Waves PDF

# Chapter 14 - Waves ## Water Contents - Introduction - Transverse and Longitudinal Waves - Displacement Relation for Progressive Wave - Speed of a Travelling Wave - Speed of a Transverse Wave on a Stretched String - Speed of a Longitudinal Wave (Sound) - The Principle of Superposition of W...

# Chapter 14 - Waves ## Water Contents - Introduction - Transverse and Longitudinal Waves - Displacement Relation for Progressive Wave - Speed of a Travelling Wave - Speed of a Transverse Wave on a Stretched String - Speed of a Longitudinal Wave (Sound) - The Principle of Superposition of Waves - Reflection of Waves - Standing Waves and Normal Modes - Normal Modes of Oscillation of an Air Column with One End Closed and Other Open(Closed Organ Pipe) - Standing waves and Normal Modes in Open Organ Pipe - Beats ## Introduction - Most people have experienced wave propagation when dropping a stone in a pond. - The waves move in expanding circles until they reach the edge of the pond. - The waves move outward because the water is moving outward from the point of disturbance. - If you watch a leaf floating on the disturbed water, you will see the leaf move up and down, but it does not move toward the shore. - This means that energy is transferred by the wave, but there is no transfer of the medium. - A _wave_ is a pattern in which energy is transmitted from one part of a medium to another part, but there is no transfer of matter as a whole. - Waves are closely connected to harmonic oscillations. - A disturbance travels to one end of the spring, then to the next, and so on. - This disturbance is called a wave. - However, each spring oscillates about its mean position. - Similarly, sound waves travel in the air by compressions and expansions of air. - When the wave propagates, it compresses or expands a small region of air. - The density increases in the compressed regions, increasing the pressure. - Pressure is directly proportional to the force. - This means that there is a restoring force acting in the compressed region, just like a spring. - The disturbance propagates through the air in the form of compressions and rarefactions. ## Transverse and Longitudinal Waves - _Mechanical waves_ require a material medium for their propagation. - They cannot travel through a vacuum. - Sound waves, water waves, and waves on a string are examples of mechanical waves. - _Electromagnetic waves_ do not require a medium for their propagation. - They can travel through a vacuum. - X-rays, radio waves, and light are electromagnetic waves. - _Matter waves_ are associated with constituents of matter such as electrons, protons, neutrons, atoms and molecules. ### Definition of the Two Types of Waves - In transverse waves, the particles of the medium oscillate perpendicular to the direction of wave propagation. - In longitudinal waves, the particles of the medium oscillate along the direction of wave propagation. ### Examples of the Different Types of Mechanical Waves - *Inside water*: **longitudinal waves** - *On the surface of water*: **longitudinal and transverse waves** ### Displacement Relation for a Progressive Wave - A single pulse is formed and travels along the rope with a fixed speed when a jerk is given to one end of a rope that has its opposite end fixed. - This is what is called a _traveling wave_. - If we give continuous periodic up and down jerks to one end of the rope, then the _harmonic wave_ is produced on the rope. - This disturbance moves from left to right as shown. - However, the particles of the string oscillate up and down, hence the name _transverse wave_. ### Speed of a Travelling Wave - *In the above equations, 'v' represents the speed of the wave and not that of the particle.* - The speed of mechanical waves depends on the inertial and elastic properties of a medium. - The speed of a mechanical wave is different for different media. - In a particular medium with a constant speed, the speed of a longitudinal wave is different from the speed of a transverse wave. - In a room, the displacement of any pendulum from its mean position depends on the position (x) of the pendulum and on time (t). - It has been shown analytically that any function of space and time which satisfies the equation $ \frac{\partial ^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial ^2 y}{\partial t^2}$ represents a wave. - Here y is the wave function and does not necessarily denote y-coordinate. - Functions $y = A sin\omega t$ or $y = A sin kx$ do not satisfy the above equation so do not represent waves, while functions $y = A sin(\omega t - kx), y = A sinkx sin\omega t$ or $y = Asin(\omega t - kx) + Bcos(\omega t + kx)$ do satisfy the above equation, thus they represent waves. ### Speed of a Transverse Wave on a Stretched String - The speed of a mechanical wave depends on the inertial and elastic properties of the medium. - For a stretched string, the inertial properties and the elastic properties are the linear mass density ($\mu$) and the tension ( ) respectively. - The speed of transverse waves on a stretched string is given by: $\boxed{ v = \sqrt{\frac{T}{\mu }}}$ ### Speed of a Longitudinal Wave (Sound) - The speed of a longitudinal wave also depends on the inertial and elastic properties of the medium. - For a longitudinal wave, such as sound, the constituents of the medium oscillate along the direction of wave propagation. - The speed of longitudinal wave is given by: $\boxed{ v = \sqrt{\frac{B}{p}}}$ Where B is the bulk modulus and p is the density of the medium. - The speed of sound is greater in solids and liquids than in gases even though they are denser than gases. This happens because solids and liquids are much more difficult to compress than gases, and so their bulk modulus is much higher. This factor more than compensates for their higher densities. - The speed of sound in a gas is given by: $\boxed{ v = \sqrt{\frac{\gamma P}{p}}}$ Where $\gamma$ is the adiabatic index, P is the pressure, and p is the density of the gas. - Newton assumed that when sound waves propagate through a gas, the change in pressure and volume of the gas remains constant. - This is also known as Newton’s formula. - Laplace pointed out that the pressure variations in the gases when sound propagates are so fast that the heat does not get enough time to flow to surroundings or from surroundings to keep the temperature constant. - Therefore the variations are adiabatic and not isothermal. - For an adiabatic process, $PV^\gamma = Constant$ - Where $\gamma$ is the ratio of molar specific heat of the gas at constant pressure ($C_p$) to its molar specific heat at constant volume ($C_v$). ### The Principle of Superposition of Waves - The principle of superposition states that when two or more waves pass through the same point in space, the resultant displacement at that point is the vector sum of the individual displacements. - This means that the waves interfere with each other, and the resultant amplitude is the sum of the amplitudes of the individual waves. - In interference of two waves, the resultant amplitude may be greater, less or even zero compared to the amplitudes of the individual waves. #### The Concept of Interferences - When two waves of the same frequency and amplitude traveling in the same direction along a stretched string overlap, then, the two waves interfere creating a new wave pattern whose amplitude is different from the amplitudes of the individual waves. - This occurs because the individual waves interfere constructively or destructively. - In _constructive interference_, the waves add together, and the amplitude of the resultant wave is larger than the amplitude of either of the individual waves. - In _destructive interference_, the waves cancel each other out, and the amplitude of the resultant wave is smaller than the amplitude of either of the individual waves. - The phase difference between the two waves, where the phase difference is given by $\phi = \frac{2 \pi \Delta x}{\lambda}$, determines whether the interference is constructive or destructive. - If the phase difference is zero or a multiple of $2\pi$, then the interference is constructive. - If the phase difference is $\pi$ or an odd multiple of $\pi$, then the interference is destructive. ### Reflection of Waves - When a progressive wave traveling along a stretched string arrives at a rigid boundary, the wave gets reflected. - The reflected wave suffers a phase change of $180^\circ$ on reflection. - At the rigid boundary, disturbance plus the reaction force from he boundary results in a reflected pulse with a phase difference of $\pi$ radian, hence a crest is reflected as a trough. - If the wave reflects from a free end, then the reflected wave has the same phase and amplitude as the incident wave. - In this case, a crest is reflected as a crest. ### Standing Waves and Normal Modes - When two waves of the same frequency and amplitude traveling along the same path but in opposite directions superimpose, the result is a stationary wave. - The amplitude of the resultant wave depends on the position and doesn’t appear separately in the equation for the resultant wave. - For a string fixed at both ends, the positions where the amplitude is zero are called _nodes_, and the positions where the amplitude is a maximum are called _antinodes._ - This means that the system cannot oscillate with any arbitrary frequency, but is characterised by a set of natural frequencies called normal modes. - The natural frequencies of vibration of the system are determined by the boundary conditions at the ends of the string. - For a string fixed at both ends, the boundary conditions are that the displacement of the string must be zero at both ends. - This leads to the following equation for the natural frequencies of vibration: $\boxed{v_n = \frac{n v}{2L}}$ Where v is the speed of the wave, L is the length of the string, and n is an integer. - The fundamental mode (first harmonic) corresponds to n = 1. - The second harmonic corresponds to n = 2, and so on. - It is called the first overtone. - The third harmonic corresponds to n = 3, and so on. - It is called the second overtone. ### Normal Modes of Oscillation of an Air Column with One End Closed and the Other Open (Closed organ pipe) - A glass tube partially filled with water is an example of such a system. - We can also see this with a closed organ pipe that is closed at one end. - The equation of the stationary wave for a closed organ pipe can be derived as follows: - Since one end is closed, a node is formed at this end; the other end is open, so an antinode is formed. - If the length of the air column is denoted by $L$, the closed end is denoted by $x=0$ and the open end is denoted by $x=L$. - The position of the nodes is given by: $\boxed{x = \frac{n\lambda}{2}}$ Where n = 1,2, 3 … - The position of the antinodes is given by: $\boxed{x = (2n-1) \frac{\lambda}{4}}$ Where n = 1, 2 … - Since an antinode is formed at the open end, the following is the equation for the wavelength: $\lambda = (2n-1) \frac{4L}{n}$ - The corresponding frequencies can be computed as: $\boxed{v_n = (2n-1)v_1}$ - In a closed organ pipe, only odd harmonics are present, i.e., odd multiples of the fundamental frequency. ### Standing Waves and Normal Modes in Open Organ Pipe - The reflecting surface in an open organ pipe is an open boundary at both ends of the pipe. - The reflecting wave is in phase with the incident wave. - Such a wave can be easily studied by taking the initial phase angle, $\phi = \frac{\pi}{2}$ - Taking the initial phase angle at $\phi =\frac{\pi}{2}$, the equation of the wave traveling along the positive direction of x-axis is given by y(x,t) = asin(kx-wt + $\frac{\pi}{2}$ ) y(x,t) = a cos(kx-wt) - The equation of the reflected wave is given by: y₂(x,t) = a cos(kx-wt) - According to the principle of superposition, the resultant wave is then given by: y(x,t) = y₁(x,t) + y₂(x,t) y(x,t) = a cos(kx-wt) + a cos(kx-wt) y(x,t) = 2a coskx coswt - Since both the ends are open, antinodes are formed at both the ends. - The position of antinodes is given by putting coskx = ± 1 kx = nπ n = 0, 1, 2, 3… $\boxed{x=\frac{n \lambda}{2}}$ - Further at $x = L$ an antinode is formed: $\frac{\lambda}{2} = \frac{L}{n}$ $\boxed{\lambda = \frac{2L}{n}}$ - And the corresponding frequencies can be computed as: $\boxed{v_n = \frac{n v}{2L}}$ Where n = 1, 2, 3 … - *For n = 1, v₁ = \frac{v}{2L} is called the fundamental mode or 1st harmonic* - *For n = 2, v₂ = \frac{2v}{2L} is called the 2nd harmonic or 1st overtone.* - *For n = 3, v₂ = \frac{3v}{2L} is called the 3rd harmonic or 2nd overtone.* ### Beats - The phenomenon that occurs when two waves of nearly equal frequencies superimpose is called _beats_. - This phenomenon can be obtained when two harmonic waves of equal amplitude but slightly different frequencies superimpose. - The amplitude variation is called a beat. - The frequency with which the amplitude rises and falls is called the beat frequency, which is equal to the difference in frequencies of the two waves. - The rise and fall of the intensity of sound is called waxing and waning. ### Summary * **Waves** are a pattern in which energy is transmitted from one part of a medium to another part, but there is no transfer of matter as a whole. * **Mechanical waves** require a material medium for their propagation. * **Electromagnetic waves** do not require a medium for their propagation. * **Matter waves** are associated with the wave-like nature of constituents of matter. * **Transverse waves** are waves in which the particles of the medium oscillate perpendicular to the direction of wave propagation. * **Longitudinal waves** are waves in which the particles of the medium oscillate along the direction of wave propagation. * **The speed of a wave** depends on the inertial and elastic properties of the medium. * **The principle of superposition** states that when two or more waves pass through the same point in space, the resultant displacement at that point is the vector sum of the individual displacements. * **Interference** occurs when two waves overlap. * **Constructive interference** occurs when the waves add together. * **Destructive interference** occurs when the waves cancel each other out. * **Reflection** occurs when a wave strikes a boundary. * **Standing waves** occur when two waves of the same frequency and amplitude traveling in opposite directions superimpose. * **Normal modes** are the natural frequencies of vibration of a system. * **Beats** are the periodic variations in the amplitude of a wave that occurs when two waves of slightly different frequencies interfere. This summary is helpful for understanding the key concepts of waves.