DISPLACEMENT RELATION IN A PROGRESSIVE WAVE -

THE GENERAL EQUATION OF WAVE MOTION(Progressive wave)

To represent a wave mathematically we need a physical quantity (Y) whose value oscillate in space and time about its equilibrium value. i.e we need a function of both position x and time t. Such a function at every instant should give the shape of the wave at that instant. Also at every given location, it should describe the motion of the constituent of the medium at that location.

For string waves, Y is the displacement. For a sound wave, Y is the displacement, pressure and density. For an electromagnetic wave, Y is the electric and magnetic field.

NOTE:- The oscillation may or may not be necessary in SHM.

Let the function f(t) represent the displacement y of the particle at x=0,

$\fn_cm \large i.e\;\;\;y(x=0,t)=f(t)$

Let $\fn_cm \large v$ is the speed of wave going in +ve x-direction.

i.e the displacement y of the particle produce at x=0, at time t reaches the point x at time $\fn_cm \large (t+\frac{x}{v})$

Similarly, the displacement of the particle at point x at time t was originated at x=0 at the time $\fn_cm \large (t-\frac{x}{v})$

$\fn_cm \large i.e\;\;\;y(x,t)=y(x=0, t-\frac{x}{v})$

$\fn_cm \large \left [ y=f(t-\frac{x}{v}) \right ]\;\;(progressive \;wave)$

This equation represented a wave travelling in the +ve x-direction with constant speed v.

If wave travels in the -ve x-axis, its equation become $\fn_cm \large \left [ y=f(t+\frac{x}{v}) \right ]\;\;\;or\;\;\;y=f(\frac{vt+x}{v})\;\;\;or\;\;\;\left [ y=f(x+vt) \right ]$

In general, wave equation can be written as $\fn_cm \large \left [ y=f(ax\pm bt) \right ]$

NOTE

The function $\fn_cm \large y=f(ax\pm bt)$ is not necessary to represent a wave function, because it is necessary for a wave function representing a travelling wave to have a finite (not undefined) value for all values of x and t. For example, $\fn_cm \large y=f(x-vt)^2$ is not represent a wave function. If the condition of the travelling wave is satisfied, then the speed of the wave (v) is given by

$\fn_cm \large \left [ v=\frac{cofficient\;of\; t}{cofficient\;of\; x} \right ]$

For example:-

(a) $\fn_cm \large y=(x-vt)^2$

if $\fn_cm \large (x-vt)\rightarrow \infty \;then\;y\rightarrow \infty$

i.e this will not represent a travelling wave.

(b) $\fn_cm \large y=\log \left ( \frac{x+vt}{x_0} \right )$

We know that the log function is not defined for the -ve value of (x+vt). i.e this will not represent a travelling wave.

(c) $\fn_cm \large y=\frac{10}{2+(2x+t)^2}$

if (2x+t)→0, then y=5 (finite) and if (2x+t)→∞, then y→0 (finite)

i.e for any value of (2x+t), y have a finite value, i.e this will represent a travelling wave.

(d) $\fn_cm \large y=A\sin(\omega t-kx)$

For any value of (ωt-kx), y lies between -1 to +1. i.e this will represent a travelling wave.

(e) $\fn_cm \large y=e^{-(x-vt)^2}$

For x-vt=0, y=0 . for x-vt→∞, y→0 . for x-vt→-∞, y→0. i.e for any value of (x-vt), y have a finite value.

i.e this function will represent a travelling wave.

Q.1. You have learnt that a travelling wave in one dimension is represented by a function y = f (x, t) where x and t must appear in the combination x – v t or x + v t, i.e. y = f (x ± v t). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave : (a) $\fn_cm \small (x-vt)^2$ (b) log [(x + vt)/ $\fn_cm \small x_0$ ] (c) 1/(x + vt)

SINE WAVE OF PROGRESSIVE WAVE (Sinusoidal wave)

Let the particle at x=0 oscillate in SHM with amplitude A and angular frequency $\fn_cm \large \omega$ . Then the equation of particle at x=0 is given by

$\fn_cm \large f(t)=A\sin \omega t$

But the equation of particle at x is given by

$\fn_cm \large y=f(t-\frac{x}{v})$

$\fn_cm \large \Rightarrow y=A\sin \omega\left ( t-\frac{x}{v} \right )$

$\fn_cm \large \Rightarrow y=A\sin \left ( \omega t-\frac{\omega}{v}x \right )$

$\fn_cm \large \Rightarrow y=A\sin \left (\omega t-kx \right )\;\;\;\left [ k=\frac{\omega}{v} \right ]$

Here $\fn_cm \large k\rightarrow$ wave number

$\fn_cm \large k=\frac{2\pi}{\lambda}=\frac{2\pi f}{v}=\frac{\omega}{v}$

$\fn_cm \large \left [ y=A\sin\left ( \frac{2\pi}{T}t -\frac{2\pi}{\lambda}x \right ) \right ]$

For a general choice of the origin of time, we will have to add a phase constant(initial phase $\fn_cm \large \phi$ , so the equation will be

$\fn_cm \large \left [ y=A\sin \left (\omega t-kx + \phi\right ) \right ]$

WAVE TERMINOLOGIES

The following figure shows the shape of the string as time passes

To see how a wave travels, we can fix attention on a crest (×) and see how it progresses with time.

TIME PERIOD(T):-

The Time Period of a wave is the time taken by the wave to complete one cycle.

AMPLITUDE(A):-

The amplitude of a wave is the maximum displacement of any particle in the medium from its equilibrium position

FREQUENCY(f or ν):-

The number of vibrations made per second by any particles of the medium is called the frequency of the wave. The frequency of a wave is a characteristic or unique property of the source and hence it only changes when the frequency of the source changes irrespective of the medium through which the wave travels. $\fn_cm \large f=\frac{1}{T}$

Each particle of the wave vibrates in SHM with the same amplitude and frequency. The phases of the particles are, however different.

PHASE:-

The state(position, direction) of a particle of a medium with respect to its mean position is called phase. From fig point p, q and r on wave vibrate on different phases. The quantity $\fn_cm \large (kx-\omega t+ \phi)$ is the phase of the wave. Clearly $\fn_cm \large \phi$ is the phase at x=0 and t=0. Hence $\fn_cm \large \phi$ is called the initial phase or initial phase angle.

WAVELENGTH(λ):–

The distance between two consecutive particles have the same phase or the distance between two consecutive crests or trough is known as wavelength.

Suppose the particle at x and x+L vibrate in the same phase.

$\fn_cm \large i.e\;\;\;A\sin(\omega t -kx)= A \sin(\omega t- k(x+L))$

$\fn_cm \large \Rightarrow \;\;A\sin(\omega t -kx)= A \sin(\omega t- kx+kL)$

$\fn_cm \large \Rightarrow \omega t-kx=\omega t-kx-kL+2n\pi$ Where n is an integer.

$\fn_cm \large \Rightarrow kL=2n\pi\;\;\;\;\;\Rightarrow L=\frac{2n\pi}{k}$

The minimum separation between the particle vibrating in the same phase is obtained by putting n=1.

Thus the wavelength is

$\fn_cm \large \left [ \lambda=\frac{2\pi}{k} \right ]$

$\fn_cm \large \lambda=\frac{2\pi v}{\omega}$

$\fn_cm \large \left [ \lambda=vT \right ]$

$\fn_cm \large v=\frac{\lambda}{T}\;\;\;or\;\;\;[v=f\lambda]$

ANGULAR WAVE NUMBER(k):-

We know that $\fn_cm \large \left [ k=\frac{2\pi}{\lambda} \right ]$ where k is known as angular wavenumber. It is defined as the number of radians per unit distance. Its S.I unit is (rad/m)

NOTE:-

1. Wave velocity of wave and particle velocity of the wave is different. Wave velocity in a medium is generally constant but particle velocity is changed.

2. $\fn_cm \large \frac{1}{\lambda}\rightarrow$ Wave-number(number of wavelengths per unit distance)

$\fn_cm \large \frac{1}{T}\rightarrow$ Frequency (Number of vibrations per second)

And when this term is expressed in the form of an angle, then this term will be written as in angular.

i.e $\fn_cm \large \frac{2\pi}{\lambda}\rightarrow\; Angular\;wave\;number\; and\; \frac{2\pi}{T}\rightarrow Angular\;frequency$

Q.1 If $\fn_cm \large y=(5 mm)\sin[(1 cm^{-1})x-(60 s^{-1})t]$ Find (a) The amplitude (b) the wavenumber (c) the wavelength (d) the frequency (e) the time period and (f) the wave velocity.

Q.2. A wave travelling along a string is described by, y(x, t) = 0.005 sin (80.0 x – 3.0 t), in which the numerical constants are in SI units (0.005 m, 80.0 rad/m, and 3.0 rad/s). Calculate (a) the amplitude, (b) the wavelength, and (c) the period and frequency of the wave. Also, calculate the displacement y of the wave at a distance x = 30.0 cm and time t = 20 s?

NOTE

1. $\fn_cm \large if \;x=0,t=0,\phi=\frac{\pi}{2}\;(i.e\;at\;extream\;position)$

Then equation becomes

$\fn_cm \large y=A\sin(\omega t-kx+\frac{\pi}{2})$

$\fn_cm \large \left [ y=A\cos(\omega t-kx) \right ]$

2. Particle velocity of the medium

$\fn_cm \large v=\frac{\mathrm{d} x}{\mathrm{d} t}$

$\fn_cm \large v=\frac{\mathrm{d} [A\sin(\omega t -kx)]}{\mathrm{d} t}$

$\fn_cm \large \Rightarrow v=A\omega \cos(\omega t-kx)$

$\fn_cm \large \left [ v_{max}=A\omega \right ]$

3. Similarly particle acceleration of medium

$\fn_cm \large a=-A\omega^2 \sin(\omega t-kx)$

$\fn_cm \large \left [ a_{max}=A\omega^2 \right ]$

VARIATION OF PHASE WITH DISTANCE

At a given time phase changes periodically with distance x. Let at a given time $\fn_cm \large \phi_1$ and $\fn_cm \large \phi_2$ be the phases of the two particles at distance $\fn_cm \large x_1$ and $\fn_cm \large x_2$ from the origin respectively.

Then, $\fn_cm \large \phi_1=\frac{2\pi}{T}t-\frac{2\pi}{\lambda}x_1$

$\fn_cm \large \phi_2=\frac{2\pi}{T}t-\frac{2\pi}{\lambda}x_2$

$\fn_cm \large \therefore \phi_2 - \phi_1=-\frac{2\pi}{\lambda}x_2 + \frac{2\pi}{\lambda}x_1$

$\fn_cm \large \Rightarrow \phi_2-\phi_1=-\frac{2\pi}{\lambda}(x_2-x_1)$

$\fn_cm \large \left [ \Delta \phi=-\frac{2\pi}{\lambda}\Delta x \right ]$

$\fn_cm \large Where\;\;\;\;\;\Delta \phi\rightarrow Phase\;difference\;\;\;\;and \;\;\;\Delta x\rightarrow Path \;difference$

-ve sign indicates the phase lag. If $\fn_cm \large \Delta x=\lambda\;\;\;\;\;then\;\;\;\;\;\left [ \Delta \phi=-2\pi \right ]$

VARIATION OF PHASE WITH TIME

At a given position the phase changes periodically with time. Let $\fn_cm \large \phi_1$ and $\fn_cm \large \phi_2$ be the phases of a particle at times $\fn_cm \large t_1$ and $\fn_cm \large t_2$ respectively, then

$\fn_cm \large \phi_1=\frac{2\pi}{T}t_1-\frac{2\pi}{\lambda}x$

and $\fn_cm \large \phi_1=\frac{2\pi}{T}t_2-\frac{2\pi}{\lambda}x$

$\fn_cm \large \therefore \left [ \Delta \phi=\frac{2\pi}{T} \Delta t \right ]$

$\fn_cm \large if\;\Delta t=T\;then\;\Delta \phi=2\pi$

TWO GRAPHS IN SINE WAVE (An Example)

We know that from the previous page (15.2) “We can look at a wave in two ways. We can fix an instant of time and picture the wave in space. This will give us the shape of the wave in space at a given instant. Another way is to fix a location i.e fix our attention on the particular element of string (wave) and see its oscillatory motion in time.”

First graph( if time is fixed)

Then the equation becomes function of x .i.e y=f(x). It gives the position (y) of every particle at a fixed time.

i.e we can say that this is a photograph (or snapshot) at a given time.

the important points in the above graph are

The amplitude of oscillation is 10mm
At a given time, y displacement of the wave-particle at x=2 m is -5mm and of the particle at x=6m is +4mm.
Two wave particles at different locations are in different phases $\fn_cm \large \left [ \Delta \phi=-\frac{2\pi}{\lambda}\Delta x \right ]$

Particles a and b have path differences of λ/2 so the phase difference between these is π.

Particles a and c have path differences of λ so the phase difference is 2π.

Particles c and d have path differences of λ/4 so the phase difference is π/2.

Second graph( if particle’s location is fixed)

Then the equation becomes function of t .i.e y=f(t). It gives the positiion(y) of a particular particle.

Fis an attention to a particular point p on the wave. This particle vibrates in SHM w.r.t its mean position. The (y-t) graph may be as shown in the figure.

The important points in this graph are

Amplitude of oscillation is 6 mm.
y displacement of the particle P is 4 mm at 4 sec. Similarly, y displacement of the particle P is -5mm at 8 sec.
Slop $\fn_cm \frac{\partial y}{\partial t}$ of this particle at any time gives the particle velocity ( $\fn_cm v=\pm \omega \sqrt{A^2 -y^2}$ )
The same particle at two different times will have a different phases. $\fn_cm \large \left [ \Delta \phi=\frac{2\pi}{T} \Delta t \right ]$

For example. t1 and t2 have a time interval of T/2, so the phase difference is π.

t2 and t3 have a time interval of T/4, so the phase difference is π/2.

t1 and t4 have a time interval of T, so the phase difference is 2π

Q.2. A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + π/4) where x and y are in cm and t in s. The positive direction of x is from left to right. (a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation? (b) What are its amplitude and frequency? (c) What is the initial phase at the origin? (d) What is the least distance between two successive crests in the wave?

Q.3. For the wave described in the previous question, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling waves differ from one point to another: amplitude, frequency or phase?

Q.4. For the travelling harmonic wave y(x, t) = 2.0 cos 2π (10t – 0.0080 x + 0.35)where x and y are in cm and t in s. Calculate the phase difference between the oscillatory motion of two points separated by a distance of (a) 4 m, (b) 0.5 m, (c) λ/2, (d) 3λ/4

Q.5. A travelling harmonic wave on a string is described by y(x, t) = 7.5 sin (0.0050x +12t + π/4) (a)what are the displacement and velocity of oscillation of a point at x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation? (b)Locate the points of the string which have the same transverse displacements and velocity as the x = 1 cm point at t = 2 s, 5 s and 11 s.

Q.2.