H. SAIBI November 11th 2015 Outline of Lecture: Reflection, Transmission, and Refraction Diffraction The Doppler Effect The Doppler shift and relativity Shock Waves Waves encountering barriers Reflection, Transmission, and Refraction Reflection Coefficient (r)=hr/hin Transmission Coefficient ()=ht/hin Heavier string Fresnel relations (valid for light and sound waves) Lighter string …(1) Reflection and Transmission coefficients Fig.1. (a) a wave pulse traveling on a string attached to a more massive string in which the wave speed is half as large. The reflected pulse is inverted, whereas the transmitted pulse is not. (b) A wave pulse traveling on a string attached to a less massive string in which the wave speed is twice as large. In this case, the reflected pulse is not inverted. ©2008 by W.H. Freeman and Company Exercise: Two Soldered Wires Two wires, different linear mass densities. Stretched under a tension FT (the tension is the same in both wires). The wave speed in the first wire is twice that in the second. A harmonic wave traveling in the first wire is incident on the junction of the wires. (A) if the amplitude of the incident wave is A, what are the amplitudes of the reflected and transmitted waves? (B) what is the ratio 2/1 of the mass densities of the wires? (C) what fraction of the incident average power is reflected at the junction and what fraction is transmitted? ©2008 by W.H. Freeman and Company Waves encountering barriers Reflection, Transmission, and Refraction Energy conservation gives another relation between the reflection and transmission coefficients. This relation is given by: …(2) Fraction of the incident power that is reflected Fraction of the transmitted In three dimensions, a boundary between two regions of differing wave speed is a surface. The transmitted ray is bent toward or away from the normal-depending on whether the wave speed in the second medium is less or greater than that in the incident medium. Fig.2. A wave striking a boundary surface between two media in which the wave speed differs. Part of the wave is reflected and part is transmitted. The change in direction of the transmitted (refracted) ray is called refraction. ©2008 by W.H. Freeman and Company Waves encountering barriers Reflection, Transmission, and Refraction Total Internal Reflection: When the incident angle is greater than the critical angle, there is no refracted ray. The amount of energy reflected from a surface depends on the surface. Fig.3. Light from a source in the water is bent away from the normal when it enters the air. For angles of incidence above a critical angle c’ there is no transmitted ray, a condition known as total internal reflection. ©2008 by W.H. Freeman and Company Figure 4. In a concert hall, a reflecting shell is placed behind the orchestra, and reflecting panels are hung from the ceiling to reflect and direct the sound back toward the listeners. ©2008 by W.H. Freeman and Company Check your understanding: Balloon Hearing Aid If you place the balloon (filled with carbon dioxide) between yourself and a sound source, the sound gets louder. Why is that? The balloon is to sound as a magnifying glass is to light. ©2008 by W.H. Freeman and Company Diffraction If a wavefront is partially blocked by an obstacle, the unblocked part of the wavefront bends behind the obstacle. This bending of the wavefronts is called diffraction. Almost all of the diffraction occurs for that part of the wavefront that passes within a few wavelengths of the edge of the obstacle. For the parts of the wavefront that pass farther than a few wavelengths from the edge, diffraction is negligible and the wave propagates in straight lines in the direction of the incident rays. When wavefronts encounter a barrier with an aperture (hole) only a few wavelengths across, the part of the wavefronts passing through the aperture all pass within a few wavelengths of an edge. Thus, flat wavefronts bend and spread out and become spherical or circular (Fig.5). Fig.5. Plane waves in a ripple tank meeting a barrier with an opening that is approximately one wavelength wide. Beyond the barrier are circular waves that are concentric about the opening, much as if there were a point source at the opening. Diffraction In contrast, for a beam of particles falling upon a barrier with an aperture, the part of the beam passing through the aperture does so with no change in the direction of the particles (Fig.6). Diffraction is one of the key characteristics that distinguish waves from particles. ©2008 by W.H. Freeman and Company Fig.6. Comparison of particles and waves passing through a narrow opening in a barrier. (a) Transmitted particles are confined to a narrow-angle beam. (b) Transmitted waves spread out (radiate widely) from the aperture, which acts like a point source of circular waves. Diffraction Although waves passing through an aperture always bend, or diffract, to some extent, the amount of diffraction depends on whether the wavelength is small or large relative to the width of the aperture. If the wavelength is greater than or equal to the width of the aperture (like in Fig.5), the diffraction effects are large, and the waves speed out as they pass through the aperture-as if the waves were originating from a point source. On the other hand, if the wavelength is small relative to the aperture, the effect of diffraction is small as shown in Fig.7. Near the edges of the aperture the wavefronts are distorted and the waves appear to bend slightly. For the most part, however, the wavefronts are not affected and the waves propagate in straight lines, much like a beam of particles. The approximation that waves propagate in straight lines in the direction of the rays with no diffraction is known as the ray approximation. Wavefronts are distorted near the edges of any obstacle blocking part of the wavefronts. By near we mean within a few wavelengths of the edges. Fig.7. Plane waves in a ripple tank meeting a barrier with an opening width that is large compared to . The wave continues in the forward direction, with only a small amount of spreading into the regions to either side of the opening. ©2008 by W.H. Freeman and Company The Doppler effect The shift in frequency caused by motion is called the Doppler effect. It occurs when a sound source is moving at speeds less than the speed of sound. The Doppler Effect Consider the source moving with speed us (Fig.8a-b) and a stationary receiver. The source has frequency fs (and period Ts=1/ fs). The received frequency fr, the number of wave crests passing the receiver per unit time, is related to wave length and wave speed v by: … (3) Stationary receiver A wave crest leaves the source at time t1 and the next wave crest leaves the source at time t2 (Fig.8c). The time between these two events is , and during this time the source and the crest leaving the source at time t1 travel distances usTs and vTs, respectively. Consequently, at time t2, the distance between the source and the crest leaving at time t1 equals the wavelength . Behind the source: In front of the source: us<v … (4) (-) minus sign is used if =f and (+) plus sign is used if =b … (5) Stationary receiver The Doppler Effect Fig.8. (a) Waves in a ripple tank produced by a point source moving to the right. The wavefronts are closer together in front of the source and farther apart behind the source. (b) Successive wavefronts emitted by a point source moving with speed us to the right. The numbers of the wavferonts correspond to the positions of the source when the wave was emitted. (c) The source vibrates one cycle in time Ts. During time ts the source moves a distance usTs and the fifth wavefront travels a distance vTs. In front of the source the wavelength f=(v-us)Ts, while behind the source b=(v+us)Ts. ©2008 by W.H. Freeman and Company The Doppler Effect When the receiver moves relative to the medium, the received frequency is different simply because the receiver moves past more or fewer wave crests in a given time. Let Tr denote the time between arrivals of successive crests for a receiver moving with a speed ur. Then during the time between the arrivals of two successive crests, each crest will have traveled a distance vTr, and during the same time the receiver will have traveled a distance urTr. If the receiver moves in the direction opposite to that of the wave (Fig.9), then during time Tr, the distance a crest moves plus the distance the receiver moves equals the wavelength. That is, or . (if the receiver moves in the same direction as the wave, then , so . Because , we have: … (6) Where, if the receiver moves in the same direction as the wave, the received frequency is lower, so we choose the negative sign. If the receiver moves in the direction opposite to that of the wave, the frequency is higher, so we choose the positive sign. Substituting for from Eq.4, we obtain: Fig.9. The time between arrivals of wave crests at the receiver is Tr. The wave crests are represented by orange lines when a wave crest reaches the receiver, and they are represented by grey lines when the next crest reaches the receiver. During time Tr the receiver travels the distance urTr, while the wave crest travels the distance vTr. … (7) ©2008 by W.H. Freeman and Company The Doppler Effect The correct choices for the plus or minus signs are most easily determined by remembering that the frequency tends to increase both when the source moves toward the receiver and when the receiver moves toward the source. For example, if the receiver is moving toward the source, the plus sign is selected in the numerator, which tends to increase the received frequency; if the source is moving away from the receiver, the plus sign is selected in the denominator so that the equation predicts a decrease in the received frequency. Equation 7 appears more symmetric, and thus is easier to remember, if expressed in the form: … (8) It can be shown that if both us and ur are much smaller than the wave speed v, then the shift in frequency is given approximately by: … (9) where is the speed of the source relative to the receiver. In a reference frame in which the medium is moving (for example, the reference frame of the ground if air is the medium and if there is a wind blowing), the wave speed v is replaced by where uw is the speed of the wind relative to the ground. Equations 3 to 9 are valid only in the reference frame of the propagating medium. The speed of sound The speed of sound in air is 343 meters per second (660 miles per hour) at one atmosphere of pressure and room temperature (21°C). An object is subsonic when it is moving slower than sound. The speed of sound We use the term supersonic to describe motion at speeds faster than the speed of sound. A shock wave forms where the wave fronts pile up. The pressure change across the shock wave is what causes a very loud sound known as a sonic boom. Doppler Shift and Relativity We see from Equations (5-6-7-8) that the magnitude of the Doppler shift in frequency depends on whether it is the source or the receiver that is moving relative to the medium. For sound, these two situations are physically different. For example, if you move relative to still air, you feel rushing past you. In your reference frame, there is no wind. For sound waves in air, therefore, we can tell whether the source or receiver is moving by noting if there is a wind in the reference frame of the source or the receiver. However, light and other electromagnetic waves propagate through empty space in which there is no propagating medium. There is no “wind” to tell us whether the source or receiver is moving. According to Einstein’s theory of relativity, absolute motion cannot be detected, and all observers measure the same speed c for light, independent of their motion relative to the source. Thus, Equations 7-8 cannot be correct for the Doppler shift for light. Two modifications must be made in calculating the relativistic Doppler effect for light. First, the speed of waves passing a receiver is c, which is independent of the motion of the receiver. Second, the time interval between the emission of successive wave crests, which is Ts=1/fs in the reference frame of the source, is different in the reference frame of the receiver when the two reference frames are in relative motion, because of relativistic time dilation and length contraction. The result is that the frequency received depends only on the relative speed of approach (or recession) u, and is related to the frequency emitted by: … (10) Choose the signs that give an up-shift in frequency when the source and receiver are approaching, and vice versa. Again, when u<<c, f/fs ≈ ± u/c as given by Equation 9. Shock Waves During our derivations of the Doppler-shift expressions, we assumed that the speed u of the source was less than the wave speed v. if a source moves with speed greater than the wave speed, then there will be no waves in front of the source. Instead, the wave pile up behind the source to form a shock wave. In the case of sound waves, this shock wave is ahead as a sonic boom when it arrives at the receiver. Figure 12 shows a source originally at point P1 moving to the right with speed u. After some time t, the wave emitted from point P1 has traveled a distance vt. The source has traveled a distance ut and will be at point P2. The line from this new position of the source to the wavefront emitted when the source was at P1 makes an angle , called the Mach angle, with the path of the source, given by: … (11) Thus, the shock wave is confined to a cone that narrows as u increases. The ratio of the source speed u to the wave speed v is called the Mach number: … (12) Shock Waves Fig.12. (a) Source moving with a speed u that is greater than the wave speed v. The envelope of the wavefronts forms a cone with the source at the apex. (b) Waves in a ripple tank produced by a source moving with a speed u>v. ©2008 by W.H. Freeman and Company Fig.10. Shock waves from a supersonic airplane. ©2008 by W.H. Freeman and Company Fig.11. Shock waves produced by a bullet traversing a helium balloon. ©2008 by W.H. Freeman and Company Exercise: A Sonic Boom A supersonic plane flying due east at an altitude of 15 km passes directly over point P. The sonic boom is heard at point P when the plane is 22 km east of point P. What is the speed of the supersonic plane? Fig. In the time that the plane moves a distance ut, the sound moves distance vt. ©2008 by W.H. Freeman and Company Doppler Effect Doppler Effect – 4 cases Source moving toward receiver Source moving away from receiver Receiver (observer) moving towards source Receiver (observer) moving away from source. Source moving case vs T Away: 1 f f 1 vs v c T Towards: vs v s 1 c c 1 c c f f v 1 vs 1 s c c Receiver (observer) moving case Towards: f c vr vr f 1 c Away: f c vr v f 1 r c Source and receiver moving vr 1 c f f v 1 s c c vr f c vS Numerator – Receiver (observer) Toward + Away – Denominator – Source Toward – Away +