answer in terms of ANSWER phase difference radians Part H Suppose that the reflected ray receives an extra half cycle phase shift when it reflects What is the new phase shift at point B Hint H 1 How many radians in a half cycle Since radians corresponds to a full cycle a half cycle must correspond to radians Express your answer in terms of ANSWER phase difference radians Whenever light reflects from a transparent interface moving from lower index of refraction to higher index of refraction it gets an extra half cycle phase difference Being able to accurately find the phase differences between waves at various points will be useful in both interference and diffraction problems Normal Modes and Resonance Frequencies Description Multiple choice questions about the definition and origin of normal modes Then compute the frequency and wavelength of the first three normal modes in a string Learning Goal To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency In general there are an infinite number of such modes each one with a distinctive frequency and associated pattern of oscillation Consider an example of a system with normal modes a string of length held fixed at and Assume that waves on this string propagate with both ends located at speed The string extends in the x direction and the waves are transverse with displacement along the y direction In this problem you will investigate the shape of the normal modes and then their frequency The normal modes of this system are products of trigonometric functions For linear systems the time dependance of a normal mode is always sinusoidal but the spatial dependence need not be Specifically for this system a normal mode is described by Part A The string described in the problem introduction is oscillating in one of its normal modes Which of the following statements about the wave in the string is correct Hint A 1 Normal mode constraints The key constraint with normal modes is that there are two spatial boundary
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