## Wave Motion Equation Of A Travelling Or Progressive Wave

A wave, that propagates through a medium in a fixed direction, is called a progressive wave in one dimension. Suppose, due to the propagation of the wave, the panicles in the medium are subjected to forced simple harmonic motions. Obviously, the wave reaches two different panicles at two different times.

- As a result, there arises a time lag, and consequently a phase lag, between the vibrations of two different particles along the direction of wave motion.
- This time lag or phase lag depends on the velocity of the wave. If the wave travels from left to right, each particle starts vibrating later than its predecessor on the left. So, the phase of a panicle lags behind that of any other panicle on its left.

Let the positive direction of the x-axis be chosen along the direction of propagation of a one-dimensional progressive wave. O is the origin.

The particles of the medium vibrate simply harmonically. The displacement of the particle at O at any instant t from its mean position is given by, y = Asinωt

where A is the amplitude of vibration and ω is the angular frequency of the SHM executed by the particle.

If n is the frequency of the particle, ω = 2πn

y = Asin2πnt…(1)

The wave is traveling along the positive x -x-direction.

So, time taken by the wave to reach P at a distance x on the right-hand side of O is \(\frac{x}{V}\) i.e., with respect to time, the point P always lags behind the point O by \(\frac{x}{V}\). If t and t’ are the times at the points O and P, respectively, t’ = t – \(\frac{x}{V}\).

So, displacement of the particle at P is given by,

y = \(A \sin \omega t^{\prime}=A \sin \omega\left(t-\frac{x}{V}\right)=A \sin 2 \pi n\left(t-\frac{x}{V}\right)\)….(2)

This is the equation of a progressive wave traveling in the positive direction of x-axis.

If the wave propagates in the opposite direction, i.e., in the negative x-direction, we put -x in place of +x. So in that case, the equation of the progressive wave will be,

y = \(A \sin \omega\left(t+\frac{x}{V}\right)=A \sin 2 \pi n\left(t+\frac{x}{V}\right)\)…(3)

**From equations (2) and (3) it is seen that the displacement of a vibrating panicle on the path of a progressive wave changes**

- With time and
- With distance.

Each of these changes, with time or with distance, is periodic. Clearly, in each of the cases, the displacement graph of the particle (y-t graph or y-x graph) is a sine graph.

Any of the harmonic functions in sine and cosine forms may be used to express a simple harmonic motion. So we can express equations (1), (2), and (3) by cosine functions.

**In equations (2) and (3), y is the displacement of a particle with respect to its mean position.**

- In the case of a transverse wave, y is perpendicular to the x-axis.
- In the case of a longitudinal wave, y is parallel to the x-axis.

**A Few Alternative Forms Of The Progressive Wave Equation:** In equation (2), the angular function is, \(\theta=\omega\left(t-\frac{x}{V}\right).\)

By expressing θ in different ways, the equation of a progressive wave can be expressed in a few alternative forms

Frequency, \(n=\frac{\omega}{2 \pi} ;\) time period, \(T=\frac{1}{n}=\frac{2 \pi}{\omega}\) wavelength, \(\lambda=\frac{V}{n}\);

Wave number, \(k=\frac{2 \pi}{\lambda}=\frac{2 \pi n}{V}=\frac{\omega}{V}\)

- \(\theta=\omega\left(t-\frac{x}{V}\right)=\omega t-\frac{\omega}{V} x=\omega t-k x\)
- \(\theta=\omega\left(t-\frac{x}{V}\right)=\frac{\omega}{V}(V t-x)=k(V t-x)=\frac{2 \pi}{\lambda}(V t-x)\)
- \(\theta=\omega t-k x=2 \pi\left(\frac{\omega}{2 \pi} t-\frac{k}{2 \pi} x\right)=2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)\)

Using these values of θ, the equation of a progressive wave can be written as

\(\left.\begin{array}{rl}

y & =A \sin \omega\left(t-\frac{x}{V}\right) \\

& =A \sin (\omega t-k x) \\

& =A \sin \frac{2 \pi}{\lambda}(V t-x) \\

& =A \sin 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)

\end{array}\right\}\)…(4)

Any of these alternative forms may be used as per convenience. Remember that if we put -x in place of x, we shall get the equation of a progressive wave moving in the negative direction of the x-axis.

**Phase**: The concept of phase has been discussed in the chapter Simple Harmonic Motion.

The equation of a progressive wave is written as y = Asinθ

where \(\theta=\frac{2 \pi}{\lambda}(V t-x)\)…(5)

If a progressive wave is not damped, the amplitude of vibration A remains constant. Except for amplitude, all other information about the wave is obtained from the angle θ. This angle θ is called the phase angle or phase. If this phase angle is known,

1. From the values of 2 and V we get,

frequency, \(n=\frac{V}{\lambda}\); time period T = \(\frac{1}{n}=\frac{\lambda}{V}\), etc.

2. For a particle in position x, the displacement at an instant t can be determined. Obviously, the phase 6 of the wave depends on x and t. At any instant, the phase changes with distance x. Again, at any point, the phase changes with time t.

**Phase Difference:** The phase difference of two particles at any two positions along the progressive wave at a particular instance of time is actually the difference of the phase angles at the two positions at that instant.

If x_{1 }and x_{2} are the positions of the two particles along the direction of propagation of the wave, the path difference on the wave between the two particles is x_{2}-x_{1}.

From equation (5), at any instant t, the phase of the two particles are, respectively,

⇒ \(\theta_1=\frac{2 \pi}{\lambda}\left(V t-x_1\right) \text { and } \theta_2=\frac{2 \pi}{\lambda}\left(V t-x_2\right)\)

∴ \(\theta_1-\theta_2=\frac{2 \pi}{\lambda}\left(x_2-x_1\right)\)

i.e., phase difference = \(\frac{2 \pi}{\lambda}\) x path difference….(6)

From equation (6) we get,

1. If the path difference of two particles is 0, λ, 2 λ,…, the phase difference becomes 0, 2π, 4π,… In this case, the particles are in the same phase.

2. If the path difference of two particles \(\frac{\lambda}{2}, \frac{3 \lambda}{2}, \frac{5 \lambda}{2}, \ldots\) the phase difference becomes π, 3π, 5π,… In this case, the particles are in opposite phases.

**Cosine form of the equation of a progressive wave:** The equation of a progressive wave can be written using a cosine function instead of the sine function. Then,

y = \(A \cos (\omega t-k x)=A \sin \left(\omega t-k x+\frac{\pi}{2}\right)\)….(7)

Comparing equations (4) and (7), it is evident that the phase difference between these two progressive waves is \(\frac{\pi}{2}\) or 90°.

**Initial Condition:** In equation (4), if x = 0 and t = 0, y = 0, i.e., at the beginning, the displacement of the particle at the origin is zero. Similarly, in equation (7), if x = 0 and t = 0, y = A, i.e., at the beginning, the displacement of the particle at the origin is maximum. Obviously, due to different initial conditions, the phase difference between the two progressive waves given by equations (4) and (7) is 90°.

Generally, the displacement of a particle at t = 0 and x = 0 may have any value between +A and -A. So, the phase of a progressive wave may be different from those in equations (4) or (7). Denoting the initial phase by ø, the general form of a progressive wave can be written as:

\(\left.\begin{array}{rl}

y & =A \sin (\omega t-k x \pm \phi) \\

& =A \sin \left[\omega\left(t-\frac{x}{V}\right) \pm \phi\right] \\

& =A \sin \left[\frac{2 \pi}{\lambda}(V t-x) \pm \phi\right] \\

& =A \sin \left[2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right) \pm \phi\right]

\end{array}\right\}\)…..(8)

According to equation (8), at t = 0 and x = 0, the phase angle of the particle is ±ø. This is called the epoch of the progressive wave at the origin.

**Partial Derivatives:** The equations in this section show that y is a function of two independent variables—x and r. So y has two independent derivatives one with respect to x and the other with respect to t. They are the partial derivatives:

1. \(\frac{\partial y}{\partial x}=\frac{d y}{d x} .\), when t is considered to be a constant = rate of change of y with respect to x when t is a constant.

2. Similarly, \(\frac{\partial y}{\partial t}\) = rate of change of y with respect to t, when x is a constant.

**Particle velocity and acceleration in a progressive wave:** Displacement of a particle in a progressive wave, y = \(A \sin (\omega t-k x \pm \phi)\)……(9)

So, the velocity of the particle,

v = \(\frac{\partial y}{\partial t}=\omega A \cos (\omega t-k x \pm \phi)\)

or, \(\nu =v_0 \cos (\omega t-k x \pm \phi)\)…..(10)

The maximum value of the velocity of the particle = \(\pm \omega A= \pm v_0\).

This v_{0} is called the velocity amplitude of the particle in a progressive wave.

From equation (9), \(\sin (\omega t-k x \pm \phi)=\frac{y}{A}\)

So, \(\cos (\omega t-k x \pm \phi)=\sqrt{1-\frac{y^2}{A^2}}=\frac{1}{A} \sqrt{A^2-y^2}\)

i.e., \(v=\omega A \cdot \frac{1}{A} \sqrt{A^2-y^2}\)

or, v = \(\omega \sqrt{A^2-y^2}\)

The phase difference between a sine function and a cosine function is 90°. So, the phase difference between displacement y and velocity v is also 90°.

From equation (11), it is also seen that if y = 0, v = ωA = v_{0} and again if y = A, v = 0,

i.e., if displacement is zero, velocity is maximum and if displacement is maximum, velocity is zero.

Acceleration of the particle in a progressive wave,

a = \(\frac{\partial^2 y}{\partial t^2}=-\omega^2 A \sin (\omega t-k x \pm \phi)\)

or, a = \(-\omega^2 y\)….(12)

From this equation it is evident that the displacement of the particle y and its acceleration a are in opposite phases, i.e., the phase difference between them is 180°.

Accordingly, the phase difference between velocity and acceleration is 90°.

The maximum acceleration of the particle = \(\pm \omega^2 A= \pm a_0\)

This a_{0} may be called the amplitude of acceleration of the particle in a progressive wave.

**Relation between particle velocity and wave velocity in a progressive wave:**

Displacement of the particle in a progressive wave, y = \(A \sin (\omega t-k x \pm \phi)\)

So, particle velocity, \(\nu=\frac{\partial y}{\partial t}=\omega A \cos (\omega t-k x \pm \phi)\)

Again, \(\frac{\partial y}{\partial x}=-k A \cos (\omega t-k x \pm \phi)\)

So, \(\frac{\partial}{\partial y}=-\frac{\omega}{k x}=-V\)

(As V is the wave velocity, \(k=\frac{\omega}{V}\) )

∴ v = -V \(\frac{\partial y}{\partial x}\)….(13)

This is the relation between particle velocity v and wave velocity V.

In case of a transverse wave, the direction of particle velocity is always perpendicular to that of wave velocity. In the case of a longitudinal wave, the direction of particle velocity is along or opposite to the direction of wave velocity.

**Differential Equation Of A Progressive Wave: **If y = \(A \sin (\omega t-k x \pm \phi)\)

⇒ \(\frac{\partial^2 y}{\partial t^2}=-\omega^2 A \sin (\omega t-k x \pm \phi)\)

and \(\frac{\partial^2 y}{\partial x^2}=-k^2 A \sin (\omega t-k x \pm \phi)\)

So, \(\frac{\frac{\partial^2 y}{\partial x^2}}{\frac{\partial^2 y}{\partial t^2}}=\frac{k^2}{\omega^2}=\frac{1}{V^2}\) (because wave velocity, \(V=\frac{\omega}{k}\))

or, \(\frac{\partial^2 y}{\partial x^2}=\frac{1}{V^2} \frac{\partial^2 y}{\partial t^2}\)…(14)

This equation is the differential equation of a progressive wave. Conversely, if any one-dimensional disturbance satisfies this equation, it is obviously a progressive wave.

**Difference Between Particle Velocity And Wave Velocity:**

- The particles of a medium in a progressive wave do not change their positions due to their velocities. Every particle only vibrates on both sides of its mean position But the progressive wave advances through the medium with its wave velocity.
- The velocities of the particles of the medium change continuously. It is maximum at the mean positions and zero at their extreme positions. However wave velocity remains constant for a particular medium. It depends only on the properties of the medium.
- A progressive wave possesses energy due to the motion of the particles of the medium. This energy propagates with the wave through the medium with the velocity of the wave.
- In a transverse wave, the direction of particle velocity is perpendicular to that of wave velocity. On the other hand, for a longitudinal wave, these two velocities are parallel.

**The velocity of progressive waves in different media:**

1. Velocity of a longitudinal wave in a solid medium, V = \(\sqrt{\frac{Y}{\rho}}\); Y = Young’s modulus of the medium, and ρ = density of the medium.

2. Velocity of a longitudinal wave in a liquid or gaseous medium, V = \(\sqrt{\frac{E}{\rho}}\); E = Bulk modulus of the medium, and ρ = density of the medium.

3. Velocity of sound wave in a gaseous medium, V = \(\sqrt{\frac{\gamma p}{\rho}}\); p = pressure of the gas, and γ = ratio of the two specific heats \(\left(\frac{c_p}{c_w}\right)\) of the gas.

4. Velocity of a transverse wave in a stretched string, V = \(\sqrt{\frac{T}{m}}\); T = tension in the string, and m = mass per unit length of the string.

5. Velocity of a longitudinal wave in a stretched string, \(V=\sqrt{\frac{Y}{\rho}}\); Y = Young’s modulus of the material of the string, and p ~ density of the material of the string.

6. Velocity of an electromagnetic wave, V = \(\frac{1}{\sqrt{\mu \epsilon}} ; \mu\) = permeability and ∈ = permittivity of the medium.

**Characteristics Of Progressive Waves:**

- Progressive wave continuously propagates through a medium, and if not damped, it can propagate to infinity.
- The progressive wave moves with a definite velocity. The wave velocity depends on the elastic properties and on the density of the medium. Energy is transferred with the wave through the medium with the velocity of the wave.
- Each particle of the medium vibrates about its mean position with identical frequency and amplitude. The direction of movement of the particles may be perpendicular (transverse wave) or parallel (longitudinal wave) with respect to the direction of wave motion.
- The velocity with which the phase of a vibrating particle of the medium is transferred to the next particle is called the wave velocity. For the same reason, it is also called the phase velocity. The phase difference between two vibrating particles is proportional to the distance of separation of the two particles along the line of wave propagation.
- The progressive wave carries energy from one point to another without displacing the particles of the medium. Energy is transferred perpendicular to the direction of the wavefronts, i.e., along the direction of the rays.
- Pressure and density in the medium, through which the progressive wave advances, follow the sinusoidal form of variation, like that of displacement, velocity, and acceleration.
- A progressive wave has double periodicity. One is time periodicity determined by the time period (T) of the wave and the other is space periodicity determined by the wavelength (λ).

**Different Properties Of Progressive Waves: **The distinctive properties of progressive waves are explained below

**Absorption:** While moving through any medium, damping of the progressive wave takes place. It means that the energy of the wave gradually decreases with the increase of distance because a part of the energy of the wave is absorbed by the medium as the wave propagates.

**Reflection:** When a progressive wave traveling in a homogeneous medium is incident on an interface with another medium, a part of the incident wave comes back to the first medium. This phenomenon is known as the reflection of a wave. The amount of energy of the incident wave that will be reflected depends on the nature of the interface of the two media.

**Refraction:** When a progressive wave traveling through a homogeneous medium is incident on an interface with another medium, a part of the incident wave is transmitted into the second medium. This phenomenon is known as the refraction of a wave. In refraction, the direction of wave motion generally changes.

**Interference:** Let us consider two progressive waves having the same wavelength and velocity. The phase difference of the two waves is always a constant. When these two waves superpose, the amplitude of the resultant wave increases at some places and decreases at some other places of the medium. This successive increase and decrease of the amplitude of the resultant wave is called interference of waves.

**Diffraction:** When a progressive wave passes through the edge of an opening or of an obstacle, the direction of the wave may change. This is called the diffraction of a wave.

**Scattering:** In the course of propagation, when a progressive wave falls on a material particle, the particle is subject to a forced vibration. So this particle also acts as a secondary source of wave, i.e., weaves propagate in all directions from the vibrating particle. This phenomenon is called the scattering of a wave.

**Polarisation:** During propagation of a transverse wave through a medium, each particle of the medium vibrates on a plane perpendicular to the direction of motion of the wave.

- The plane is called the normal plane. If the vibrations of the particles on such planes arc somehow are restricted to a particular direction, then this phenomenon is called the polarisation of a wave. Obviously, polarization does not take place in case of longitudinal waves.
- It is to be noted that, polarisation does not take place in the case of sound waves. This shows that sound waves are longitudinal. On the other hand, light waves can be polarised by suitable arrangements. So light waves are transverse.

In optics, these properties of waves are discussed in detail. In this chapter, the phenomena of reflection and refraction of sound waves are mainly discussed.