Waves and Oscillations (Waves and their Properties)

What is a Wave?

A wave is a disturbance that travels through a medium or through space, transferring energy from one point to another without causing the permanent transfer of matter along with it.

The disturbance may be produced by the vibration or oscillation of particles of a medium, as in mechanical waves, or by variations in electric and magnetic fields, as in electromagnetic waves.

As a wave travels, the particles of the medium generally oscillate about their mean positions while the disturbance itself propagates from one region to another.

Therefore, the particles of the medium do not move along with the wave over large distances; rather, they transfer energy to neighbouring particles.

For example, when a stone is dropped into still water, a disturbance is produced at the point where the stone strikes the surface.

This disturbance travels outward in the form of circular ripples, carrying energy away from the point of disturbance.

Similarly, when one end of a stretched string is disturbed, the disturbance travels along the string as a wave.

Sound waves are another example in which vibrating particles of the medium transfer energy from one region to another.

Waves may differ in the manner in which the particles oscillate relative to the direction of propagation.

In a transverse wave, the particles of the medium vibrate perpendicular to the direction in which the wave travels.

In a longitudinal wave, the particles of the medium vibrate parallel to the direction of propagation of the wave.

Waves can also be classified according to whether they require a material medium for their propagation.

Mechanical waves require a material medium, whereas electromagnetic waves can propagate even through vacuum.

A wave is characterized by several physical quantities such as amplitude, wavelength, frequency, time period, phase and wave velocity.

The amplitude represents the maximum displacement of a vibrating particle from its mean position.

The wavelength is the distance between two successive particles that are in the same phase of vibration.

The frequency represents the number of complete oscillations made by a particle per second.

Thus, a wave can be understood as the propagation of a disturbance through a medium or space, resulting in the transfer of energy from one place to another without the net transfer of matter.

Basic Terms Associated with Waves

To describe the motion and properties of a wave, several physical quantities are used. The most important quantities are amplitude, wavelength, time period, frequency, wave number, angular frequency, phase and wave velocity.

Amplitude

The amplitude of a wave is the maximum displacement of a particle of the medium from its mean or equilibrium position during the propagation of the wave.

\[ \boxed{A=\text{maximum displacement}} \]

The SI unit of amplitude is metre (m). Amplitude determines the maximum extent of vibration of the particles of the medium. For many waves, the energy carried by the wave is proportional to the square of its amplitude.

Wavelength

The wavelength is the minimum distance between two successive particles of a wave that are vibrating in the same phase.

For a transverse wave, the distance between two successive crests or two successive troughs is one wavelength. For a longitudinal wave, the distance between two successive compressions or two successive rarefactions is one wavelength.

\[ \boxed{\lambda=\text{distance travelled by the wave during one time period}} \]

The SI unit of wavelength is metre (m). Wavelength is represented by the Greek letter \(\lambda\).

Time Period

The time period of a vibrating particle is the time required by it to complete one complete vibration or one complete cycle about its mean position.

\[ \boxed{T=\text{time taken for one complete vibration}} \]

The SI unit of time period is second (s). The time period is represented by \(T\).

The time period and frequency are related by:

\[ \boxed{T=\frac{1}{f}} \]

Frequency

The frequency of a wave is the number of complete vibrations or cycles made by a particle in one second.

\[ \boxed{f=\frac{1}{T}} \]

The SI unit of frequency is hertz (Hz). One hertz represents one complete vibration per second.

\[ 1\,\text{Hz}=1\,\text{s}^{-1} \]

The frequency of a progressive wave is equal to the frequency of vibration of the particles of the medium through which the wave travels.

Wave Number

The wave number is the number of radians of phase change occurring per unit distance along the direction of propagation of a wave.

It is represented by the symbol \(k\) and is related to wavelength by:

\[ \boxed{k=\frac{2\pi}{\lambda}} \]

The SI unit of wave number is radian per metre (rad m^-1). Since the radian is dimensionless, it may also be expressed as m^-1.

Angular Frequency

The angular frequency is the rate of change of phase of a vibrating particle with respect to time. It is represented by the symbol \(\omega\).

One complete vibration corresponds to a phase change of \(2\pi\) radians. Therefore:

\[ \boxed{\omega=2\pi f} \]

Since \(f=1/T\), angular frequency can also be written as:

\[ \boxed{\omega=\frac{2\pi}{T}} \]

The SI unit of angular frequency is radian per second (rad s^-1).

Phase

The phase of a vibrating particle specifies its state of vibration at a particular instant. It indicates the instantaneous position and direction of motion of the particle in its vibration.

For a sinusoidal wave, the phase is generally represented by an angular quantity measured in radians. For example, a progressive wave may be represented as:

\[ y=A\sin(\omega t-kx) \]

In this equation, the quantity \((\omega t-kx)\) represents the phase of the particle at position \(x\) and time \(t\).

Wave Velocity

The wave velocity is the speed with which a particular phase of the wave, such as a crest, trough, compression or rarefaction, travels through the medium.

\[ \boxed{v=\frac{\text{distance travelled by the wave}}{\text{time taken}}} \]

The SI unit of wave velocity is metre per second (m s^-1).

During one complete time period \(T\), the wave travels a distance equal to one wavelength \(\lambda\). Therefore:

\[ v=\frac{\lambda}{T} \]

Since \(f=1/T\):

\[ \boxed{v=f\lambda} \]

This relation connects the wave velocity, frequency and wavelength and is one of the most important relations in wave motion.

Relation Between Angular Frequency, Wave Number and Wave Velocity

The fundamental wave relation is:

\[ v=f\lambda \]

Using:

\[ \omega=2\pi f \]

and:

\[ k=\frac{2\pi}{\lambda} \]

We obtain:

\[ \boxed{v=\frac{\omega}{k}} \]

Thus, wave velocity can be expressed either as \(v=f\lambda\) or as \(v=\omega/k\).

Important Wave Quantities

The most important quantities used to describe a wave are therefore amplitude \(A\), wavelength \(\lambda\), time period \(T\), frequency \(f\), wave number \(k\), angular frequency \(\omega\), phase and wave velocity \(v\).

These quantities are closely related and are used throughout the study of progressive waves, stationary waves and vibrations of stretched strings.

Understanding these basic quantities is essential before studying the mathematical equation of a progressive wave and the formation of stationary waves.

What is a Progressive Wave?

A progressive wave is a wave in which the disturbance travels continuously from one point to another through a medium or through space, transferring energy from one region to another.

As the wave propagates, the particles of the medium undergo oscillations about their respective mean positions.

The disturbance therefore moves forward continuously, while the particles of the medium do not travel along with the wave over a large distance.

The energy associated with the wave is transferred progressively from one particle to the next through the medium.

At any particular instant, different particles of the medium are generally at different stages of their oscillations.

Consequently, the particles at different positions have different displacements, velocities and phases of vibration.

A progressive wave can propagate in a definite direction and carries energy away from the source of the disturbance.

The distance between two successive particles that are in the same phase is equal to one wavelength of the wave.

In a progressive transverse wave, the particles vibrate perpendicular to the direction of propagation of the wave.

In a progressive longitudinal wave, the particles vibrate parallel to the direction of propagation of the wave.

A progressive wave may be represented mathematically by a wave equation that gives the displacement of a particle as a function of its position and time.

For a progressive wave travelling along the positive x-direction, a general form of the wave equation is y = A sin(ωt − kx + φ).

Here, A is the amplitude, ω is the angular frequency, k is the wave number, and φ is the initial phase constant.

The wave velocity represents the speed with which the disturbance or wave profile travels through the medium.

For a periodic progressive wave, the wave velocity is related to its wavelength and frequency by the relation v = fλ.

Examples of progressive waves include waves travelling along a stretched string, sound waves travelling through air and ripples travelling across the surface of water.

Thus, a progressive wave is a travelling disturbance that continuously propagates through a medium or space and transfers energy from one place to another.

Mechanical Waves

Mechanical waves are waves that require a material medium for their propagation. They are produced when the particles of a medium are disturbed from their equilibrium positions and begin to oscillate.

The oscillating particles of the medium transfer the disturbance and its associated energy to neighbouring particles, causing the wave to propagate through the medium.

Mechanical waves cannot propagate through vacuum because the presence of interacting particles of a material medium is essential for the transfer of the disturbance.

The nature and speed of a mechanical wave depend on the mechanical properties of the medium through which it travels.

For example, the speed of a wave travelling along a stretched string depends on the tension and the mass per unit length of the string.

Similarly, the speed of sound in a medium depends on the elastic and inertial properties of that medium.

Mechanical waves can be classified mainly according to the direction in which the particles of the medium vibrate relative to the direction of propagation of the wave.

Types of Mechanical Waves

Mechanical waves are mainly of two types: transverse mechanical waves and longitudinal mechanical waves.

1. Transverse Mechanical Waves

In a transverse mechanical wave, the particles of the medium vibrate perpendicular to the direction in which the wave propagates.

As the wave travels, the particles move alternately above and below their mean positions, producing crests and troughs in the medium.

Waves travelling along a stretched string are common examples of transverse mechanical waves.

2. Longitudinal Mechanical Waves

In a longitudinal mechanical wave, the particles of the medium vibrate parallel to the direction in which the wave propagates.

The particles move alternately closer together and farther apart, producing regions of compression and rarefaction.

Sound waves travelling through air are a common example of longitudinal mechanical waves.

Thus, mechanical waves require a material medium for propagation and can be broadly classified into transverse and longitudinal waves according to the direction of particle vibration relative to the direction of wave propagation.

Non-Mechanical Waves

Non-mechanical waves are electromagnetic waves that do not require a material medium for their propagation and can travel through vacuum. They consist of oscillating electric and magnetic fields that are mutually perpendicular and also perpendicular to the direction of propagation.

Examples: Light waves, radio waves, microwaves, infrared waves, ultraviolet waves, X-rays and gamma rays.

Transverse Progressive Wave

A transverse progressive wave is a progressive wave in which the particles of the medium vibrate perpendicular to the direction of propagation of the wave. The disturbance travels continuously through the medium, transferring energy from one region to another.

As the wave travels, each particle of the medium oscillates about its mean position, while the wave disturbance moves forward. Different particles of the medium are generally at different stages of vibration and therefore have different phases at a given instant.

Properties of a Transverse Progressive Wave

1. Phase: Phase describes the state of vibration of a particle at a particular instant. In a progressive wave, particles at different positions generally have different phases.

2. Amplitude: Amplitude is the maximum displacement of a particle from its mean position during vibration. It is represented by A.

3. Time Period: Time period is the time taken by a particle to complete one full oscillation. It is represented by T and is measured in seconds.

4. Wave Velocity: Wave velocity is the speed with which the disturbance travels through the medium. It is represented by v and is given by v = fλ.

5. Crest: A crest is the point on a transverse wave where the displacement of a particle is maximum in the positive direction. It represents the highest point of the wave.

6. Trough: A trough is the point on a transverse wave where the displacement of a particle is maximum in the negative direction. It represents the lowest point of the wave.

7. Wavelength: Wavelength is the distance between two successive particles of the medium that are in the same phase. In a transverse wave, the distance between two successive crests or two successive troughs is equal to one wavelength. It is represented by λ and is measured in metres.

Figure: Wavelength (λ) is the distance between two successive crests of a transverse wave.

8. Energy: A transverse progressive wave transfers energy from one region of the medium to another without causing a net transfer of matter in the direction of propagation.

9. Examples: Waves travelling along a stretched string are common examples of transverse progressive mechanical waves. Electromagnetic waves such as light are also transverse waves, but they are non-mechanical.

Thus, a transverse progressive wave is characterized by perpendicular particle vibration and continuous propagation of the disturbance, with quantities such as amplitude, phase, time period, velocity and wavelength describing its motion.

Longitudinal Progressive Waves

A longitudinal progressive wave is a progressive wave in which the particles of the medium vibrate to and fro parallel to the direction of propagation of the wave. The disturbance travels continuously through the medium, transferring energy from one region to another.

During the propagation of a longitudinal wave, the particles of the medium form alternate regions of compression and rarefaction. The particles oscillate about their mean positions, while the compressions and rarefactions travel through the medium.

Properties of a Longitudinal Progressive Wave

1. Phase: Phase describes the state of vibration of a particle at a particular instant. In a longitudinal progressive wave, particles at different positions generally have different phases.

2. Amplitude: Amplitude is the maximum displacement of a particle of the medium from its mean position during vibration. It is represented by A.

3. Time Period: Time period is the time taken by a particle of the medium to complete one full oscillation about its mean position. It is represented by T and is measured in seconds.

4. Compressions and Rarefactions: A longitudinal progressive wave consists of alternate compressions and rarefactions that travel continuously through the medium in the direction of wave propagation.

Figure: Alternate compressions and rarefactions in a longitudinal progressive wave.

5. Compression: A compression is a region in a longitudinal wave where the particles of the medium are closer together than their normal separation.

6. Rarefaction: A rarefaction is a region in a longitudinal wave where the particles of the medium are farther apart than their normal separation.

7. Pressure and Density: In a compression, the pressure and density of the medium are greater than their respective normal values. In a rarefaction, the pressure and density are lower than their respective normal values.

8. Wavelength: Wavelength is the distance between two successive compressions or two successive rarefactions. It is represented by λ and is measured in metres.

9. Particle Velocity at the Mean Position: When a particle of the medium passes through its mean position, its particle velocity is maximum.

10. Particle Velocity at the Extreme Position: When a particle reaches either of its extreme positions, its particle velocity becomes zero momentarily before reversing its direction of motion.

Wave Velocity and Particle Velocity

1. Wave Velocity: Wave velocity is the velocity with which a wave disturbance travels through the medium. It is represented by v and is given by:

v = fλ

2. Particle Velocity: Particle velocity is the instantaneous velocity with which a particle of the medium moves about its mean position while the wave passes through it.

Relation Between Wave Velocity, Frequency and Wavelength

The velocity of a progressive wave is related to its frequency and wavelength by a fundamental relation. This relation is applicable to both mechanical and non-mechanical waves.

Wave Velocity and Wavelength

Consider a progressive wave travelling through a medium. Let its wavelength be \(\lambda\) and its time period be \(T\).

During one complete time period, the wave advances through a distance equal to one wavelength.

\[ \text{Distance travelled in one time period}=\lambda \]

The wave velocity is defined as the distance travelled by the wave per unit time.

\[ v=\frac{\text{distance travelled}}{\text{time taken}} \]

Therefore, during one time period:

\[ v=\frac{\lambda}{T} \]

Derivation of the Relation \(v=f\lambda\)

The frequency \(f\) of a wave is the number of complete vibrations made per second. Therefore, frequency is the reciprocal of time period.

\[ f=\frac{1}{T} \]

Hence:

\[ T=\frac{1}{f} \]

Substituting this value of \(T\) in the expression for wave velocity:

\[ v=\frac{\lambda}{T} \]

\[ v=\frac{\lambda}{1/f} \]

Therefore:

\[ \boxed{v=f\lambda} \]

Thus, the velocity of a progressive wave is equal to the product of its frequency and wavelength.

Meaning of the Relation \(v=f\lambda\)

The relation \(v=f\lambda\) shows that the wave velocity depends on the frequency and wavelength of the wave. However, frequency and wavelength are not generally independent quantities for a given wave travelling in a particular medium.

For a particular medium under fixed physical conditions, the wave velocity is determined by the properties of the medium. Therefore, if the frequency of the source is increased, the wavelength changes in such a way that the wave velocity remains unchanged.

For example, if the wave velocity remains constant:

\[ v=f\lambda=\text{constant} \]

Therefore:

\[ \boxed{f\propto\frac{1}{\lambda}} \]

Thus, for a given medium, frequency and wavelength are inversely proportional when the wave velocity is constant.

Relation in Terms of Time Period

Since \(f=1/T\), the fundamental wave relation can also be written as:

\[ v=f\lambda \]

\[ \boxed{v=\frac{\lambda}{T}} \]

This expression means that the wave travels one wavelength \(\lambda\) during one time period \(T\).

Relation with Angular Frequency and Wave Number

The angular frequency is related to frequency by:

\[ \omega=2\pi f \]

The wave number is related to wavelength by:

\[ k=\frac{2\pi}{\lambda} \]

Starting with:

\[ v=f\lambda \]

Substituting \(f=\omega/2\pi\) and \(\lambda=2\pi/k\):

\[ v=\frac{\omega}{2\pi}\frac{2\pi}{k} \]

Therefore:

\[ \boxed{v=\frac{\omega}{k}} \]

SI Units of the Quantities

The wave velocity \(v\) is measured in metre per second (m s^-1), frequency \(f\) is measured in hertz (Hz), and wavelength \(\lambda\) is measured in metre (m).

The dimensional formula of wave velocity is:

\[ \boxed{[v]=[LT^{-1}]} \]

The dimensional formula of frequency is:

\[ \boxed{[f]=[T^{-1}]} \]

The dimensional formula of wavelength is:

\[ \boxed{[\lambda]=[L]} \]

Important Forms of the Wave Relation

The fundamental relation between wave velocity, frequency and wavelength can be expressed in several useful forms:

\[ \boxed{v=f\lambda} \]

\[ \boxed{f=\frac{v}{\lambda}} \]

\[ \boxed{\lambda=\frac{v}{f}} \]

\[ \boxed{v=\frac{\lambda}{T}} \]

\[ \boxed{v=\frac{\omega}{k}} \]

The relation \(v=f\lambda\) is one of the fundamental equations of wave motion and is widely used in the study of progressive waves, sound waves, electromagnetic waves and stationary waves.

Equation for a Progressive Wave

Consider a progressive wave travelling along the positive x-axis with wave velocity v. Let the displacement of a particle at the origin be represented by a simple harmonic function.

Let A be the amplitude of the wave and ω be its angular frequency.

Derivation of the Equation of a Progressive Wave

At the origin, where x = 0, let the particle begin its oscillation from the mean position at time t = 0.

\[ y=A\sin\omega t \]

Now consider another particle at a distance x from the origin. Since the wave travels with velocity v, the disturbance takes a certain time to reach this particle.

\[ t’=\frac{x}{v} \]

Therefore, the particle at distance x starts its oscillation with a time delay of x/v compared with the particle at the origin.

Hence, the time available for the particle at position x to oscillate is:

\[ t-\frac{x}{v} \]

Therefore, the displacement of the particle at position x and at time t is:

\[ y=A\sin\omega\left(t-\frac{x}{v}\right) \]

This is the equation of a progressive wave travelling along the positive x-axis in terms of wave velocity.

Derivation of the Equation in Terms of Wave Number

The equation obtained above can be expanded by multiplying ω into the terms inside the bracket.

\[ y=A\sin\left(\omega t-\frac{\omega x}{v}\right) \]

The wave number k is defined as the angular phase change per unit distance.

\[ k=\frac{2\pi}{\lambda} \]

The angular frequency of the wave is related to its frequency by:

\[ \omega=2\pi f \]

The wave velocity is related to frequency and wavelength by:

\[ v=f\lambda \]

Therefore:

\[ \frac{\omega}{v}=\frac{2\pi f}{f\lambda} \]

\[ \frac{\omega}{v}=\frac{2\pi}{\lambda} \]

Since:

\[ k=\frac{2\pi}{\lambda} \]

Therefore:

\[ \frac{\omega}{v}=k \]

Substituting this relation into the equation obtained earlier:

\[ y=A\sin\left(\omega t-kx\right) \]

Hence, the equation of a progressive wave travelling along the positive x-axis is:

\[ \boxed{y=A\sin(\omega t-kx)} \]

Alternative Form of the Progressive Wave Equation

Since the wave number is related to angular frequency and wave velocity by:

\[ k=\frac{\omega}{v} \]

Substituting this relation in the progressive wave equation:

\[ y=A\sin\left(\omega t-\frac{\omega x}{v}\right) \]

Taking ω common:

\[ \boxed{y=A\sin\omega\left(t-\frac{x}{v}\right)} \]

Thus, the two equations represent the same progressive wave travelling along the positive x-axis.

\[ \boxed{y=A\sin(\omega t-kx)} \]

\[ \boxed{y=A\sin\omega\left(t-\frac{x}{v}\right)} \]

Where: y is the displacement of the particle, A is the amplitude, ω is the angular frequency, t is the time, x is the distance from the origin, v is the wave velocity, λ is the wavelength, and k is the wave number.

For a wave travelling in the positive x-direction, the negative sign occurs between the time-dependent term and the position-dependent term. For a wave travelling in the negative x-direction, the corresponding equation contains a positive sign.

Particle Velocity and Particle Acceleration in a Progressive Wave

When a progressive wave travels through a medium, the particles of the medium oscillate about their mean positions. The particles do not travel along with the wave; instead, they undergo periodic motion while the disturbance propagates through the medium.

The particle velocity is the velocity with which an individual particle of the medium moves about its mean position, whereas the wave velocity is the velocity with which the disturbance or a particular phase of the wave travels through the medium.

Equation of a Progressive Wave

Consider a sinusoidal progressive wave travelling in the positive \(x\)-direction. Its displacement can be written as:

\[ y=A\sin(\omega t-kx) \]

where \(A\) is the amplitude, \(\omega\) is the angular frequency, \(k\) is the wave number, \(x\) is the position of the particle and \(t\) is the time.

Particle Velocity

The particle velocity is the rate of change of displacement of a particle with respect to time at a fixed position.

\[ v_p=\frac{\partial y}{\partial t} \]

Starting with the equation of the progressive wave:

\[ y=A\sin(\omega t-kx) \]

Differentiating with respect to time while keeping \(x\) constant:

\[ v_p=A\omega\cos(\omega t-kx) \]

Therefore, the particle velocity is:

\[ \boxed{v_p=A\omega\cos(\omega t-kx)} \]

Maximum Particle Velocity

The maximum value of \(\cos(\omega t-kx)\) is \(1\). Therefore, the maximum particle velocity is:

\[ \boxed{v_{p,\max}=A\omega} \]

Using \(\omega=2\pi f\):

\[ \boxed{v_{p,\max}=2\pi fA} \]

Thus, the maximum particle velocity depends on the amplitude and frequency of vibration.

Particle Velocity in Terms of Displacement

The displacement of the particle is:

\[ y=A\sin(\omega t-kx) \]

The particle velocity is:

\[ v_p=A\omega\cos(\omega t-kx) \]

Squaring both expressions gives:

\[ y^2=A^2\sin^2(\omega t-kx) \]

\[ v_p^2=A^2\omega^2\cos^2(\omega t-kx) \]

Using \(\sin^2\theta+\cos^2\theta=1\):

\[ \frac{y^2}{A^2}+\frac{v_p^2}{A^2\omega^2}=1 \]

Therefore:

\[ \boxed{v_p^2=\omega^2(A^2-y^2)} \]

Hence:

\[ \boxed{v_p=\pm\omega\sqrt{A^2-y^2}} \]

The positive or negative sign indicates the direction of motion of the particle.

Particle Velocity at the Mean Position

At the mean position, the displacement of the particle is zero.

\[ y=0 \]

Using:

\[ v_p=\pm\omega\sqrt{A^2-y^2} \]

We obtain:

\[ \boxed{|v_p|=\omega A} \]

Thus, the particle velocity has its maximum magnitude at the mean position.

Particle Velocity at the Extreme Position

At an extreme position, the displacement of the particle is equal to the amplitude.

\[ y=\pm A \]

Therefore:

\[ v_p=\pm\omega\sqrt{A^2-A^2} \]

\[ \boxed{v_p=0} \]

Thus, the particle is momentarily at rest at its extreme positions.

Particle Acceleration

The particle acceleration is the rate of change of particle velocity with respect to time at a fixed position.

\[ a_p=\frac{\partial v_p}{\partial t} \]

The particle velocity is:

\[ v_p=A\omega\cos(\omega t-kx) \]

Differentiating with respect to time:

\[ a_p=-A\omega^2\sin(\omega t-kx) \]

Since:

\[ y=A\sin(\omega t-kx) \]

Therefore:

\[ \boxed{a_p=-\omega^2y} \]

This is the standard equation of simple harmonic motion. Therefore, each particle of the medium in a sinusoidal progressive wave performs simple harmonic motion about its mean position.

Maximum Particle Acceleration

The magnitude of particle acceleration is maximum when the magnitude of displacement is maximum.

\[ |y|=A \]

Using:

\[ a_p=-\omega^2y \]

The maximum magnitude of acceleration is:

\[ \boxed{a_{p,\max}=\omega^2A} \]

Using \(\omega=2\pi f\):

\[ \boxed{a_{p,\max}=4\pi^2f^2A} \]

Particle Acceleration at the Mean Position

At the mean position:

\[ y=0 \]

Therefore:

\[ \boxed{a_p=0} \]

Thus, the particle acceleration is zero when the particle passes through its mean position.

Particle Acceleration at the Extreme Positions

At the extreme positions:

\[ y=\pm A \]

Therefore:

\[ a_p=-\omega^2(\pm A) \]

Hence, the magnitude is:

\[ \boxed{|a_p|=\omega^2A} \]

The acceleration is always directed towards the mean position, which is characteristic of simple harmonic motion.

Difference Between Wave Velocity and Particle Velocity

The wave velocity is the velocity with which the wave disturbance travels through the medium, whereas the particle velocity is the velocity with which an individual particle oscillates about its mean position.

For a progressive wave:

\[ \boxed{v=\frac{\omega}{k}} \]

The particle velocity is:

\[ \boxed{v_p=A\omega\cos(\omega t-kx)} \]

Therefore, wave velocity and particle velocity are different physical quantities. The wave velocity is associated with the propagation of the disturbance, while particle velocity is associated with the oscillatory motion of the particles.

Important Results

For a sinusoidal progressive wave represented by:

\[ y=A\sin(\omega t-kx) \]

The particle velocity is:

\[ \boxed{v_p=A\omega\cos(\omega t-kx)} \]

The maximum particle velocity is:

\[ \boxed{v_{p,\max}=A\omega} \]

The particle acceleration is:

\[ \boxed{a_p=-\omega^2y} \]

The maximum particle acceleration is:

\[ \boxed{a_{p,\max}=A\omega^2} \]

At the mean position, particle velocity is maximum and particle acceleration is zero. At the extreme positions, particle velocity is zero and particle acceleration has maximum magnitude.

Thus, although a progressive wave travels through the medium with a definite wave velocity, each particle of the medium executes oscillatory motion about its equilibrium position. Understanding particle velocity and acceleration is essential for studying the energy carried by a wave and the motion of particles in progressive and stationary waves.

Relation Between Phase Difference and Path Difference

The phase difference between two particles of a progressive wave is directly proportional to the path difference between them. The relation between phase difference and path difference is given below.

Let Δφ be the phase difference between two particles separated by a path difference Δx, and let λ be the wavelength of the wave.

A path difference of one wavelength corresponds to a phase difference of 2π radians.

\[ \lambda \longrightarrow 2\pi \]

Therefore, for a path difference of Δx:

\[ \Delta x \longrightarrow \Delta\phi \]

Hence, the phase difference corresponding to a path difference of Δx is:

\[ \boxed{\Delta\phi=\frac{2\pi}{\lambda}\Delta x} \]

Since the wave number is defined as:

\[ k=\frac{2\pi}{\lambda} \]

The relation can also be written as:

\[ \boxed{\Delta\phi=k\Delta x} \]

Thus, the phase difference is directly proportional to the path difference and inversely proportional to the wavelength.

Special cases: When the path difference is λ, the phase difference is 2π radians, or 360°, and the two particles are in the same phase.

When the path difference is λ/2, the phase difference is π radians, or 180°, and the two particles are in opposite phases.

Therefore, the fundamental relation between phase difference and path difference is:

\[ \boxed{\frac{\Delta\phi}{2\pi}=\frac{\Delta x}{\lambda}} \]

Principle of Superposition of Waves

The principle of superposition of waves states that when two or more waves travel simultaneously through the same medium, the resultant displacement of a particle at any instant is equal to the algebraic sum of the displacements produced by the individual waves at that instant.

The waves continue to propagate through the medium independently, while their displacements combine at points where they overlap. This principle is fundamental to the understanding of interference, stationary waves and beats.

Mathematical Statement of the Principle

Suppose two waves produce displacements \(y_1\) and \(y_2\) at the same point of a medium at the same instant.

According to the principle of superposition, the resultant displacement is:

\[ \boxed{y=y_1+y_2} \]

If three waves are present, the resultant displacement becomes:

\[ \boxed{y=y_1+y_2+y_3} \]

For \(n\) waves, the general expression is:

\[ \boxed{y=\sum_{i=1}^{n}y_i} \]

Thus, the resultant displacement is obtained by adding the individual displacements algebraically at the point of superposition.

Superposition of Two Waves in the Same Direction

Consider two sinusoidal waves having the same frequency and wavelength and travelling in the same direction. Let their displacements be:

\[ y_1=A_1\sin(\omega t-kx+\phi_1) \]

\[ y_2=A_2\sin(\omega t-kx+\phi_2) \]

The resultant displacement is:

\[ y=y_1+y_2 \]

Therefore:

\[ y=A_1\sin(\omega t-kx+\phi_1)+A_2\sin(\omega t-kx+\phi_2) \]

The resultant wave is also sinusoidal when the two waves have the same frequency and wave number.

Superposition of Two Waves of Equal Amplitude

Consider two waves having equal amplitudes \(A\), equal angular frequencies and equal wave numbers. Let their phase difference be \(\phi\).

The two waves may be written as:

\[ y_1=A\sin(\omega t-kx) \]

\[ y_2=A\sin(\omega t-kx+\phi) \]

The resultant displacement is:

\[ y=y_1+y_2 \]

Using the trigonometric identity:

\[ \sin C+\sin D=2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right) \]

The resultant displacement becomes:

\[ y=2A\cos\left(\frac{\phi}{2}\right)\sin\left(\omega t-kx+\frac{\phi}{2}\right) \]

Therefore, the amplitude of the resultant wave is:

\[ \boxed{A_R=2A\left|\cos\frac{\phi}{2}\right|} \]

Thus, the resultant amplitude depends on the phase difference between the two waves.

Constructive Interference

When two waves meet in the same phase, their displacements reinforce each other. This phenomenon is called constructive interference.

For constructive interference, the phase difference is:

\[ \boxed{\phi=2n\pi} \]

where \(n=0,\pm1,\pm2,\ldots\).

The resultant amplitude is:

\[ A_R=2A\left|\cos\frac{2n\pi}{2}\right| \]

\[ \boxed{A_R=2A} \]

Thus, when two waves of equal amplitude interfere constructively, the resultant amplitude is twice the amplitude of either wave.

In constructive interference, crest meets crest and trough meets trough, producing a larger resultant displacement.

Destructive Interference

When two waves meet in opposite phases, their displacements oppose each other. This phenomenon is called destructive interference.

For destructive interference, the phase difference is:

\[ \boxed{\phi=(2n+1)\pi} \]

For two waves of equal amplitude, the resultant amplitude becomes:

\[ A_R=2A\left|\cos\frac{(2n+1)\pi}{2}\right| \]

\[ \boxed{A_R=0} \]

Thus, two waves of equal amplitude and opposite phase can completely cancel each other at a point.

In destructive interference, a crest of one wave coincides with a trough of the other wave, reducing the resultant displacement.

Resultant Amplitude for Two Waves of Unequal Amplitudes

If two waves have amplitudes \(A_1\) and \(A_2\), the amplitude of their resultant wave is:

\[ \boxed{A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}} \]

For waves in the same phase, \(\phi=0\):

\[ A_R=A_1+A_2 \]

For waves in opposite phase, \(\phi=\pi\):

\[ \boxed{A_R=|A_1-A_2|} \]

Complete cancellation occurs only when the two interfering waves have equal amplitudes and opposite phases.

Superposition and Stationary Waves

The formation of a stationary wave is a direct application of the principle of superposition. A stationary wave is produced when two waves of the same amplitude, frequency and wavelength travel through the same medium in opposite directions.

Let the two waves be:

\[ y_1=A\sin(\omega t-kx) \]

\[ y_2=A\sin(\omega t+kx) \]

According to the principle of superposition:

\[ y=y_1+y_2 \]

Using the trigonometric identity for the sum of two sine functions:

\[ \boxed{y=2A\sin\omega t\cos kx} \]

The amplitude therefore varies with position, producing fixed nodes and antinodes. This is the mathematical basis of the formation of stationary waves.

Superposition and Beats

Beats are also a consequence of the superposition of two waves having slightly different frequencies. When the waves overlap, their resultant amplitude periodically increases and decreases, producing alternate maxima and minima of sound intensity.

Thus, the principle of superposition provides the foundation for understanding several important wave phenomena, including interference, stationary waves and beats.

Important Points

The resultant displacement produced by superposition is the algebraic sum of the individual displacements.

Constructive interference occurs when waves meet in the same phase, producing maximum resultant amplitude.

Destructive interference occurs when waves meet in opposite phases, producing minimum resultant amplitude.

For two waves of equal amplitude \(A\), the maximum resultant amplitude is \(2A\), while the minimum resultant amplitude is zero.

The principle of superposition is responsible for the formation of stationary waves when two identical waves travel in opposite directions.

Therefore, the principle of superposition is one of the fundamental principles of wave motion and forms the basis for understanding interference, stationary waves and beats.

Reflection of Waves at a Boundary

When a wave travelling through a medium reaches the boundary of another medium or a fixed end, it may return into the original medium. This phenomenon is called reflection of a wave.

During reflection, the reflected wave travels in the direction opposite to that of the incident wave. Depending on the nature of the boundary, the reflected wave may or may not undergo a phase change.

Incident Wave and Reflected Wave

The wave travelling towards the boundary is called the incident wave, while the wave returning from the boundary is called the reflected wave.

The behaviour of the reflected wave depends on whether the boundary is fixed or free.

Reflection at a Fixed Boundary

Consider a transverse wave travelling along a string whose end is fixed to a rigid support. When the incident wave reaches the fixed end, the end cannot move. Therefore, the displacement of the string at the fixed end must always be zero.

The reflected wave is therefore inverted relative to the incident wave. A crest is reflected as a trough, and a trough is reflected as a crest.

Thus, reflection at a fixed boundary produces an inversion of the transverse displacement.

Phase Change at a Fixed Boundary

When a transverse wave is reflected from a fixed boundary, the reflected wave undergoes a phase change of \(\pi\) radians, or \(180^\circ\).

\[ \boxed{\Delta\phi=\pi} \]

A phase change of \(\pi\) radians means that a crest becomes a trough and a trough becomes a crest upon reflection.

The reflected wave is therefore said to be inverted at a fixed boundary.

Equation of Reflection at a Fixed Boundary

Suppose the incident wave travelling in the positive \(x\)-direction is represented by:

\[ y_i=A\sin(\omega t-kx) \]

For a reflected wave travelling in the negative \(x\)-direction, the general form is:

\[ y_r=A\sin(\omega t+kx+\phi) \]

At a fixed boundary, let the boundary be at \(x=0\). The displacement at the boundary must be zero at every instant.

\[ y_i+y_r=0 \]

This condition requires a phase difference of \(\pi\) between the incident and reflected waves.

\[ \boxed{\phi=\pi} \]

Hence, the reflected wave is inverted relative to the incident wave.

Reflection at a Free Boundary

Now consider a string whose end is free to move vertically. When a transverse wave reaches the free end, the end can move freely. The reflected wave is not inverted.

Therefore, a crest is reflected as a crest and a trough is reflected as a trough.

Thus, reflection at a free boundary occurs without inversion.

Phase Change at a Free Boundary

At a free boundary, the reflected wave does not undergo a phase reversal.

\[ \boxed{\Delta\phi=0} \]

Therefore, the incident and reflected waves have no phase change due to reflection at the free end.

Comparison of Reflection at Fixed and Free Boundaries

At a fixed boundary, the reflected wave is inverted and undergoes a phase change of \(\pi\) radians.

\[ \boxed{\Delta\phi=\pi} \]

At a free boundary, the reflected wave is not inverted and undergoes no phase change.

\[ \boxed{\Delta\phi=0} \]

This distinction is particularly important in understanding the formation of stationary waves.

Reflection and Formation of Stationary Waves

When a wave travelling along a stretched string reaches a fixed end, it is reflected with a phase reversal. The incident and reflected waves then travel in opposite directions along the same string.

When these waves have the same frequency, wavelength and amplitude, their superposition produces a stationary wave.

The resultant displacement can be represented by:

\[ \boxed{y=2A\sin\omega t\cos kx} \]

The resulting pattern contains fixed points called nodes and points of maximum amplitude called antinodes.

Important Characteristics of Wave Reflection

The wave returning from a boundary is called the reflected wave.

At a fixed boundary, the reflected transverse wave is inverted and undergoes a phase change of \(\pi\) radians.

At a free boundary, the reflected transverse wave is not inverted and undergoes no phase change.

The frequency of the reflected wave remains equal to the frequency of the incident wave.

The wavelength of the reflected wave remains equal to that of the incident wave when both waves travel in the same medium.

The direction of propagation of the reflected wave is opposite to that of the incident wave.

Reflection of waves is fundamental to the formation of stationary waves because the reflected wave travels opposite to the incident wave and can superpose with it to produce a fixed pattern of nodes and antinodes.

Energy of a Wave

A progressive wave carries energy from one point to another without causing a net transfer of matter. The energy associated with a wave is due to the oscillatory motion of the particles of the medium.

Consider a progressive wave travelling along the positive x-axis whose displacement is given by:

\[ y=A\sin(\omega t-kx) \]

Here, A is the amplitude, ω is the angular frequency, k is the wave number, x is the position of the particle and t is the time.

Derivation of Energy Density of a Wave

The particle velocity is obtained by differentiating the displacement with respect to time.

\[ V_p=\frac{dy}{dt} \]

Differentiating the wave equation with respect to time:

\[ V_p=A\omega\cos(\omega t-kx) \]

Consider a small volume V of the medium having density ρ. The mass of this volume is:

\[ m=\rho V \]

The kinetic energy of the particles contained in this volume is:

\[ K.E.=\frac{1}{2}mV_p^2 \]

Substituting \(m=\rho V\):

\[ K.E.=\frac{1}{2}\rho V V_p^2 \]

Substituting the expression for particle velocity:

\[ K.E.=\frac{1}{2}\rho V\left[A\omega\cos(\omega t-kx)\right]^2 \]

Therefore:

\[ \boxed{K.E.=\frac{1}{2}\rho V A^2\omega^2\cos^2(\omega t-kx)} \]

Energy Density of a Wave

Kinetic energy per unit volume of the medium is called the kinetic energy density.

\[ U=\frac{K.E.}{V} \]

Substituting the expression for kinetic energy:

\[ U=\frac{1}{V}\left[\frac{1}{2}\rho V A^2\omega^2\cos^2(\omega t-kx)\right] \]

Therefore:

\[ \boxed{U=\frac{1}{2}\rho A^2\omega^2\cos^2(\omega t-kx)} \]

The maximum value of \(\cos^2(\omega t-kx)\) is 1. Therefore, the maximum kinetic energy density is:

\[ U_{\max}=\frac{1}{2}\rho A^2\omega^2 \]

\[ \boxed{U_{\max}=\frac{1}{2}\rho A^2\omega^2} \]

Energy Associated with a Small Volume of the Medium

Consider a small element of the medium having cross-sectional area S and thickness Δx. Its volume is:

\[ \Delta V=S\Delta x \]

The energy associated with this small volume is obtained by multiplying the energy density by the volume.

\[ \Delta E=U\Delta V \]

For the maximum energy density:

\[ \Delta E=U_{\max}\Delta V \]

Substituting the expressions for \(U_{\max}\) and \(\Delta V\):

\[ \Delta E=\left(\frac{1}{2}\rho A^2\omega^2\right)(S\Delta x) \]

Therefore:

\[ \boxed{\Delta E=\frac{1}{2}\rho A^2\omega^2S\Delta x} \]

Thus, the energy associated with a small volume \(S\Delta x\) of the medium is directly proportional to the density of the medium, the square of the amplitude, the square of the angular frequency, the cross-sectional area and the length of the considered element.

Power of a Wave

The power of a wave is the rate at which the wave transfers energy through a given cross-sectional area of the medium. For a progressive wave, the average power transmitted depends on the density of the medium, amplitude, angular frequency, wave velocity and cross-sectional area.

Derivation of the Expression for Power of a Wave

Consider a progressive wave travelling through a medium with wave velocity v. Let ρ be the density of the medium, A be the amplitude of the wave and ω be its angular frequency.

The maximum kinetic energy density of the wave is:

\[ U_{\max}=\frac{1}{2}\rho A^2\omega^2 \]

For a sinusoidal progressive wave, the average kinetic energy density is:

\[ U_K=\frac{1}{4}\rho A^2\omega^2 \]

The average potential energy density is equal to the average kinetic energy density:

\[ U_P=U_K \]

Therefore, the total average energy density of the wave is:

\[ U=U_K+U_P \]

Substituting the average kinetic and potential energy densities:

\[ U=\frac{1}{4}\rho A^2\omega^2+\frac{1}{4}\rho A^2\omega^2 \]

Therefore:

\[ U=\frac{1}{2}\rho A^2\omega^2 \]

Now consider a cross-sectional area S perpendicular to the direction of wave propagation.

In a time interval Δt, the wave travels a distance:

\[ \Delta x=v\Delta t \]

Therefore, the volume of the medium through which the wave travels during this time is:

\[ \Delta V=S\Delta x \]

Substituting \(\Delta x=v\Delta t\):

\[ \Delta V=Sv\Delta t \]

The energy transmitted through this volume is equal to the product of energy density and volume:

\[ \Delta E=U\Delta V \]

Substituting the expressions for energy density and volume:

\[ \Delta E=\left(\frac{1}{2}\rho A^2\omega^2\right)(Sv\Delta t) \]

Therefore:

\[ \Delta E=\frac{1}{2}\rho A^2\omega^2vS\Delta t \]

Power is defined as the rate of transmission of energy:

\[ P=\frac{\Delta E}{\Delta t} \]

Substituting the expression for \(\Delta E\):

\[ P=\frac{\frac{1}{2}\rho A^2\omega^2vS\Delta t}{\Delta t} \]

Cancelling \(\Delta t\):

\[ \boxed{P=\frac{1}{2}\rho A^2\omega^2vS} \]

Hence, the average power transmitted by a progressive wave through a cross-sectional area S is:

\[ \boxed{P=\frac{1}{2}\rho A^2\omega^2vS} \]

Thus, the average power carried by a wave is directly proportional to the density of the medium, the square of the amplitude, the square of the angular frequency, the wave velocity and the cross-sectional area through which the wave propagates.

Intensity of a Wave

The intensity of a wave is defined as the amount of energy transmitted by the wave per unit area of cross-section per unit time. It is equal to the average power transmitted by the wave per unit area perpendicular to the direction of propagation.

If P is the average power transmitted through a cross-sectional area S, then the intensity of the wave is:

\[ I=\frac{P}{S} \]

From the expression for the average power of a progressive wave:

\[ P=\frac{1}{2}\rho A^2\omega^2vS \]

Substituting this expression for power in the intensity equation:

\[ I=\frac{\frac{1}{2}\rho A^2\omega^2vS}{S} \]

Cancelling the cross-sectional area S:

\[ \boxed{I=\frac{1}{2}\rho A^2\omega^2v} \]

Thus, the intensity of a progressive wave is directly proportional to the density of the medium, the square of the amplitude, the square of the angular frequency and the wave velocity.

Intensity of a Wave from a Point Source

Consider a point source emitting waves uniformly in all directions. At a distance r from the source, the wave energy is distributed over the surface of a sphere.

The area of a spherical surface of radius r is:

\[ S=4\pi r^2 \]

The intensity at a distance r from the source is the power transmitted by the source divided by the area of the spherical surface.

\[ I=\frac{P}{S} \]

Substituting the area of the spherical surface:

\[ I=\frac{P}{4\pi r^2} \]

Since the power P and \(4\pi\) are constant for a given source:

\[ \boxed{I\propto\frac{1}{r^2}} \]

Therefore, the intensity of a wave from a point source is inversely proportional to the square of the distance from the source.

Dependence of Intensity on Amplitude

From the expression for intensity:

\[ I=\frac{1}{2}\rho A^2\omega^2v \]

For a given medium and a wave of constant frequency, the quantities \(\rho\), \(\omega\) and \(v\) remain constant.

\[ \boxed{I\propto A^2} \]

Thus, the intensity of a wave is directly proportional to the square of its amplitude. If the amplitude is doubled, the intensity becomes four times the original intensity.

SI Unit of Intensity

Since intensity is power transmitted per unit area, its SI unit is watt per square metre.

\[ \boxed{\mathrm{SI\ unit\ of\ intensity}=W\,m^{-2}} \]

Therefore, the intensity of a wave represents the rate of energy transmission per unit area and, for a uniformly radiating point source, decreases inversely as the square of the distance from the source.

Speeds of a Transverse Wave in Different Cases and Their Derivations

The speed of a transverse wave travelling along a stretched string depends mainly on the tension in the string and its linear mass density. Different forms of the wave-speed equation can be obtained depending on the physical quantities involved.

Speed of a Transverse Wave on a Stretched String

Consider a string of length l and mass m, stretched between two rigid supports. Let the tension in the string be T.

The linear mass density of the string is defined as the mass per unit length of the string.

\[ \mu=\frac{m}{l} \]

The speed of a transverse wave travelling along the stretched string is:

\[ \boxed{v=\sqrt{\frac{T}{\mu}}} \]

Thus, the speed of a transverse wave increases with the tension in the string and decreases with its linear mass density.

\[ v\propto\sqrt{T} \]

\[ v\propto\frac{1}{\sqrt{\mu}} \]

Speed in Terms of Density, Cross-Sectional Area and Stress

Let S be the cross-sectional area of the string and ρ be the density of the material of the string.

The volume of the string is:

\[ V=Al \]

Therefore, the mass of the string is:

\[ m=\rho V \]

Substituting the expression for volume:

\[ m=\rho Al \]

The linear mass density is:

\[ \mu=\frac{m}{l} \]

Substituting \(m=\rho Al\):

\[ \mu=\frac{\rho Al}{l} \]

Therefore:

\[ \boxed{\mu=\rho A} \]

The speed of the transverse wave is:

\[ v=\sqrt{\frac{T}{\mu}} \]

Substituting \(\mu=\rho A\):

\[ v=\sqrt{\frac{T}{\rho A}} \]

Stress is defined as the restoring force acting per unit cross-sectional area.

\[ S=\frac{T}{A} \]

Therefore:

\[ T=SA \]

Substituting \(T=SA\) in the wave-speed equation:

\[ v=\sqrt{\frac{SA}{\rho A}} \]

Cancelling the cross-sectional area:

\[ \boxed{v=\sqrt{\frac{S}{\rho}}} \]

Thus, the speed of a transverse wave can also be expressed as the square root of the ratio of stress to density.

Speed When a String is Stretched by a Suspended Load

Suppose a load of mass M is suspended from the string so that the tension in the string is produced by the weight of the load.

The tension in the string is:

\[ T=Mg \]

The speed of the transverse wave is:

\[ v=\sqrt{\frac{T}{\mu}} \]

Substituting \(T=Mg\):

\[ \boxed{v=\sqrt{\frac{Mg}{\mu}}} \]

Therefore, for a given string, the wave speed increases as the square root of the suspended load.

\[ v\propto\sqrt{M} \]

Speed When the Load is Immersed in a Liquid

Now suppose the suspended body of mass M is completely immersed in a liquid of density ρ_L. Let the density of the body be ρ_b.

When the body is immersed in the liquid, it experiences an upward buoyant force. Therefore, the effective tension in the string is less than the weight of the body.

The weight of the body is:

\[ Mg \]

The buoyant force acting on the immersed body is equal to the weight of the liquid displaced by the body.

\[ F_b=\rho_L Vg \]

Since the mass of the body is related to its density and volume:

\[ M=\rho_bV \]

Therefore, the volume of the body is:

\[ V=\frac{M}{\rho_b} \]

Substituting this volume in the expression for buoyant force:

\[ F_b=\rho_L\frac{M}{\rho_b}g \]

The effective tension in the string is equal to the weight of the body minus the buoyant force.

\[ T=Mg-F_b \]

Substituting the expression for buoyant force:

\[ T=Mg-\rho_L\frac{M}{\rho_b}g \]

Taking \(Mg\) common:

\[ \boxed{T=Mg\left(1-\frac{\rho_L}{\rho_b}\right)} \]

The speed of the transverse wave is:

\[ v=\sqrt{\frac{T}{\mu}} \]

Substituting the effective tension:

\[ v=\sqrt{\frac{Mg\left(1-\frac{\rho_L}{\rho_b}\right)}{\mu}} \]

\[ \boxed{v=\sqrt{\frac{Mg\left(1-\frac{\rho_L}{\rho_b}\right)}{\mu}}} \]

Thus, when the load is immersed in a liquid, the wave speed decreases because the buoyant force reduces the effective tension in the string.

Determination of Relative Density from Wave Velocities in Air and Water

Let v₁ be the speed of the transverse wave when the load is suspended in air and v₂ be the speed when the same load is completely immersed in water.

Let ρ_b be the density of the body and ρ_w be the density of water.

When the load is suspended in air, neglecting the buoyancy of air, the tension is:

\[ T_1=Mg \]

Therefore, the wave speed in air is:

\[ v_1=\sqrt{\frac{Mg}{\mu}} \]

When the load is immersed in water, the effective tension is:

\[ T_2=Mg\left(1-\frac{\rho_w}{\rho_b}\right) \]

Therefore, the wave speed in water is:

\[ v_2=\sqrt{\frac{Mg\left(1-\frac{\rho_w}{\rho_b}\right)}{\mu}} \]

Taking the ratio of the two wave velocities:

\[ \frac{v_1}{v_2} = \sqrt{\frac{1}{1-\frac{\rho_w}{\rho_b}}} \]

Squaring both sides:

\[ \frac{v_1^2}{v_2^2} = \frac{1}{1-\frac{\rho_w}{\rho_b}} \]

Taking the reciprocal:

\[ \frac{v_2^2}{v_1^2} = 1-\frac{\rho_w}{\rho_b} \]

Rearranging:

\[ \frac{\rho_w}{\rho_b} = 1-\frac{v_2^2}{v_1^2} \]

Taking the reciprocal relation:

\[ \frac{\rho_b}{\rho_w} = \frac{1}{1-\frac{v_2^2}{v_1^2}} \]

Multiplying the numerator and denominator by \(v_1^2\):

\[ \frac{\rho_b}{\rho_w} = \frac{v_1^2}{v_1^2-v_2^2} \]

The relative density of the material is defined as the ratio of the density of the material to the density of water.

\[ d=\frac{\rho_b}{\rho_w} \]

Therefore, the relative density of the material is:

\[ \boxed{d=\frac{v_1^2}{v_1^2-v_2^2}} \]

Thus, the relative density of the material of the suspended body can be determined by measuring the transverse-wave velocity in air and in water.

Important: The above immersed-load derivation assumes that the body is completely immersed, the string remains taut, the buoyant force is the only significant additional force, and the effect of air buoyancy is negligible.