WAVE MOTION

In simple terms, we can say that wave motion involves transfer of disturbance (energy) from one point to the other with the particles of medium oscillating about their mean positions. That is, the particles of the medium do not themselves travel along with the wave. Instead, they oscillate back and forth about some equilibrium position as the wave passes by. Only the disturbance is propagated. In this chapter, we will limit our discussion to mechanical waves (elastic waves) which require a medium to travel. There are also electromagnetic waves which do not require any medium and can travel in vacuum.

Transverse Wave

In transverse waves the particle of the medium oscillate perpendicular to the direction in which the wave travels. Travelling waves on a tight rope are transverse waves. If one end of the rope is rigidly fixed and the other end is given periodic up and down jerks, the disturbance propagates along the length of the rope but the particles of the rope oscillate up and down. Disturbance travels along the rope in form of crests (upward peaks) and troughs (downward peaks).

Longitudinal Waves

In longitudinal waves, the oscillation of the particles is parallel to the direction in which the wave travels. Disturbance travelling in a ring parallel to its length a pressure variation propagating in a liquid, sound waves travelling in a medium are examples of longitudinal waves.

Longitudinal waves do not require shearing stress and hence can travel in any elastic medium: solid liquid and gases.

speed of longitudinal waves in a liquid = $\sqrt{\frac{Bulk\mathrm{mod}ulus}{density}}=\sqrt{\frac{B}{d}}$

speed of longitudinal waves in a gas = $\sqrt{\frac{\gamma P}{\rho }}$

speed of longitudinal waves in solid rod = $\sqrt{\frac{Y}{d}}$

Where B: Bulk Modulus d: density $\rho$: density of gas

$\gamma$P: adiabatic bulk modulus Y: Young's modulus P: pressure

WAVE PROPERTIES

Wave speed (c)

The speed of a wave is the distance it covers in one second. It should be carefully noted that wave speed is completely different from particle speed. Particle speed is the speed of the vibrating particles in the medium. On the other hand, wave speed is the speed with which the disturbance (or wave) propagates in the medium.

Wave Frequency ($\upsilon$)

The frequency with which the particles of the medium (through which the wave is passing) oscillate is known as wave frequency. In transverse waves, frequency is the number of crests (or troughs) that pass through a point in one second. In longitudinal waves frequency is the number of compressions (or rarefactions) that pass through a point in one second.

Time period (T)

The time period of the oscillation of the particles in the medium is the time period of the wave.

Amplitude (A)

Amplitude of the wave is same as the amplitude of the oscillating particles.

Wavelength ($\lambda$)

Wavelength is the distance between two consecutive crests (or compression) in a wave.

Wavelength, wave speed and frequency are related as follows

$c=\upsilon \lambda$

Phase

When a wave passes through a medium, all particles oscillate with same frequency, but they reach the correonding particles in their path at different time instants.

For example, in the above figure, the particle at P is at its top extreme; the particle at Q is passing through its mean position; the particle at R is at its bottom extreme. These relative positions represents the phase of motion.

(i) If two particles have same position and same velocities at all time instants, they are said to be in same phase (or in–phase).

(ii) Two particles are said to be in opposite phase (or exactly out of phase) if their dilacements from mean position and their velocities are equal in magnitude but opposite in direction.

$\bullet$the distance between particles in same phase = 0, $\lambda$, 2$\lambda$, 3$\lambda$, ……..

$\bullet$the distance between particles in opposite phase = $\lambda$/2, 3$\lambda$/2, 5$\lambda$/2, …..

When we describe the wave equation, we will give a mathematical meaning to the phase of an oscillating particle.

EQUATION OF A TRAVELLING WAVE

Suppose, man holding a stretched string starts snapping his hand at t = 0 and finishes his job at t = $\Delta$t. The vertical dilacement y of the left end of the string is a function of time. It is zero for t < 0, has non-zero value of 0 < t < $\Delta$ t and is again zero for t > $\Delta$t. Let us represent this function by f(t). Take the left end of the string a the origin and take the X-axis along the string towards right. The function f(t) represents the dilacement of the left end at time t – x / v is f(t – x/v). Hence,

y (x, t) = y (x = 0, t – x/v)

= f(t – x/v).

The dilacement of the particle at x at time t i.e., y(x, t) is generally abbreviated as y and the wave equation is written as

y = f(t – x/v) …… (i)

Equation (i) represents a wave travelling in the positive x-direction with a constant speed v. such a wave is called a travelling wave or a progressive wave. The function f is arbitrary and depends on how the source moves. The time t and the position x must appear in the wave equation in the combination t – x/v only.

For example,

$y=A\sin \frac{\left(t-x/v\right)}{T},y=A{e}^{\frac{\left(t-x/v\right)}{T}}$

etc. are valid wave equations. They represent waves travelling in positive x-direction with constant speed. The equation $y=A\sin \frac{\left(x^2-v^2{t}^2\right)}{L^2}$ does not represent a wave travelling speed.

If a wave travels in negative s-direction with speed v, its general equation may be written as

y = f(t + x/v) …. (ii)

The wave travelling in position x-direction (equation (i)) can also be written as

$y=f\left(\frac{vt-x}{v}\right)$

or, y = g(x – vt), …. (iii)

Where g is some other function having the following meaning. If we put t = 0 in equation (iii), we get the dilacement of various particles at t = 0 i.e.,

y(x, t = 0) g(x)

Thus, g(x) represents the shape of the string at t = 0. If the dilacement of the different particles at t = 0 is represented by the function g(x), the dilacement of the particle at x at time t will be y = g(x – vt). Similarly, if the wave is travelling along the negative x-direction and the dilacement of different particles at t = 0 is g(x), the dilacement of the particle at x at time t will be

y = g(x + vt) ….(iv)

Thus, the function f in equation (i) and represents the dilacement of the point x = 0 as time passes and g in (iii) and (iv) represents the dilacement at t = 0 of different particles.

The travelling wave moving with constant speed v towards positive x direction must satisfy must satisfy the following wave function condition.

$\frac{d^{2}y}{d{t}^2}=v^2\frac{d^{2}y}{d{x}^2}$ …. (v)

EQUATION OF A SIMPLE HARMONIC PLANE WAVE

In case of harmonic wave the dilacement of successive particles of the medium is given by a sine wave or cosine function of position.

The dilacement y for different values of x at t = 0 is given by

y = A sin kx …. (vi)

Where A and k are constants.

Suppose this disturbance is propagating along positive x-direction then

y = A sin k(x –vt) …. (vii)

Since the waveform represented by equation (vi) is based on sine function, it would repeat itself at regular distances. The first repetition would take place when

kx = 2$\pi$ or x = $\frac{2\pi }{k}$

This distance after which the repetition takes place is called the wavelength and denoted by $\lambda$. Hence $\lambda$ = $\frac{2\pi }{k}$ or k = $\frac{2\pi }{\lambda }$

This constant k is called propagation constant or wave number. Now equation (vii) turns into

y = A sin $\frac{2\pi }{\lambda }$ (x – vt)

At t = 0 y = A sin $\frac{2\pi }{\lambda }$ x

Relation Between Wavelength and Velocity of Propagation

Time taken for one complete cycle of wave to pass any point is the time period (T).

This is also the time taken by the disturbance in propagating a distance $\lambda$.

v = $\frac{\lambda }{T}$ = f$\lambda$ where f = frequency (Hz)

$\omega$ = $\frac{2\pi }{T}$ = 2$\pi$f = circular frequency (rad/s)

Different Forms of Simple Harmonic Wave Equation

y = A sin($\omega$t – kx – $\varphi$)

= A sin $\left[2\pi \left(\frac{t}{T}-\frac{x}{\lambda }-\frac{\varphi }{2\pi }\right)\right]$ = A sin $\left[2\pi f\left(t-\frac{x}{v}\right)-\varphi \right]$

Where $\varphi$ = phase angle.

Energy of a plane progressive wave

Consider a plane wave propagative with a velocity v in x-direction across an area S. An element of material medium (density = $\rho$ (Sdx).

The dilacement of a particle from its equilibrium position is given by the wave equation

y = A sin($\omega$t – kx)

Total energy of this element is dE = $\frac{1}{2}(dm)V_{\max }^2=\frac{1}{2}$($\rho$Sdx) (A$\omega$)²

= $\rho$Sdx (2$\pi$²f²A²)

$\Rightarrow$ Energy density = $\frac{dE}{(Sdx)}$ = 2$\pi$²f²A²$\rho$ (J/m³)

Energy per unit length = 2$\pi$²f²A²$\rho$S

$\therefore$ Power transmitted = 2h$\pi$²f²A²$\rho$S (Watt = J/s)

Intensity of the Wave (I)

Intensity of the wave is defined as the power crossing per unit area

= 2$\pi$²f²A²2$\rho$v ….Watt/m²

For wave propagation through a taut string,

$\rho$S = $\mu$, the linear density in kg/m

$\therefore$ Energy per unit length = 2$\pi$²f²A²$\mu$

TRANSVERSE WAVE IN A STRETCHED STRING

Consider a transverse pulse produced in a taut string of linear mass density $\mu$. Consider a small segment of the pulse, of length $\Delta$l, forming an arc of a circle of radius R. A force equal in magnitude to the tension T pulls tangentially on this segment at each end.

Let us set an observer at the centre of the pulse which moves along with the pulse towards rights. For the observer any small length dl of the string as shown will appear to move backward with a velocity v.

Now the small mass of the string is in a circular path of radius R moving with speed v. Therefore, the required centripetal force is provided by the only force acting, (neglecting gravity) is the component of tension along the radius.

The net restoring force on the element is

F = 2T sin($\Delta$$\theta$) $\approx$ T (2$\Delta$$\theta$) = T $\frac{\Delta \ell }{R}$

The mass of the segment is $\Delta$m = $\mu$$\Delta$$\ell$

The acceleration of this element toward the centre of the circle is

a = $\frac{v^2}{R}$, where v is the velocity of the pulse.

Using second law of motion,

T $\frac{\Delta \ell }{R}$ = ($\mu$$\Delta$$\ell$)$\left(\frac{v^2}{R}\right)$ or, v = $\sqrt{\frac{T}{\mu }}$

Laws of Transverse Vibrations of A string: Sonometer

The fundamental frequency of vibration of a stretched string fixed at both ends is given by v = $\frac{1}{2\ell }\sqrt{\frac{T}{\mu }}$. From this equation, one can immediately write the following statements known as "Laws of transverse vibrations of a string"

(a) Law of length – the fundamental frequency of vibration of a string (fixed at both ends) is inversely proportional to the length of the string provided its tension and its mass per unit length remain the same.

v $\propto$ 1 / L if T and $\mu$ are constants.

(b) Law tension – The fundamental frequency of a string is proportional to the square root of its tension provided its length and the mass per unit length remain the same.

v $\propto$ $\sqrt{T}$ if L and $\mu$ are constants.

(c) Law of mass – The fundamental frequency of a string is inversely proportional to the square root of the linear mass density, i.e., mass per unit length provided the length and the tension remain the same.

v $\propto$ $\frac{1}{\sqrt{\mu }}$ if L and T are constants.

Illustration 1: A transverse wave of wavelength 50 cm is travelling towards +ve X axis along a string whose inner density is 0.05 g/cm. The tension in the string is 450N. At t = 0, the particle through its mean position with an upward velocity. Form an equation describing the wave. The amplitude of the wave is 2.5 cm.

Solution: Let the wave be described by: y (x, t) = A sin (kx – $\omega t$+${\varphi }_0$)

Where k = $\frac{2\pi }{\lambda }=\frac{2\pi }{0.5}=4\pi$ and A = 2.5 cm

y (0,0) = 0 Velocity of transverse wave in the string is

A sin${\varphi }_0$ = 0 given by :

${\varphi }_0=0,\pi$ .... (i)

$\sqrt{\frac{T}{m}}=\sqrt{\frac{450}{0.05\times {10}^{-1}}}=300m/s$

We also have ${\frac{dy}{dt}]}_{\left(0,0\right)}$> 0

$\Rightarrow$ $\omega =2\pi \nu =\frac{2\pi c}{\lambda }=\frac{2\pi (300)}{0.5}=1200\pi$rad/s.

–A$\omega$ cos${\varphi }_0$ > 0 ...... (ii)

Using all the quantities, the equation is:

From I and II, we get ${\varphi }_0=\pi$

Y =2.5 sin (4 $\pi$x – 1200 $\pi$ t + $\pi$)