REFLECTION OF WAVES -

Reflection of waves is the change in the direction of a wave upon striking the interface between two materials.

REFLECTION OF TRANSVERSE WAVE

REFLECTION AT RIGID BOUNDARY(closed boundary/ denser boundary)

The phenomenon of echo is an example of reflection by rigid boundary. From fig, a pulse travelling along a stretched string and being reflected by the boundary. If there is no absorption of energy by the boundary, the reflected wave has the same shape but suffers a phase change of π or 180°. Because the boundary is rigid and the disturbance must have zero displacements at all times at the boundary. By the principle of superposition, this is possible only if the reflected and incident wave differs by a phase of π.

NOTE:- It can also be understood in this way. As the pulse arrives at the wall, it exerts a force on the wall. By Newton’s third law, the wall exerts an equal and opposite force on the string, generating a reflected pulse that differs by a phase of π.

Let incident travelling wave is $\dpi{120} \fn_cm y= A sin(kx-\omega t)$

At a rigid boundary, the reflected wave is given by $\dpi{120} \fn_cm y^{'}=A sin(kx-\omega t+\pi)=-A sin(kx-\omega t)$

REFLECTION AT FREE BOUNDARY(open boundary/ rarer boundary)

If the boundary point is not rigid but completely free to move (as shown in fig), the reflected pulse has the same phase and amplitude (assuming no energy dissipation) as the incident pulse.

The net maximum displacement at the boundary is then twice the amplitude of each pulse.

If the incident travelling wave is $\dpi{120} \fn_cm y= A sin(kx-\omega t)$

At an open boundary, the reflected wave is given by $\dpi{120} \fn_cm y^{'}=A sin(kx-\omega t+0^0)=A sin(kx-\omega t)$

REFLECTION OF LONGITUDINAL WAVE

REFLECTION AT RIGID BOUNDARY(denser medium)

A longitudinal wave travels in a medium in the form of compression and rarefaction. Suppose a compression travelling from left to right reaches the rigid support. The compression exerts a force F on the rigid support. In turn, the support gives an equal and opposite reaction. The result is that the direction of compression is reversed. i.e a phase change of 180° takes place in the direction of compression.

REFLECTION AT FREE BOUNDARY(rarer medium)

Suppose a compression travelling from left to right reaches the free boundary. The compression exerts a force F on the free end towards the right. Being free, it has smaller resistance for expansion. As a result, the medium behind gets rarefied and a rarefaction travels from right to left. i.e compression is reflected as rarefaction and vice-versa.

NOTE

If a wave enters a region where the wave velocity is smaller, the reflected wave is inverted. If it enters a region where the wave velocity is larger, the reflected wave is not inverted. The transmitted wave is never inverted.

STANDING WAVES

When two progressive waves of the same wavelength and amplitude travel with the same speed along with the same straight line in the opposite direction, then these waves interfere to produce standing waves or stationary waves.

In a standing wave, the amplitude varies from place to place. There are points where the amplitude of particles of the medium is zero. These points are called NODES and midway between these nodes, there are points, where the amplitude of particles of the medium is maximum. These points are called ANTINODES.

In between the nodes and the antinodes, the amplitude of particles varies from zero to maximum.

NOTE:

These waves are called stationary waves because there is no transfer of energy along with the wave. The reason is simple, the particle at the nodes do not move at all. Therefore energy cannot be transmitted across them.

DIFFERENCE BETWEEN A STANDING WAVE AND A TRAVELLING WAVE

In a travelling wave, the disturbance produced in a region propagates with a definite velocity but in a standing wave, it is confined to the region where it is produced.
In a travelling wave, the motion of all the particles is similar in nature(same amplitude). In a standing wave, different particles move with different amplitudes.
In a standing wave, the particles at nodes always remain at rest. In travelling wave, there are no particles which always remains at rest.
In a standing wave, all particles cross their mean positions together. In a travelling wave, there is no instance when all the particles are at the mean position together.
In a standing wave, all the particles between two successive nodes reach their extreme positions together, thus moving in a phase. In travelling waves, the phases of nearby particles are always different.
In a travelling wave, energy is transmitted from one region of space to other but in a standing wave, the energy of one region is always confined in that region.

STANDING WAVES ON A STRING FIXED AT BOTH ENDS

Suppose a string of length L is kept fixed at the ends x=0 and x=L and sine waves are produced on it. For certain wave frequencies, standing waves are set up in the string due to multiple reflections at the ends and damping effects.

fig……………………………………….

Consider a wave travelling along the positive direction of the x-axis and a reflected wave of the same amplitude and wavelength in the negative direction of the x-axis are given by (let Φ=0)

$\dpi{120} \fn_cm y_1=A sin(kx-\omega t)\;\;\;\;\;and\;\;\;\;\;y_2=A sin(kx+\omega t)$

The resultant wave on the string is, according to the principle of superposition

$\dpi{120} \fn_cm y=y_1 +y_2$

$\dpi{120} \fn_cm =A sin(kx-\omega t)+A sin(kx+\omega t)$

$\dpi{120} \fn_cm =A 2 sin(\frac{kx-\omega t+kx+\omega t}{2})cos(\frac{kx+\omega t-kx+\omega t}{2})$

$\dpi{150} \fn_cm \left [ y=2A sinkxcos \omega t \right ]$ ————————————(A)

Definitely, this is not the equation of travelling waves.

Here amplitude is a function of x.

at $\dpi{120} \fn_cm x=0$ , amplitude $\dpi{120} \fn_cm 2Asin 0^0=0$

i.e the first boundary condition is satisfied

The 2nd boundary condition will be satisfied if

$\dpi{120} \fn_cm sinkL=0$

$\dpi{120} \fn_cm \Rightarrow kL=n\pi\;\;\;\;\;where\;\;n=1,2,3,.......$

$\dpi{120} \fn_cm \Rightarrow \frac{2\pi}{\lambda_n}L=n\pi$

$\dpi{120} \fn_cm \left [ L=n\frac{\lambda_n}{2} \right ]$

If the length of the string is an integral multiple of $\dpi{120} \fn_cm \frac{\lambda}{2}$ , standing waves are produced.

Thus, the possible wavelength of standing waves is constrained by the relation.

$\dpi{120} \fn_cm \left [ \lambda_n=\frac{2L}{n} \right ]\;\;\;\;n=1,2,3,......$

with corresponding frequencies

$\dpi{120} \fn_cm f=\frac{v}{\lambda}\;\;\;\;\;(v\rightarrow speed\;of\;wave\;on\;string)$

$\dpi{120} \fn_cm \left [ f_n=\frac{nv}{2L} \right ]\;\;\;for\;n=1,2,3......$

The lowest possible frequency is

$\dpi{120} \fn_cm f_0=\frac{v}{2L}$ (Fundamental modes of frequency or first Harmonic)

$\dpi{120} \fn_cm f_1=\frac{2v}{2L}=2f_0$ (second harmonic/ 1st overtone)

$\dpi{120} \fn_cm f_3=\frac{3v}{2L}=3f_0$ (third harmonic/ 2nd overtone)

NOTE

A string need not vibrate in one of these modes only. Generally, the vibration of a string will be a superposition of different modes. Some modes may be more strongly excited and some less.

Musical instruments like sitar or violin are based on this principle. Where the string is plucked or bowed, determines which modes are more prominent than others.

STANDING WAVES ON A STRING FIXED AT ONE ENDS

If the vibrations are produced by a source of the correct frequency, standing waves are produced.

The equation is given by

$\dpi{120} \fn_cm y=2A sin kx cos \omega t$

at x=0, Amplitude= $\dpi{120} \fn_cm 2Asin kx=2A sin 0^0 =0$

i.e the first boundary condition is satisfied.

The second boundary condition that x=L is an antinode. Where L is the length of the string.

$\dpi{120} \fn_cm i.e\;\;sin kL=\pm 1$

$\dpi{120} \fn_cm \Rightarrow kL=\left ( n+\frac{1}{2} \right )\pi\;\;\;(where\;n=0,1,2,3.....)$

$\dpi{120} \fn_cm \Rightarrow \frac{2\pi}{\lambda}=\left ( n+\frac{1}{2} \right )\pi$

$\dpi{120} \fn_cm \left [ L=\left ( n+\frac{1}{2} \right )\frac{\lambda_n}{2} \right ]$

$\dpi{120} \fn_cm \Rightarrow L=\left ( n+\frac{1}{2} \right )\frac{v}{2f_n}\;\;\;\;\;(v=f\lambda)$

The normal mode -the natural frequency of the system is

$\dpi{120} \fn_cm \left [ f_n=\left ( n+\frac{1}{2} \right )\frac{v}{2L} \right ]$

The fundamental frequency is obtained when n=0

$\dpi{120} \fn_cm i.e\;\;\left [ f_0=\frac{v}{4L} \right ]$

The higher frequencies are odd harmonics. i.e odd multiples of the fundamental frequency.

$\dpi{120} \fn_cm f_1=3\frac{v}{4L}=3f_0\;\;\;\;\;(first\;overtone)$

$\dpi{120} \fn_cm f_2=5\frac{v}{4L}=5f_0\;\;\;\;\;(second\;overtone)$

$\dpi{120} \fn_cm f_3=7\frac{v}{4L}=7f_0\;\;\;\;\;(third\;overtone)$

STANDING LONGITUDINAL WAVES AND VIBRATIONS OF AIR COLUMN

When the particles of the medium vibrate, the pressure variations in the layer is also changed in a simple harmonic fashion. i.e the equation of sound wave may be written as

$\dpi{120} \fn_cm p=p_0 sin(kx-\omega t)$

where $\dpi{120} \fn_cm p\rightarrow$ Excess pressure developed above the equilibrium pressure.

$\dpi{120} \fn_cm p_0\rightarrow$ The maximum change in the pressure

But pressure variation and particle vibration are in out of phase. i.e if the particle’s amplitude is maximum then pressure variation is minimum and vice versa.

CLOSED ORGAN PIPE

A closed organ pipe is a cylindrical tube having an air column with one end closed

OPEN ORGAN PIPE

An open organ pipe is a cylindrical tube containing an air column open at both ends.

Here both ends of the tube are open, there are pressure nodes (or displacement antinodes) at both ends.

From above Fig. The length of the organ pipe is

$\dpi{120} \fn_cm \left [ L=\frac{n\lambda_n}{2} \right ]$ (where n=1,2,3,……………………..)

$\dpi{120} \fn_cm \Rightarrow L=\frac{n}{2}.\frac{v}{f_n} \;\;\;\;\;(v=f\lambda)$

$\dpi{120} \fn_cm \left [ f_n=\frac{n}{2}.\frac{v}{L} \right ]\;\;\;\;\;(where\;n=1,2,3.......)$

Here fundamental frequency is

$\dpi{120} \fn_cm f_1=\frac{v}{2L}$

2nd harmonic

$\dpi{120} \fn_cm f_2=2\left ( \frac{v}{2L} \right )=2f_1$

3rd harmonic

$\dpi{120} \fn_cm f_3=3\left ( \frac{v}{2L} \right )=3f_1$

And so on…………………..

NOTE

The systems above, string and air column can also undergo forced oscillations. If the external frequency is close to one of the natural frequencies, the system shows resonance.
The frequencies of normal modes of the Table is more complex. This problem involves wave propagation in two dimensions.