PHY103: Vibrational Waves, Sound and Optics Module 03-Behaviour of Waves PDF

# PHY103: Vibrational Waves, Sound and Optics ## 1. VIBRATION AND WAVES ### 1.3 Behaviour of Waves As seen in the animation on the virtual learning class, the wave does not stop when it reaches the end of the medium. Rather, a wave will undergo certain behaviours when it encounters the end of the...

# PHY103: Vibrational Waves, Sound and Optics ## 1. VIBRATION AND WAVES ### 1.3 Behaviour of Waves As seen in the animation on the virtual learning class, the wave does not stop when it reaches the end of the medium. Rather, a wave will undergo certain behaviours when it encounters the end of the medium. Waves exhibit a variety of properties that describe their characteristics and behaviour. We are already familiar with some wave terminology like amplitude, wavelength, frequency, period, velocity, and phase. The study of waves in two dimensions is often done using a ripple tank. A ripple tank (Figure 1.3) is a large glass-bottomed tank of water that is used to study the behavior of water waves. A light typically shines upon the water from above and illuminates a white sheet of paper (Figure 1.3, screen) placed directly below the tank. A portion of light is absorbed by the water as it passes through the tank. A crest of water will absorb more light than a trough. So, the bright spots represent wave troughs and the dark spots represent wave crests. As the water waves move through the ripple tank, the dark and bright spots move as well. As the waves encounter obstacles in their path, their behavior can be observed by watching the movement of the dark and bright spots on the sheet of paper. Ripple tanks are commonly used in experiments to demonstrate the principles underlying the reflection, refraction, and diffraction of waves. **Figure 1.3** Ripple Tank (Courtesy Katie M at CIE IGCSE Physics) * **Light Source** * **Support** * **Wooden Bar Supported By Elastic Bands** * **Water** * **Wavefronts** * **Screen** ### 1.3.1 Reflection of waves Reflection is simply the bouncing back of a wave when it encounters a boundary or obstacle. In Figure 1.3, the translational motion of the wooden bar is the source of straight waves. These straight waves have alternating crests and troughs. The crests are the dark lines stretching across the screen and the troughs are the bright lines. These waves will travel through the water until they encounter an obstacle - such as the wall of the tank or an object placed within the water. Figure 1.4 depicts a series of straight waves approaching a long barrier extending at an angle across the tank of water. The direction that these wavefronts (straight-line crests) are traveling through the water is represented by the blue arrow (a ray). The ray and is perpendicular to the wavefronts. Upon reaching the barrier placed within the water, these waves bounce off the water and head in a different direction. Regardless of the angle at which the wavefronts approach the barrier, one general law of reflection holds true: the waves will always reflect in such a way that the angle at which they approach the barrier equals the angle at which they reflect off the barrier. This is known as the law of reflection. **Figure 1.4** Reflection of waves *(a) Before Reflection* * barrier * Incident Ray *(b) After Reflection* * barrier * Reflected Ray ### 1.3.2 Refraction of waves Refraction of waves involves a change in the direction of waves as they pass from one medium to another. Refraction, or the bending of the path of the waves, is accompanied by a change in speed and wavelength of the waves. Refraction can be shown by placing a glass block in the tank. The glass block should sit below the surface of the water and cover only some of the tank floor. The depth of water becomes shallower here the glass block is placed. Since speed depends on depth, the ripples slow down when travelling over the block. This is a good model of refraction showing how waves slow down when passing from deep water into shallow water. The speed of a wave is dependent upon the properties of the medium through which the waves travel. The most significant property of water that would affect the speed of waves traveling on its surface is the depth of the water. Water waves travel fastest when the medium is the deepest. Thus, if water waves are passing from deep water into shallow water, they will slow down. This decrease in speed will also be accompanied by a decrease in wavelength (Figure 1.5). **Figure 1.5** Refraction at different water depths (Courtesy Katie M at CIE IGCSE Physics) * **λ** * **DEEP WATER** * **SHALLOW WATER** ### 1.3.3 Diffraction of waves Diffraction refers to the deviation from straight-line propagation that occurs when a wave passes beyond a partial obstruction. It usually corresponds to the bending or spreading of waves around the edges of apertures and obstacles. Water waves have the ability to travel around corners, around obstacles and through openings. This ability is most obvious for water waves with longer wavelengths. Diffraction can be demonstrated by placing small barriers and obstacles in a ripple tank and observing the path of the water waves as they encounter the obstacles. As the water waves encounter two obstacles with a gap between them, the waves can be seen to spread out as seen in Figure 1.6 (a). Meanwhile, as the water waves encounter the edge of an obstacle, the waves can be seen to spread out differently (Figure 1.6 b). The waves are seen to pass around the barrier into the regions behind it; subsequently the water behind the barrier is disturbed. The amount of diffraction (the sharpness of the bending) increases with increasing wavelength and decreases with decreasing wavelength. In fact, when the wavelength of the waves is smaller than the obstacle, no noticeable diffraction occurs. **Figure 1.6** Diffraction of water waves (a) through a gap, (b) after passing an edge. *(a)* *(b)* In fact, any wave passing through an opening experiences diffraction. Diffraction is most noticeable when an opening is about the same size as the wavelength of the wave. Another example of diffraction phenomenon is shown in Figure 1.7. When both the source of light and the screen are relatively close to the obstacle forming the diffraction pattern. This situation is described as near-field diffraction or Fresnel diffraction, pronounced "Freh-nell" (after the French scientist Augustin Jean Fresnel, 1788-1827). By contrast, we use the term Fraunhofer diffraction (after the German physicist Joseph von Fraunhofer, 1787-1826) for situations in which the source, obstacle, and screen are far enough apart that we can consider all lines from the source to the obstacle to be parallel, and can likewise consider all lines from the obstacle to a given point on the screen to be parallel. **Figure 1.7** An example of diffraction * **Photograph of a razor blade illuminated by monochromatic light from a point source (a pinhole). Notice the fringe around the blade outline** * **Enlarged view of the area outside the geometric shadow of the blade's edge** * **Position of geometric shadow** ### Diffraction and Huygens's Principle Huygens's principle states that we can consider every point of a wave front as a source of secondary wavelets. These spread out in all directions with a speed equal to the speed of propagation of the wave. The position of the wave front at any later time is the envelope of the secondary wavelets at that time. Huygens construction can be used to quantify the diffraction phenomenon. ### Single-Slit Diffraction When parallel rays of light of wavelength λ are incident normally upon a slit of width D, a diffraction pattern is observed beyond the slit. On a far-away screen, complete darkness is observed at angles θm' to the straight-through beam, where: _m'λ = D sinθm' (m' = ±1, ±2, ±3, ...)_ (1.9) The pattern consists of a broad central bright band flanked on both sides by an alternating succession of faint narrow light and dark bands. ### Limit of Resolution of two objects due to diffraction: If two objects are viewed through an optical instrument, the diffraction patterns caused by the aperture of the instrument limit our ability to distinguish the objects from each other. For distinguishability, the angle θ subtended at the aperture by the objects must be larger than a critical value θcr, given by _sin θcr = (1.22)λ/D_ (1.10) where D is the diameter of the circular aperture of the instrument (be it an eye, telescope, or camera). ### Interference The term interference refers to any situation in which two or more waves overlap in space. When this occurs, the total wave at any point at any instant of time is governed by the principle of superposition. Therefore, interference of waves is the combination of two or more waves to produce a resultant wave. Sunlight is composed of light containing a broad range of frequencies and corresponding wavelengths. We often see different colors separated out of sunlight after it refracts and reflects in raindrops, forming a rainbow. We also sometimes see various colors from sunlight due to constructive and destructive interference phenomena in thin transparent layers of materials, such as soap bubbles or thin films of oil floating on water. ### The Principle of Superposition The Principle of Superposition is a fundamental concept in wave theory that describes how multiple waves interact when they overlap in a given medium. According to this principle, the resultant displacement at any point and at any instant in a medium, due to the presence of two or more waves, is the algebraic sum of the individual displacements produced by each wave if they were present alone. Key points about the Principle of Superposition include: 1. **Addition of Displacements:** When two or more waves overlap in a medium, the displacement of the medium at any point and at any time is the sum of the displacements caused by each individual wave. Mathematically, if y1 and y2 are the displacements caused by two waves, the resultant displacement y is given by y = y1 + y2 2. **Superposition of Waves in Different Phases:** * Waves can be in different phases (in-phase or out-of-phase) when they overlap. * In-phase waves reinforce each other, leading to constructive interference. * Out-of-phase waves may cancel each other, leading to destructive interference. 3. **Applicability to Different Types of Waves:** The Principle of Superposition is applicable to various types of waves, including mechanical waves and electromagnetic waves. ### Conditions for interference 1. The sources must be coherent, which means the waves they emit must maintain a constant phase with respect to each other. 2. The waves must have identical wavelengths. When waves from two or more sources arrive at a point in phase, they reinforce each other: The amplitude of the resultant wave is the sum of the amplitudes of the individual waves. This is called **constructive interference.** * **Total constructive interference** occurs when two coherent waves of the same amplitude are superposed reinforcement (or in the case of light, brightness) occurs when they are in-phase. * **Total destructive interference** is a situation that occurs when two coherent waves of the same amplitude are superposed, cancellation (or in the case of light, darkness) occurs when the waves are 180° out-of-phase. Interference effects are most easily seen when we combine sinusoidal waves with a single frequency f and wavelength λ. Figure 1.8 (a) shows a "snapshot” of a single source of sinusoidal waves and some of the wave fronts produced by this source. Figure 1.8 (b) shows two identical sources of monochromatic coherent wave sources S₁ and S2, (both of which produce waves of the same amplitude and the same wavelength) and are permanently in phase (vibrate in unison). In Figure 1.8 (b), the point a (on the x-axis) is of the same distance from S₁ and S2. Hence waves that leave S₁ and S2 in phase arrive at a in phase. The two waves add, and the total amplitude at a is twice the amplitude of each individual wave. This is true for any point on the x-axis. **Figure 1.8** (a) Sinusoidal waves of frequency f and wavelength λ spreading out from source S₁ in all directions. (b) Sinusoidal waves spreading out from two coherent sources S₁ and S2. Constructive interference occurs at point a and at point b (equidistant from the two sources). Destructive interference occurs at point c. * **Wave fronts: crests of the wave (frequency f) separated by one wavelength λ** * **y** * **S₁** * **a** * **S₂** * **c** * **x** * **The wave fronts move outward from source S₁ at the wave speed v = fλ** * **(a)** * **(b)** **Conditions for constructive and destructive interference** In Figure 1.9 (a), the distance from S2 to point b is exactly two wavelengths greater than the distance from S₁ to b. A wave crest from S₁ arrives at b exactly two cycles earlier than a crest emitted at the same time from S2. The two waves arrive in phase. As at point a, the total amplitude is the sum of the amplitudes of the waves from S₁ and S2. Generally, when waves from two or more sources arrive at a point in phase, they reinforce each other: The amplitude of the resultant wave is the sum of the amplitudes of the individual waves. This is called **constructive interference.** Something different occurs at point c in Figure 1.9 (b). At this point, the path difference r2 - r₁ = -2.50λ which is a half-integral number of wavelengths. Waves from the two sources arrive at point c exactly a half-cycle out of phase. A crest of one wave arrives at the same time as a crest in the opposite direction (a "trough") of the other wave. The resultant amplitude is the difference between the two individual amplitudes. If the individual amplitudes are equal, then the total amplitude is zero. This cancellation or partial cancellation of the individual waves is called **destructive interference.** **Constructive interference occurs for** _r2 - r₁ = mλ (m=0, ±1,±2, ±3, ...)_ (1.11) **Destructive interference occurs for** _r2 - r₁ = (m + 1)λ (m=0, ±1, 2, 3, ...)_ (1.12) **Figure 1.9** (a) condition for constructive interference; (b) condition for destructive interference *(a)* * **S₁** * **S₁** * **S₂** * **r₁ = 7λ** * **r₂ - r₁ = 2λ** * **λ** * **S₂** *(b)* * **b** * **S₁** * **r₂ - r₁ = -2.50λ** * **S₂** * **r₂ = 9λ** * **r₂ = 7.25λ** * **A** * **λ** * **c** ### Young's double-slit experiment Perhaps the most fundamental arrangement for producing and studying interference is Young's experiment (also known as double-beam interference), depicted in Figure 1.10. The sources So, S₁, and S₂ are either small holes or narrow slits perpendicular to the barrier. With slits, a cylindrical wave from So illuminates both S₁, and S2 so that they, in turn, act as in-phase sources of coherent waves that propagate on to the observing screen. **Figure 1.10** (a) Schematic diagram of Young's double-slit experiment. Slits S₁, and S₂ behave as coherent sources of light waves that produce an interference pattern on the viewing screen. (b) An enlargement of the center of a fringe pattern formed on the viewing screen with many slits could look like this. (c) An interference pattern involving water waves is produced by two vibrating sources at the water's surface. The pattern is analogous to that observed in Young's double-slit experiment. Note the regions of constructive (A) and destructive (B) interference. * **S₁** * **S₀** * **S₂** * **First barrier** * **Second barrier** * **(a)** * **Viewing screen** * **(b)** * **(c)** * **A.** * **B.** The pattern on the screen of Figure 1.10 (b) is a succession of bright and dark bands, or **interference fringes**, parallel to the slits S₁, and S2., _For small angles ym = Rma/d (Constructive interference in Young's experiment)_ (1.13) where ym is the distance from the center of the pattern to the center of the mth bright band of the interference fringes, λ is the wavelength, R is the distance between the slits and the screen. d is the separation of the slits. Q1. The figure below shows a two-slit interference experiment in which the slits are 0.200 mm apart and the screen is 1.00 m from the slits. The m = 3 bright fringe in the figure is 9.49 mm from the central fringe. Find the wavelength of the light. * **Slits** * **d = 0.200 mm** * **9.49 mm** * **↑** * **m = 3** * **m = 2** * **m = 1** * **X** * **m = -1** * **m = -2** * **m = -3** * **R = 1.00 m -** * **Screen** A1. From Eqn 6. λ = Ymd/mR = (9.49 × 10-3 m) (0.20 × 10-3 m)/(3)(1.0 m) = 6.33 × 10-9 m = 6.33 nm Q2. It is often desirable to radiate most of the energy from a radio transmitter in particular directions rather than uniformly in all directions. Pairs or rows of antennas are often used to produce the desired radiation pattern. As an example, consider two identical vertical antennas 400 m apart, operating at 1500 kHz = 1.5 × 106 Hz (near the top end of the AM broadcast band) and oscillating in phase. At distances much greater than in what directions is the intensity from the two antennas greatest? * **m=-1** * **θ = -30°** * **m = 0** * **θ = 0°** * **m = +1** * **θ = +30°** * **30°** * **30°** * **m = -2** * **θ=-90°** * **90°1** * **m = +2** * **θ = +90°** * **S₁K** * **S₂** * **400 m** A1. Use sin θ = mλ/d = m(200 m)/400 m = m/2. θ = 0, ±30°, ±90° ### Practice Problems 1) State a condition for interference. 2) The atoms in an HCl molecule vibrate like two charged balls attached to the ends of a spring. If the wavelength of the emitted electromagnetic wave is 3.75 µm, what is the frequency of the vibrations? [hint c = fλ, 1µm = 10-6 m]. 3) An observer on the west-facing beach of a large lake is watching the beginning of a sunset. The water is very smooth except for some areas with small ripples. The observer notices that some areas of the water are blue and some are pink. Why does the water appear to be different colors in different areas? 4) A typical compact disc stores information in tiny pits on the disc's surface. A typical pit size is 1.2 µm. What is the frequency of electromagnetic waves that have a wavelength equal to the typical CD pit size? [hint c = fλ, 1µm = 10-6 m]. 5) As wave travels, what happens to the intensity? ### 1.3.6 The Doppler Effect The Doppler effect is a phenomenon that describes the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the source of the wave. It is named after the Austrian physicist Christian Doppler who first proposed it in 1842. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for the observer and the source are approaching and an apparent downward shift in frequency when the observer and the source is receding. The Doppler effect is commonly observed in various types of waves, including sound waves, light waves, and electromagnetic waves. **Doppler Effect for Sound:** **Moving Source Towards a Stationary Observer:** * If a sound source is moving toward a stationary observer, the waves are compressed, leading to a higher perceived frequency (higher pitch). * This effect is commonly experienced when a vehicle with a sounding horn approaches an observer. **Moving Source Away from a Stationary Observer:** * If a sound source is moving away from a stationary observer, the waves are stretched, leading to a lower perceived frequency (lower pitch). * This is akin to the decreasing pitch of a siren as an emergency vehicle moves away. **Stationary Source and Moving Observer:** * If the sound source is stationary and the observer is moving towards the source, the same compression and higher frequency occur. * If the observer is moving away, the stretching and lower frequency occur. In the diagram below, an observer _O_ (the cyclist) moves with a speed _vo_ toward a stationary point source _S_, the horn of a parked truck. The observer hears a frequency _fo_ that is greater than the source frequency. **Figure 1.11** A schematic diagram to illustrate the Doppler effect Let _fs_ be the frequency of the source, the wavelength to be λ, and the speed of sound to be _v_. Let the source approach the observer at velocity _vs_ measured relative to the medium conducting the sound. Suppose the observer is moving toward the source at velocity _v_ also measured relative to the medium. Then the observer will hear a sound of frequency _fo_ given by the general Doppler-shift expression _fo = (u + vo)fs / (v + vs)_ (1.14) If either the source or the observer is moving away from each other, the sign on its velocity in Eqn. (7) must be changed. **Exercise:** A submarine (Sub A) travels through water at a speed of 8.00 m/s, emitting a sonar wave at a frequency of 1400 Hz. The speed of sound in the water is 1533 m/s. A second submarine (Sub B) is located such that both submarines are traveling directly toward each other. The second submarine is moving at 9.00 m/s. (a) What frequency is detected by an observer riding on Sub B as the subs approach each other? [Answer 1416 Hz]. (b) The subs barely miss each other and pass. What frequency is detected by an observer riding on Sub B as the subs recede from each other? [Answer 1385 Hz]. ### Doppler Effect for Light and Electromagnetic Waves: **Redshift and Blueshift:** * For light and other electromagnetic waves, a similar effect occurs with a change in color. * When a source of light is moving away from an observer, the light waves are stretched, resulting in a shift towards the red end of the spectrum, known as redshift. * Conversely, if the source is moving towards the observer, the waves are compressed, resulting in a shift towards the blue end of the spectrum, known as blueshift. ### Assignment: 1. Polarisation of waves means the orientation of the oscillations of transverse waves in a particular direction. Seek resources to discuss more on polarisation of waves stating the significance. 2. Dispersion is the separation of a wave into its individual components based on their frequencies. Discuss the significance of dispersion. 3. Explain the concept of scattering of waves 4. Discuss how waves transport energy without transporting matter.