WAVES-Notes
Physics - Notes
WAVE
A wave is a physical phenomenon by which a disturbance or oscillation travels through a medium or space, transferring energy from one point to another without any net transfer of matter. In a wave motion, the individual particles of the medium do not move along with the wave; instead, they execute oscillatory motion about their mean or equilibrium positions while the disturbance propagates forward.
waves arise when a system that possesses inertia and restoring forces is disturbed from its equilibrium state. The restoring force attempts to bring the system back to equilibrium, while inertia causes it to overshoot, resulting in oscillations. When such oscillations are communicated from one particle of the medium to the neighboring particles, a wave is formed. Thus, wave motion represents the collective and organized motion of particles, even though each particle itself remains confined to a small region around its equilibrium position.
Waves can exist only in a medium that can transmit disturbances through inter-particle interactions, as in the case of mechanical waves such as sound and water waves. However, certain waves like electromagnetic waves do not require a material medium and can propagate through vacuum. In all cases, the essential feature of a wave is the transport of energy and information through space and time, without the permanent displacement of the medium as a whole.
Transverse and Longitudinal Waves
Waves are commonly classified on the basis of the direction of vibration of particles of the medium relative to the direction in which the wave propagates. From this point of view, waves are mainly of two types: transverse waves and longitudinal waves.
Transverse Waves
A transverse wave is a wave in which the particles of the medium vibrate in a direction perpendicular to the direction of propagation of the wave. When such a wave travels through a medium, each particle oscillates up and down about its mean position while the wave itself moves forward along the medium.
In transverse waves, alternate regions of maximum upward displacement and maximum downward displacement are produced. These are known as crests and troughs, respectively. The distance between two successive crests or two successive troughs represents the wavelength of the wave.
Transverse waves can propagate only in media that possess elasticity of shape, such as solids and the surface of liquids. For example, waves produced on a stretched string or ripples on the surface of water are transverse in nature. Electromagnetic waves, although they do not require a material medium, are also transverse waves, with electric and magnetic fields oscillating perpendicular to the direction of propagation.
Longitudinal Waves
A longitudinal wave is a wave in which the particles of the medium vibrate in a direction parallel to the direction of propagation of the wave. In this type of wave motion, particles of the medium oscillate back and forth along the same line in which the wave travels.
As a longitudinal wave passes through a medium, it produces alternate regions of compression and rarefaction. A compression is a region where particles are closer together and pressure is higher, whereas a rarefaction is a region where particles are farther apart and pressure is lower. The distance between two successive compressions or two successive rarefactions is equal to the wavelength of the wave.
Longitudinal waves can propagate in solids, liquids, and gases, since all these media can support changes in pressure and density. Sound waves in air are the most common example of longitudinal waves.
Capillary waves and Gravity waves
Surface waves produced on the free surface of a liquid are an important class of mechanical waves discussed in the context of wave motion. Depending on the dominant restoring force responsible for the propagation of these waves, surface waves are broadly classified into capillary waves and gravity waves
Capillary Waves
Capillary waves are surface waves in which the restoring force is provided mainly by surface tension of the liquid. These waves are generally produced when the wavelength of the disturbance is very small, typically of the order of a few millimetres or less.
In capillary waves, the effect of gravity is negligible compared to surface tension. When a small disturbance is created on the liquid surface, surface tension acts to minimize the surface area and tends to restore the surface to its equilibrium shape. As a result, the disturbed particles of the liquid execute oscillatory motion, leading to the propagation of capillary waves along the surface.
Capillary waves are commonly observed as fine ripples on the surface of water, such as those formed when a small object gently touches the surface or when light wind begins to blow. These waves play an important role in phenomena involving surface properties of liquids.
Gravity Waves
Gravity waves are surface waves in which the restoring force is primarily due to gravity. These waves have relatively large wavelengths, typically much greater than a few centimetres, for which the effect of surface tension becomes insignificant.
When the surface of a liquid is disturbed over a larger region, gravity tends to pull the raised portions of the liquid back down and lift the depressed portions upward. This gravitational restoring force causes oscillations of the liquid surface, which propagate as gravity waves.
Common examples of gravity waves include ocean waves, waves produced by strong winds, and waves generated by disturbances such as earthquakes under the sea. In these waves, the motion of liquid particles is mainly circular near the surface and decreases with depth.
Difference between Capillary Waves and Gravity Waves
| Aspect | Capillary Waves | Gravity Waves |
|---|---|---|
| Restoring Force | Surface tension of the liquid | Gravity (weight of water) |
| Wave Characteristics | Small amplitude and wavelength | Large amplitude and wavelength |
| Common Examples | Ripples on water surface, small disturbances | Ocean waves, large waves in oceans |
| Wavelength Range | Typically less than a few centimeters | Typically greater than a few centimeters |
| Effect of Gravity | Gravity has a negligible effect | Gravity plays a dominant role in wave propagation |
| Amplitude | Low amplitude waves | High amplitude waves |
| Propagation Medium | Liquids with high surface tension | Liquids, mainly oceans and seas |
Displacement Relation in a Progressive Wave
A progressive wave (or travelling wave) is a wave that moves continuously through a medium, transferring energy from one point to another without any net transfer of matter. To describe such a wave quantitatively, the displacement of a particle of the medium is expressed as a function of both position and time. This mathematical expression is called the displacement relation or wave equation of a progressive wave.
Consider a harmonic wave travelling along the positive x-direction of a stretched string or elastic medium. Each particle of the medium executes simple harmonic motion with the same amplitude and frequency, but particles at different positions differ in phase.
Displacement Relation
Let the wave have:
\(\begin{aligned}\text{amplitude} &&a\\
\text{wavelength} &&\lambda\\
\text{frequency} &&f\\
\text{angular frequency} &&\omega = 2\pi f\\
\text{and wave number} && k = \dfrac{2\pi}{\lambda}\end{aligned}\)
If the wave travels along the positive x-direction, the displacement \(y\) of a particle at position \(x\) and time \(t\) is given by
\[ y(x,t) = a \sin (kx - \omega t + \phi) \]The term φ in the argument of sine function means equivalently that we are considering a linear combination of sine and cosine functions:
\[ y(x,t) = A \sin (kx - \omega t ) + B \cos(kx-\omega t) \]The choice of sine or cosine depends on the initial conditions of the wave.
Amplitude
\[a=\sqrt{A^2+B^2}\]Initial phase angle (a+x = 0, t = 0)
\[\phi=\tan \left(\dfrac{B}{A}\right) \]Physical Significance
The term \((kx - \omega t)\) represents the phase of the wave. As time increases, the phase advances in space, indicating that the disturbance travels through the medium.
The velocity of propagation of the wave is given by
\[ v = \frac{\omega}{k} = f\lambda \]Although the wave travels with a definite speed, each particle of the medium only oscillates about its equilibrium position. Thus, the displacement relation clearly shows that wave motion involves transfer of energy without transfer of matter.
Amplitude and Phase
Amplitude
The amplitude of a wave is defined as the maximum displacement of a particle of the medium from its mean (equilibrium) position during wave motion. It is usually denoted by the symbol \(a\).
Amplitude represents the extent of oscillation of particles and is a measure of the energy carried by the wave. A larger amplitude corresponds to greater energy transfer through the medium, while a smaller amplitude indicates lower energy. For example, in sound waves, greater amplitude corresponds to louder sound, and in water waves, higher waves indicate larger amplitudes.
It is important to note that amplitude is the same for all particles in a progressive wave propagating through a uniform medium, provided there is no loss of energy.
Phase
The phase of a wave specifies the state of oscillation of a particle at a given position and time. It tells us whether a particle is at its maximum displacement, minimum displacement, or passing through its mean position, and in which direction it is moving.
Mathematically, for a progressive wave represented by the equation
\[y(x,t)=A\sin\,(kx-\omega t + \phi)\]
the quantity \((kx-\omega t + \phi)\) is called the phase of the wave at position \(x\) and time \(t\).
\(\phi\) is the phase at \(x = 0\text{ and }t = 0\). Hence, \(\phi\) is called the initial phase angle.
Two particles are said to be:
- In the same phase if their phase difference is zero or an integral multiple of \(2\pi\).
- Out of phase if their phase difference is not an integral multiple of \(2\pi\).
Phase plays a crucial role in wave phenomena such as interference and beats, where the relative phase of interacting waves determines whether the resultant amplitude increases or decreases.
Wavelength and Angular Wave Number
The wavelength of a wave is defined as the minimum distance between two points that are in the same phase of oscillation. It is usually denoted by the symbol \( \lambda \). For simplicity, points of the same phase may be chosen as successive crests or successive troughs. Hence, the wavelength is equal to the distance between two consecutive crests or two consecutive troughs of the wave.
Consider a progressive wave travelling along the positive \(x\)-direction, represented by the equation
\[ y(x,t) = a \sin (kx - \omega t + \phi) \]Taking the initial phase \( \phi = 0 \), the displacement of a particle at position \(x\) at time \(t = 0\) is given by
\[ y(x,0) = a \sin kx \]Since the sine function is periodic, it repeats its value whenever its argument changes by an integral multiple of \(2\pi\). Therefore, the displacements at positions
\[ x \quad \text{and} \quad x + \frac{2n\pi}{k}, \quad \text{where } n = 1,2,3,\ldots \]are the same at any given instant of time. The least distance between two such points (having the same displacement and hence the same phase) is obtained by taking \( n = 1 \). This distance is defined as the wavelength \( \lambda \).
\[ \lambda = \frac{2\pi}{k} \]or equivalently,
\[ k = \frac{2\pi}{\lambda} \]The quantity \(k\) is known as the angular wave number or propagation constant. It represents the rate of change of phase of the wave with distance.
The SI unit of angular wave number is radian per metre (rad m\(^{-1}\)).
Thus, wavelength describes the spatial periodicity of a wave, while the angular wave number provides a convenient angular measure of this periodicity. Both quantities are fundamental to the mathematical description of wave motion.
Period, Angular Frequency and Frequency
In wave motion, every particle of the medium executes oscillatory motion about its mean position as the wave passes through it. The time-related characteristics of this oscillatory motion are described in terms of period, frequency, and angular frequency, as discussed in the NCERT Class XI Physics textbook.
The period of a wave, denoted by \(T\), is defined as the time taken by a particle of the medium to complete one full oscillation. It is the time interval after which the motion of the particle repeats itself. The SI unit of period is the second (s).
The frequency of a wave, denoted by \(f\) or \(\nu\), is defined as the number of complete oscillations made by a particle in one second. Frequency is the reciprocal of the period and is given by
\[ f = \frac{1}{T} \]The SI unit of frequency is hertz (Hz), where one hertz corresponds to one oscillation per second. Frequency is a characteristic property of the source producing the wave and remains unchanged when the wave travels from one medium to another.
To describe wave motion in angular terms, the concept of angular frequency is introduced. Angular frequency, denoted by \(\omega\), is defined as the rate of change of phase of the wave with respect to time.
Mathematically, angular frequency is related to frequency by
\[ \omega = 2\pi f \]Using the relation between frequency and period, angular frequency can also be written as
\[ \omega = \frac{2\pi}{T} \]The SI unit of angular frequency is radian per second (rad s\(^{-1}\)). In the equation of a progressive wave,
\[ y(x,t) = a \sin (kx - \omega t), \]the term \(\omega t\) represents the phase change of the wave with time.
Thus, the period specifies the time taken for one complete oscillation, frequency gives the number of oscillations per second, and angular frequency provides an angular measure of oscillation in time. These quantities are fundamental for the mathematical description and analysis of wave motion.
THE SPEED OF A TRAVELLING WAVE
To determine the speed of propagation of a travelling wave, we fix our attention on any particular point on the wave that is characterised by a definite value of phase. As discussed in the NCERT textbook, it is convenient to consider the motion of a crest of the wave, although the same reasoning applies to any other fixed phase point.
Consider the wave pattern at two instants of time that differ by a small time interval \(\Delta t\). During this interval, the entire wave pattern is observed to shift along the positive \(x\)-direction by a small distance \(\Delta x\). Hence, the crest moves a distance \(\Delta x\) in time \(\Delta t\). The speed \(v\) of the travelling wave is therefore given by
\[ v = \frac{\Delta x}{\Delta t} \]The same speed must be associated with every point of fixed phase on the wave; otherwise, the shape of the wave would change as it propagates.
The equation of a progressive wave travelling along the positive \(x\)-direction may be written as
\[ y(x,t) = a \sin (kx - \omega t) \]For a point of fixed phase on the wave, the quantity \((kx - \omega t)\) remains constant. Therefore, the motion of a fixed phase point is described by
\[ kx - \omega t = \text{constant} \]As time changes, the position of this phase point must change in such a way that the phase remains unchanged. Hence, for a small change in position \(\Delta x\) and time \(\Delta t\),
\[ k(x + \Delta x) - \omega (t + \Delta t) = kx - \omega t \]Simplifying, we get
\[ k \Delta x - \omega \Delta t = 0 \]or
\[ \frac{\Delta x}{\Delta t} = \frac{\omega}{k} \]Taking the limit as \(\Delta x \rightarrow 0\) and \(\Delta t \rightarrow 0\), the ratio becomes a derivative. Thus, the speed \(v\) of the travelling wave is
\[ v = \frac{dx}{dt} = \frac{\omega}{k} \]Using the relations
\[ \begin{aligned} \omega &= 2\pi \nu \\\\ k &= \frac{2\pi}{\lambda} \end{aligned} \]we obtain
\[ \begin{aligned} v &= \frac{2\pi \nu}{2\pi / \lambda} \\\\&= \lambda \nu \end{aligned} \]Since the frequency \(\nu\) is the reciprocal of the time period \(T\),
\[ \nu = \frac{1}{T}, \]the wave speed may also be written as
\[ v = \frac{\lambda}{T} \]Thus, using the calculus method and the condition of constant phase, we arrive at the important result that the speed of a travelling wave is given by
\[ \begin{aligned} v &= \frac{\omega}{k} \\\\&= \lambda \nu \\\\&= \frac{\lambda}{T} \end{aligned} \]This derivation clearly shows that the wave speed depends on the wavelength and frequency of the wave and is determined by the properties of the medium through which the wave propagates.
Speed of a Transverse Wave on Stretched String
When a transverse wave travels along a stretched string, the particles of the string vibrate perpendicular to the direction of propagation of the wave. The speed of such a wave depends on the tension in the string and the mass per unit length of the string
Physical Model
Consider a uniform string stretched tightly between two fixed supports. Let
- \(T\) be the tension in the string,
- \(\mu\) be the mass per unit length of the string.
Derivation of Wave Speed
Take a small element of the string of length \(\Delta x\). If the string is slightly displaced, the tension at the two ends of the element acts along the tangents to the string. The vertical components of the tension provide the restoring force responsible for wave propagation.
Using Newton’s second law for the transverse motion of the string element and considering small slopes, it can be shown that the transverse acceleration of the string element depends on the curvature of the string and the tension \(T\).
This analysis leads to the wave equation for a stretched string, from which the speed vof the transverse wave is obtained as
\[\boxed{\;v=\sqrt{\frac{T}{\mu}}\;}\]Meaning of the Terms
- \(T\) is the tension in the string (in newtons).
- \(\mu\) is the linear mass density of the string, given by \[\mu=\frac{m}{L}\]
- where \(m\) is the mass of the string and \(L\) its length. \(v\) is the speed of the transverse wave along the string.
Speed of a Longitudinal Wave (Speed of Sound)
A longitudinal wave is a wave in which the particles of the medium oscillate parallel to the direction of wave propagation. Sound waves are the most common examples of longitudinal waves. The speed of a longitudinal wave, particularly the speed of sound, depends on the elastic and inertial properties of the medium through which it travels.
General Expression for Speed of Longitudinal Waves
the speed vof a longitudinal wave in a medium is given by
\[v=\sqrt{\frac{\mathrm{Elastic\ modulus}}{\mathrm{Density}}}\]This is a general result applicable to solids, liquids, and gases, with the appropriate elastic modulus for each medium.
Speed of Sound in Solids
In solids, longitudinal waves propagate due to changes in length, and the relevant elastic constant is Young’s modulus \(Y\). If \rhois the density of the solid, the speed of sound is
\[v=\sqrt{\frac{Y}{\rho}}\]This explains why sound generally travels faster in solids than in liquids and gases.
Speed of Sound in Liquids
In liquids, longitudinal waves involve changes in volume, and the relevant elastic constant is the bulk modulus \(B\). The speed of sound in a liquid is given by
\[v=\sqrt{\frac{B}{\rho}}\]Liquids have much higher bulk modulus than gases, hence sound travels faster in liquids than in gases.
Speed of Sound in Gases
In gases, sound propagation occurs through compressions and rarefactions, and the elastic behavior is governed by the bulk modulus of the gas. For gases, sound propagation is an adiabatic process, and the bulk modulus is given by
\[B=\gamma P\]where
- \(P\) is the pressure of the gas,
- \(\gamma\) is the ratio of specific heats \(\left(C_p/C_v\right)\).
Substituting this into the general formula, the speed of sound in a gas is
\[v=\sqrt{\frac{\gamma P}{\rho}}\]Using the ideal gas relation, this can also be written as
\[v=\sqrt{\frac{\gamma RT}{M}}\]where
- \(T\) is the absolute temperature,
- \(R\) is the universal gas constant,
- \(M\) is the molar mass of the gas.
Laplace correction
The Laplace correction refers to the important modification made to Newton’s formula for the speed of sound in gases. It explains why sound waves in gases propagate under adiabatic conditions rather than isothermal conditions, thereby bringing theoretical results into agreement with experimental observations.
Newton’s Formula
Newton assumed that the compressions and rarefactions produced during the propagation of sound in a gas occur under isothermal conditions. Using the isothermal bulk modulus, which is equal to the pressure \(P\), he obtained the expression
\[ v = \sqrt{\frac{P}{\rho}} \]where \(\rho\) is the density of the gas. This formula gives a value of the speed of sound that is lower than the experimentally observed value.
Laplace’s Explanation
Laplace pointed out that in the propagation of sound waves, compressions and rarefactions take place very rapidly. As a result, there is no sufficient time for heat exchange between the gas and its surroundings. Hence, the process is adiabatic rather than isothermal.
For an adiabatic process in a gas, the bulk modulus \(B\) is given by
\[ B = \gamma P \]where \(\gamma = \dfrac{C_p}{C_v}\) is the ratio of specific heats of the gas.
Corrected Formula for Speed of Sound
Substituting the adiabatic bulk modulus into the general expression for the speed of a longitudinal wave,
\[ v = \sqrt{\frac{B}{\rho}} \]we obtain the corrected expression for the speed of sound in a gas as
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]This modified expression is known as the Laplace correction.
Significance of Laplace Correction
The Laplace correction resolves the discrepancy between Newton’s theoretical value and experimental measurements. It establishes that sound propagation in gases is an adiabatic process and provides a more accurate theoretical expression for the speed of sound.
THE PRINCIPLE OF SUPERPOSITION OF WAVES
When two wave pulses travelling in opposite directions meet or cross each other, they do not get destroyed or permanently altered. As shown in Fig. 14.9 of the NCERT textbook, each pulse continues to move forward with its original shape and speed after crossing the other. However, during the interval of overlap, the wave pattern is different from that of either pulse taken alone.
Consider two wave pulses of equal magnitude and opposite shapes moving towards each other. When they overlap, the displacement of any particle of the medium at that instant is obtained by taking the algebraic sum of the displacements produced by each pulse individually. Since displacements may be positive or negative, they may reinforce or cancel each other. In the situation shown in Fig. 14.9(c), the displacements cancel completely, giving zero displacement throughout the region of overlap.
This rule governing the combination of wave displacements is known as the principle of superposition of waves. According to this principle, each wave moves as if the other waves are not present.
Mathematical Statement
Let \(y_1(x,t)\) and \(y_2(x,t)\) be the displacements produced at a point \((x,t)\) by two wave disturbances. If the waves overlap, the resultant displacement \(y(x,t)\) is given by
\[ y(x,t) = y_1(x,t) + y_2(x,t) \]If several waves propagate simultaneously in the same medium, and their wave functions are
\[ \begin{aligned} y_1 &= f_1(x - vt), \\ y_2 &= f_2(x - vt), \quad \ldots \\ y_n &= f_n(x - vt), \end{aligned} \]then the resultant disturbance is
\[ y = f_1(x - vt) + f_2(x - vt) + \cdots + f_n(x - vt) \] \[ y = \sum_{i=1}^{n} f_i(x - vt) \]The principle of superposition forms the basis of the phenomenon of interference.
Superposition of Two Harmonic Waves
Consider two harmonic travelling waves on a stretched string, both moving in the positive \(x\)-direction. Let both waves have the same angular frequency \(\omega\), wave number \(k\), wavelength \(\lambda\), and equal amplitudes \(a\). They differ only in their initial phase.
The wave equations are
\[ y_1(x,t) = a \sin (kx - \omega t) \] \[ y_2(x,t) = a \sin (kx - \omega t + \phi) \]According to the principle of superposition, the resultant displacement is
\[ y(x,t) = a \sin (kx - \omega t) + a \sin (kx - \omega t + \phi) \]Using the trigonometric identity \(\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}\), we obtain
\[ y(x,t) = 2a \cos \frac{\phi}{2} \sin \left(kx - \omega t + \frac{\phi}{2}\right) \]This represents a harmonic travelling wave with the same frequency and wavelength as the original waves, but with an amplitude
\[ A(\phi) = 2a \cos \frac{\phi}{2} \]Constructive and Destructive Interference
For \(\phi = 0\), the waves are in phase and the resultant amplitude is maximum:
\[ A = 2a \]This is the case of constructive interference.
For \(\phi = \pi\), the waves are completely out of phase and the resultant displacement is zero everywhere:
\[ y(x,t) = 0 \]This corresponds to destructive interference, where the amplitudes cancel out completely.
Thus, the principle of superposition explains the interaction of waves and provides the foundation for understanding interference phenomena in wave motion.
Reflection of Waves
Reflection of waves is the phenomenon in which a wave, on striking a boundary or obstacle, returns back into the same medium instead of passing through it. This behaviour of waves is analogous to the reflection of light and arises due to the interaction of the wave with the boundary conditions imposed by the medium.
Reflection of Transverse Waves on a Stretched String
Consider a transverse wave travelling along a stretched string.
Reflection at a Fixed End
When a wave pulse reaches a fixed end of a string, the end point cannot move. Hence, the displacement at the boundary must be zero at all times. To satisfy this condition, the reflected wave undergoes a phase change of \(\pi\) radians (or 180°).
- A crest is reflected as a trough.
- A trough is reflected as a crest.
As a result:
Thus, reflection at a fixed end leads to inversion of the wave.
Reflection at a Free End
When a wave pulse reaches a free end of a string, the end point is free to move. In this case, there is no phase change on reflection.
- A crest is reflected as a crest.
- A trough is reflected as a trough.
As a result:
Thus, reflection at a free end occurs without inversion.
Reflection of Longitudinal Waves
In the case of longitudinal waves such as sound waves, reflection depends on whether the boundary is rigid or flexible.
- At a rigid boundary, a compression is reflected as a compression and a rarefaction as a rarefaction.
- At an open boundary, reflection may lead to a change in pressure conditions, depending on the constraints.
Mathematical Representation
If an incident wave travelling along the positive x-direction is represented by
\[y_i(x,t)=a\sin\,(kx-\omega t)\]then the reflected wave travelling along the negative x-direction can be written as
\[y_r(x,t)=a\sin\,(kx+\omega t+\phi)\]where \(\phi\) represents the phase change upon reflection.
- For reflection at a fixed end, \(\phi=\pi\).
- For reflection at a free end, \(\phi=0\).
Important Conclusions
- Reflection changes the direction of wave propagation but not its frequency.
- Reflection at a fixed end causes phase reversal, while reflection at a free end does not.
- Reflection of waves plays a key role in the formation of stationary waves, echoes, and resonance phenomena.
Standing Waves and Normal Modes
When two waves having the same frequency, wavelength, and amplitude travel through a medium in opposite directions, their superposition produces a special wave pattern known as a standing wave or stationary wave. Unlike progressive waves, standing waves do not carry energy from one point to another. Instead, the energy remains confined within the medium.
Standing Waves
Standing waves are formed due to the superposition of two identical progressive waves travelling in opposite directions with the same speed. This situation commonly arises due to the reflection of waves at boundaries, such as in a stretched string fixed at both ends or in air columns.
Mathematical Description
Let the two progressive waves be
\[ y_1(x,t) = a \sin (kx - \omega t) \] \[ y_2(x,t) = a \sin (kx + \omega t) \]According to the principle of superposition, the resultant displacement is
\[ y(x,t) = y_1 + y_2 \]Using trigonometric identities, we obtain
\[ y(x,t) = 2a \sin kx \cos \omega t \]This equation represents a standing wave.
Nodes and Antinodes
From the standing wave equation:
Nodes are points where the displacement is always zero. This occurs when
\[ \sin kx = 0 \Rightarrow kx = n\pi \]or
\[ x = \frac{n\lambda}{2}, \quad n = 0,1,2,\ldots \]Antinodes are points where the displacement is maximum. This occurs when
\[ \sin kx = \pm 1 \Rightarrow kx = \frac{(2n+1)\pi}{2} \]or
\[ x = \frac{(2n+1)\lambda}{4} \]The distance between two consecutive nodes or two consecutive antinodes is \(\frac{\lambda}{2}\), while the distance between a node and the nearest antinode is \(\frac{\lambda}{4}\).
Characteristics of Standing Waves
Nodes and antinodes remain fixed in position. All particles between two adjacent nodes vibrate with the same frequency but with different amplitudes. There is no net transfer of energy along the medium. Standing waves are possible only in bounded media.
Normal Modes of Vibration
A normal mode is a specific pattern of standing wave formed in a system under given boundary conditions. Each normal mode corresponds to a definite frequency, called a normal frequency.
Consider a stretched string of length \(L\) fixed at both ends.
Fundamental Mode (First Harmonic)
In the fundamental mode, the string has nodes at both ends and one antinode at the centre. The wavelength is
\[ \lambda_1 = 2L \]and the frequency is
\[ f_1 = \frac{v}{2L} \]This frequency is called the fundamental frequency.
Higher Normal Modes (Harmonics)
For higher modes of vibration,
\[ \lambda_n = \frac{2L}{n} \] \[ f_n = \frac{nv}{2L}, \quad n = 1,2,3,\ldots \]These frequencies are integral multiples of the fundamental frequency and are known as harmonics.
Physical Significance
Standing waves and normal modes explain the vibrations of strings in musical instruments, formation of sound in air columns, resonance phenomena, and energy distribution in oscillating systems.
Thus, standing waves arise due to the superposition of two identical waves travelling in opposite directions, and the allowed vibration patterns under given boundary conditions are called normal modes.
BEATS
Beats are a phenomenon associated with sound waves and arise due to the superposition of two sound waves having nearly equal but slightly different frequencies, travelling in the same direction. As a result of this superposition, the sound intensity heard at a point varies periodically with time, producing alternate loud and feeble sounds. This periodic variation in loudness is known as beats.
Explanation of Beats
Consider two sound waves of equal amplitude \(a\) and slightly different frequencies \(\nu_1\) and \(\nu_2\), such that \(\nu_1 \approx \nu_2\). Their corresponding angular frequencies are
\[ \omega_1 = 2\pi \nu_1, \qquad \omega_2 = 2\pi \nu_2 \]The equations of the two waves may be written as
\[ y_1 = a \sin (\omega_1 t), \qquad y_2 = a \sin (\omega_2 t) \]According to the principle of superposition, the resultant displacement is
\[ y = y_1 + y_2 \]Using the trigonometric identity
\[ \sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}, \]we obtain
\[ y = 2a \sin \left(\frac{\omega_1 + \omega_2}{2} t\right) \cos \left(\frac{\omega_1 - \omega_2}{2} t\right) \]Physical Interpretation
The resultant wave consists of a rapidly varying sine term, corresponding to the average frequency, and a slowly varying cosine term that acts as an envelope and controls the amplitude of the wave. Since the loudness of sound depends on amplitude, the sound alternates between loud and feeble.
Maximum loudness occurs when
\[ \cos \left(\frac{\omega_1 - \omega_2}{2} t\right) = \pm 1 \]Minimum loudness occurs when
\[ \cos \left(\frac{\omega_1 - \omega_2}{2} t\right) = 0 \]Thus, the periodic rise and fall of sound intensity gives rise to beats.
Beat Frequency
The beat frequency is defined as the number of beats heard per second. It is given by
\[ \nu_b = |\nu_1 - \nu_2| \]Hence, the beat frequency is equal to the absolute difference between the frequencies of the two interfering sound waves.
Conditions for Beats
For beats to be distinctly heard, the frequencies of the two waves must be nearly equal, their amplitudes should be equal or nearly equal, and the waves must reach the observer simultaneously while travelling in the same direction.
Applications of Beats
Beats are commonly used for tuning musical instruments, determining unknown frequencies by comparison with a known source, and for understanding the phenomenon of interference of sound waves.
