Wave Motion and Waves on a string PDF

BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil PHYSICS – I (BPHY – 102) Dr. ISMAYIL Associate Professor Department of Physics Manipal Institute of Technology, Manipal...

BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil PHYSICS – I (BPHY – 102) Dr. ISMAYIL Associate Professor Department of Physics Manipal Institute of Technology, Manipal 1  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil PHYSICS - I Syllabus (contd..) Wave Motion and Waves on a String 1. Wave Motion 2. Sine Wave on a String 3. Standing Waves 4. Transverse and Longitudinal Waves 5. Laws of Transverse Vibrations of a string: Sonometer Text Book: Paul G Hewitt, Conceptual Physics,12th Edn., Pearson Education, Inc., USA 2  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Some applications of the above topics in the medical field: 1. Wave Motion: 1. Ultrasound Imaging: Uses high-frequency sound waves (a type of mechanical wave) to create images of organs and tissues inside the body. It's a non-invasive diagnostic tool widely used in medical practice. 2. Therapeutic Ultrasound: Applies sound waves to generate heat or vibration within tissues, aiding in physical therapy and tissue healing. 2. Sine Wave on a String: 1. Electrocardiography (ECG): The sine wave is fundamental in understanding the electrical activity of the heart, which is recorded as a waveform. Each heartbeat generates a wave that resembles a sine wave, used to diagnose heart conditions. 2. Medical Instrument Calibration: The sine wave is often used in the calibration of medical devices that require precise waveforms, such as oscilloscopes or heart monitors. 3. Standing Waves: 1. MRI Imaging: Magnetic Resonance Imaging (MRI) exploits standing waves of radiofrequency waves in the human body to create detailed images of tissues. The interaction of these waves with the body's hydrogen atoms is crucial for generating the MRI signal. 2. Acoustic Resonance: In certain therapeutic applications, standing waves are used to focus energy at specific points, such as in lithotripsy for breaking kidney stones using focused ultrasound waves. 3  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil 4. Transverse and Longitudinal Waves: 1. Diagnostic Ultrasound: Longitudinal waves are used in ultrasound to penetrate the body and reflect off tissues, providing internal images. Transverse waves are also important in specific diagnostic techniques. 2. Hearing Aids: Understanding transverse and longitudinal sound waves helps in designing and tuning hearing aids that amplify sound to aid hearing-impaired individuals. 5. Laws of Transverse Vibrations of a String: Sonometer: 1. Hearing Tests (Audiometry): Principles from the sonometer are applied in testing the frequencies and amplitudes that can be heard by a patient. This helps in diagnosing hearing loss and in designing hearing aids. 2. Vibration Therapy: The understanding of transverse vibrations is used in vibration therapy, where controlled vibrations are applied to the body for therapeutic effects, such as improving circulation and muscle recovery. 4  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Vibrations Anything moving back and forth, side to side, or up and down is vibrating. A vibration is a periodic oscillation in time, and a wave is a periodic oscillation in space and time. Light and sound are vibrations that move through space as waves, but they differ greatly. Sound is a mechanical wave needing a material medium; it can't travel through a vacuum. Light is an electromagnetic wave that can travel through a vacuum, like between the Sun and Earth. 5  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Harmonic or Periodic motion Periodic motion: Any motion that repeats after equal intervals of time is called periodic or harmonic motion. Examples: Revolution of earth around the sun is periodic motion with period 1 year. Motion of hands of clock Motion of electron round the nucleus of an atom Motion of a simple pendulum is also periodic. 6  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Simple Harmonic Motion (SHM) Simple harmonic motion (SHM) is a type of periodic motion where an object moves back and forth along a path, and the restoring force acting on it is directly proportional to its displacement from its equilibrium position and is always directed towards that position. This motion is characterized by the following features: 1. Oscillatory Motion: The object moves in a regular, repeating pattern, such as the back-and-forth swing of a pendulum. 2. Restoring Force: The force that brings the object back to its equilibrium position is proportional to the displacement from that position (e.g., Hooke's law in springs, F = -kx). 3. Sinusoidal Waveform: The displacement of the object over time forms a sine or cosine wave, indicating smooth and continuous oscillations. 4. Constant Frequency: The time taken for each complete cycle of motion (period) is constant, meaning the motion is uniform and predictable. 5. Energy Conservation: The total mechanical energy (kinetic + potential) in SHM remains constant if no damping forces are present. An example of simple harmonic motion is the motion of a mass on a spring or the swing of a simple pendulum under small displacements. 7  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Wave Description Crest A sine curve is a pictorial representation of a wave. Trough The high points of a sine wave are called A sine curve can be traced by a bob attached to a spring undergoing vertical simple harmonic motion. crests and the low points are called troughs. The term amplitude refers to the distance from the midpoint to the crest (or trough) of the wave. So the amplitude equals the maximum displacement from equilibrium. The wavelength (λ) of a wave is the distance from the top of one crest to the top of the next crest. Or, equivalently, the wavelength is the distance between any successive identical parts of the wave. The wavelengths of waves are measured in meters. The number of oscillations per second is called frequency (f). It is expressed in Hertz (Hz). Time taken to complete one oscillation is called time period (T). It is expressed in ‘seconds’. 1 Frequency = Time period 8  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil P1. An electric toothbrush completes 90 cycles every second. What are (a) its frequency and (b) its period? Ans: a) Frequency, f = 90 Hz b) Time period, T = 1/f = 1/90 = 0.011 s P2. Gusts of wind make the Willis Tower in Chicago sway back and forth, completing a cycle in 10 s. What are (a) its period and (b) its frequency? Ans: a) Time period, T = 10 s b) Frequency, f = 1/T = 1/10 = 0.1 Hz 9  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Simple Pendulum: l Periodic time ‘T’ of a simple pendulum: T = 2 g T of a simple pendulum depends on a) ‘l’ - length of the pendulum. Tmore if ‘l’ is more Length is measured from center of the bob. l b) ‘g’ - acceleration due to gravity. Tless if ‘g’ is more 10  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil P3: You have a simple pendulum with a length of 0.5 meters. Calculate the period (T) of the pendulum's swing. Assume the acceleration due to gravity (g) is 9.8 m/s². Ans : Given : l = 0.5 m, g = 9.8 m/s2 l Now, Period T = 2 g 0.5 = 2 × 3.14 9.8 = 1.42 s 11  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Wave Motion Wave motion refers to the transfer of energy through a medium without the permanent displacement of the particles (or matter) in the medium. Most information reaches us through wave motion, like sound, light, and radio signals. A rope shaken up and down demonstrates wave motion, where the disturbance travels. In a pond, waves from a stone propagate outward, disturbing water but not moving it. A leaf on water bobs up and down, returning to its starting point after waves pass. In tall grass, waves from the wind move the stems, but they return to their original position. 12  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Sine Wave on a String: Fasten one end of a rope to a wall and hold the free end in your hand. If you suddenly shake the free end up and then down, a pulse will travel along the rope and back. In this case, the motion of the rope (up and down arrows) is at right angles to the direction of the wave speed. Now shake the rope with a regular, continuing up-and-down motion, and the series of pulses will produce a wave. 13  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Transverse and Longitudinal Waves In transverse waves, the particle oscillation is perpendicular to the direction of wave propagation. In these waves, the crests and troughs are the highest and lowest points of the wave, respectively. Example: Waves on a string, electromagnetic waves etc. Both waves transfer energy from left to right. In longitudinal waves, the particle oscillation is parallel to the direction of wave propagation. These waves have compressions (where particles are close together) and rarefactions (where particles are spread out). Example: Sound waves, pressure waves in fluids. 14  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Mechanical and Non-mechanical waves: Mechanical waves need a material medium (like air, water, or solid) for their propagation. They transfer energy by causing particles in the medium to vibrate or oscillate. Examples: Sound waves (in air), water waves (in water), and seismic waves (in the Earth). Non-mechanical waves, also known as electromagnetic waves, doesn’t require material medium for their propagation and they can travel through vacuum. They transfer energy through oscillating electric and magnetic fields. Examples: Light, radio waves, X-rays, and microwaves. 15  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Wave Speed (v): The speed of periodic wave motion is related to the frequency and wavelength of the waves. We know that speed is defined as distance divided by time. In this case, the distance is one wavelength and the time is one period. So, Wave speed = wavelength/period Since period is the inverse of frequency, Wave speed = frequency × wavelength v=f×λ 16  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil P4: If a water wave oscillates up and down three times each second and the distance between wave crests is 2 m, what is its frequency? What is its wavelength? What is its wave speed? Ans: The frequency of the wave = 3 Hz Its wavelength = 2 m Wave speed = frequency × wavelength =3×2 = 6 m/s 17  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil P5: On an average, a human heart is found to beat 75 times in a minute. Calculate its frequency and period. Ans: The beat frequency of heart f = 75/(1 min) = 75/(60 s) = 1.25 /s = 1.25 Hz The time period T = 1/(1.25) = 0.8 s 18  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Wave Interference Wave interference occurs when two or more waves overlap in the same space. This overlapping causes the waves to combine, resulting in a new wave pattern. In phase Constructive Interference: If the crest (high point) of one wave overlaps with the crest of another, their amplitudes add together, creating a wave with a higher amplitude. This results in a stronger wave. Destructive Interference: If the crest of one wave overlaps with the out of phase trough (low point) of another, their amplitudes subtract from each other. This can reduce the overall amplitude, and in some cases, completely cancel out the wave, creating a region of zero amplitude. Wave interference is a key characteristic of all types of waves, whether they are sound waves, light waves, or water waves. This phenomenon can create interesting patterns, such as the ripples you see when two stones are thrown into a pond at the same time. 19  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Standing Waves (or Stationary Waves) Standing waves are a type of wave pattern that occurs when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. Unlike traveling waves, which move through space, standing waves appear to be stationary, with certain points on the wave, called nodes, remaining still while other points, called antinodes, vibrate with maximum amplitude. Antinode Node The incident and reflected waves interfere to produce a standing wave. Examples: Standing waves can be seen in vibrating strings (like guitar strings), in air columns (like in a flute or organ pipe), and even on the surface of water when waves are reflected back and forth. 20  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Antinode Node Characteristics of Stationary waves: 1. Stationary waves do not travel in a medium, so no transfer of energy. 2. Some particles in a stationary wave are always fixed in their positions, are called ‘Nodes’. At nodes displacement of vibration is zero. 3. Exactly at midway between two nodes there is an ‘Antinode’ which has maximum displacement of vibration. 4. Distance between two consecutive nodes or antinodes is equal to ‘λ/2’. 5. Distance between a node and the next immediate antinode is equal to ‘λ/4’. 21  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Laws of Transverse Vibrations of a string: The Laws of Transverse Vibrations of a string describe how the frequency of vibrations in a stretched string (fixed at both ends) depends on certain factors. There are three main laws: (i) Law of length : The fundamental frequency (f) of vibration of a string (fixed at both ends) is inversely proportional to the length (L) of the string provided its tension (T) and its mass per unit length () remain the same. Here tension (T) is the force that pulls the string tight. 1 i.e., 𝑓 ∝ if T and  are constants 𝐿 (ii) Law of tension : The fundamental frequency of a string is proportional to the square root of its tension provided its length and the mass per unit length remain the same. i.e., 𝑓 ∝ 𝑇 if L and  are constants (iii) Law of mass: The fundamental frequency of a string is inversely proportional to the square root of the linear mass density, i.e., mass per unit length provided the length and the tension remain the same. 1 i.e., 𝑓 ∝ if T and L are constants 𝜇 22  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil The three laws can be combined into a single equation that gives the frequency of a vibrating string: 𝟏 𝑻 𝒇= 𝟐𝑳 𝝁 Where, f is the frequency of vibration. L is the length of the string. T is the tension in the string. μ is the mass per unit length of the string. 23  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Sonometer: A sonometer is a device used to study the relationship between the frequency of vibrations of a stretched string and its physical properties, such as length, tension, and mass per unit length. It's often used to demonstrate the principles of sound waves and resonance in strings. Structure of a Sonometer: Wooden Base: The sonometer typically consists of a hollow rectangular wooden box that acts as a resonator. This box amplifies the sound produced by the vibrating string. Wires or Strings: A wire or string is stretched over the length of the box between two fixed bridges (supports). The string can be made of steel, brass, or any other material. Pulley and Weights: One end of the string is attached to a fixed support, while the other end passes over a small, smooth pulley and is connected to a pan where weights can be added. The weights create tension in the string. Bridges or Knife Edges: Two movable bridges or knife edges are placed under the string. These can be adjusted to change the vibrating length of the string. Tuning Fork: A tuning fork is often used alongside the sonometer. It helps in tuning the string to a specific frequency and demonstrating resonance. 24  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Working of Sonometer: Adjusting Length: By moving the bridges closer or farther apart, one can change the length of the vibrating portion of the string. According to the Law of Length, this affects the frequency of the sound produced by the string. Changing Tension: Adding or removing weights from the pan changes the tension in the string. According to the Law of Tension, increasing the tension increases the frequency, and vice versa. Resonance: When the frequency of the string matches the frequency of the tuning fork, resonance occurs, causing the sound to be 𝟏 𝑻 𝒇= amplified. This resonance condition helps in precisely determining 𝟐𝑳 𝝁 the frequency of the string. 25  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil P6: A 50 cm long wire of mass 20 g supports a mass of 1.6 kg as shown in figure. Find the fundamental frequency of the portion of the string between the wall and the pulley. Take g = 10 m/s2. Solution: The tension in the string is, T = mg = 1.6 × 10 = 16 N The linear mass density of the string is,  = ms/l = 0.02/0.5 = 0.04 kg/m 𝟏 𝑻 The fundamental frequency is, 𝒇 = 𝟐𝑳 𝝁 1 16 = 2×0.4 = 𝟐𝟓 𝐇𝐳 26 0.04  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil P7: A guitar string is 90 cm long and has a fundamental frequency of 124 Hz. Where should it be pressed to produce a fundamental frequency of 186 Hz ? Solution : 𝟏 𝑻 The fundamental frequency of a string fixed at both ends is given by, 𝒇= 𝟐𝑳 𝝁 As T and  are fixed, 1 𝑓∝ 𝐿 𝑓1 𝐿2 Now, = 𝑓2 𝐿1 𝑓1 Or, 𝐿2 = 𝐿 𝑓2 1 124 = × 90 = 60 cm 186 Thus, the string should be pressed at 60 cm from an end. 27  BSc (HS) | Physics - I | BPHY – 102 | Dr. Ismayil Questions: 1. What is a wave? 2. What is harmonic motion? Explain simple harmonic motion. 3. Define the following: a) wavelength b) frequency c) time period d) amplitude. 4. What is wave motion? 5. Distinguish between transverse and longitudinal waves. 6. What are mechanical and non-mechanical waves? 7. Explain wave interference. 8. What are stationary waves? Explain. 9. List the characteristics of stationary waves. 10.Explain the laws of Transverse Vibrations of a string. 11.Explain the construction and working of sonometer. 28