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Waves¶

Wave is due to disturbance created in a medium.
Waves transport energy without transporting matter.

Classification of Waves¶

Visible Waves (Water waves, String waves)
Invisible Waves (Sound waves, Matter waves, Radio waves)

Info

In case of mechanical waves we deal with the cooperative motion of a collection of particles.

Classification on the Basis of Nature¶

Mechanical waves (require medium for propagation)
- Sound waves
- String waves
- Water waves
Electromagnetic waves (do not require medium for propagation)
- Radio waves
- Heat waves
- Light waves

Progressive / Traveling Waves¶

Traveling wave is that which propagates or distributes specific its pulses in space along specific direction. For example:

Waves on a string
Waves on a water surface

Periodic Waves¶

Periodic waves are those, which are repeated in regular interval of time

periodic wave may be transverse or longitudinal.

Transverse Waves¶

Displacement of medium is perpendicular to the direction of propagation of the waves.
Transverse waves cannot propagate in a gas or a liquid because there is no mechanism for driving motion perpendicular to the propagation of the wave.
Transverse waves may occur on a string, on the surface of a liquid. and throughout a solid.
A ripple in a pond and a wave on a string are easily visualized as transverse waves.
In fluids, transverse waves die out very quickly and usually cannot be produced at all.

Longitudinal Waves¶

Displacement of the medium is parallel to the propagation of the wave.
A wave in a "slinky" is a good visualization.
Sound waves in air are longitudinal waves.

Info

The waves transport both energy and momentum in a medium.

Transverse Periodic Waves¶

In a time interval equal to time period, a particle in the wave travels a distance equal to wavelength.
For all waves $v=f\lambda$
The particles in the wave separated by a distance which is integral multiple of $\lambda$ i.e. $n\lambda$ are in phase motion with each other.
The particles separated by a distance which is odd multiple of $\lambda$ i.e: $$ \left(n+\frac{1}{2}\right)\lambda $$ are out phase to each other.

Sound¶

A vibrating body produces sound waves ($\lambda = 1m$).
Three things are essential for the detection of sound:
- Vibrating body for production of sound
- Medium for propagation of sound
- Listener (ear) for the detection of sound
Sound waves are longitudinal waves having three dimensional propagation in air.
Longitudinal sound waves consist of compressions and rarefactions.
Compression is a region where crowding of articles of medium is maximum
Rarefaction is region where crowding of particles of medium is minimum.
Sound waves produce Reflection, Refraction, Diffraction, Interference but not polarization because sound waves are longitudinal.

Wave	Speed ($m/s$)
EM Waves	$300,000,000$
Sound in Air	$340$
Sound in water	$1500$
Sound in steel	$5000$

Sound Intensity¶

Sound intensity is defined as the sound power per unit area. The usual context is the measurement of sound intensity in the air at a listener's location. The basic units are $watt/m^2$ or $watt/cm^2$

The most common approach to sound intensity measurement is to use the decibel scale $$ 1(dB)=10\log_{10} \left[\frac{I}{I_o}\right] $$

Velocity in Air¶

Speed of any mechanical wave is found by the following formula $$ v=\sqrt{\frac{E}{\rho}} $$ where $E=$ modulus if elasticity of medium , $\rho=$ density of medium.
In gases, sound travels in the form of compressional wave; and gases have bulk modulus of elasticity. So for gases, we get $$ v=\sqrt{\frac{E_{\text{bulk}}}{\rho}} $$
Newton proved that for air the isothermal modulus of elasticity is $E_{\text{bulk}}=P$, and speed of sound is $$ V=\sqrt{\frac{P}{\rho}} $$
At STP the speed of sound in sir is $$ v=\sqrt{\frac{hdg}{\rho}} $$ $$ v=\sqrt{\frac{76 \times 13.6 \times 981}{0.001293}} $$ $$ \;=28100cm/s=281m/s $$ Whereas experimental value is $332m/s$
Percentage error in Newtons' Calculation was $16$% because of assumption that sound propagate through sir isothermally.
Laplace corrected Newton's formula by proposing that "Sound wave does not propagates in air isothermally but adiabatically"
Laplace's formula is given as: $$ v=\sqrt{\frac{\gamma P}{\rho}} $$ where $\gamma P=$$E_{\text{bulk}}=$ adiabatic elastic modulus of fluid and $\gamma=\frac{C_p}{C_v}=1.42$ (for air)
Experimentally it is found that speed of sound increases by $0.61m/s$, or $61 cm/s$ for each $1^oC$ rise in temperature.

Effect of moisture¶

water vapours are lighter than air, thus the presence of moisture in air reduces density and hence the speed of sound increases in such cases.

Dependence of velocity of sound¶

$v$ is independent of pressure
$$ v\propto \frac{1}{\sqrt{\rho}} $$
$$ v\propto \sqrt{T} $$
$$ \frac{v_1}{v_2}=\sqrt{\frac{T_1}{T_2}} $$
$v_t=v_o+0.61t$

Info

The speed of sound in hydrogen is four times more than the speed of sound in oxygen at same temperature

Principle of Superposition¶

When two or more waves reach a point of the medium simultaneously then the resultant displacement at that point of the medium is equal to sum of the individual displacements produced by each wove. This is called principle of superposition $$ Y_{\text{total}}= Y_1+Y_2+Y_3+....+Y_n $$

Applications¶

Interference¶

It is produced by superposition of two waves of same frequencies, which are traveling in the same direction.

Beats¶

These are produced by two waves of slightly different frequencies traveling in the same direction.

Stationary Waves¶

These are produced by two waves of equal frequency and equal amplitude traveling along same line in opposite direction.

Interference Of Sound¶

Superposition (mixing up) of two identical sound waves while passing through same medium propagating along same direction is called their interference.

Conditions for Interference¶

Coherent waves
Same medium
Same direction
Identical waves
Sources of sound should be close to each other
In constructive interference, two interfering sound waves reinforce each other, so that the resultant is a louder sound.

Condition for Constructive Interference¶

Paths different $=n\lambda$ where $n=1,\pm1,\pm2,.....$
Echoing zone is region of constructive interference
In destructive interference, two interfering sound cancel each other's effect, so that the resultant loudness of sound wave is become fainter.

Condition for Destructive Interference¶

Path difference $$ \left(n+\frac{1}{2}\right)\lambda $$ where $n=0,\pm1,\pm2,....$
Silence zone is region of destructive interference
Path difference is the difference between lengths of paths traveled by two waves in reaching the same point.

Beats¶

The periodic alterations of sound between maximum and minimum loudness are called beats. * Beats are produced by the super position of two waves having slightly different frequencies traveling in same medium along same direction. * Beat frequency is defined as number of beats per second. * Absolute difference between frequencies of producing beats is equal to beat frequency $f_{\text{beat}}=|f_{A}-f_{B}|$

Info

If frequency of a tuning fork is 32 Hz then after ¾ sec, it will complete 24 vibrations.

Maximum detectable beat frequency for normal human ear is 10Hz.
Beats are used to:
- Determine unknown frequencies
- Tune musical instruments

Reflection of waves¶

All kinds of wave shows reflection
When reflection of wave takes place from denser boundary, the phase of waves reverses. In this case, reflection co-efficient is maximum and transmission co-efficient is almost zero except for quantum mechanical transmission of particle from potential barrier.

Info

The reflected wave has the same wavelength and frequency but its phase may change depending on the nature of reflecting medium.

When reflection takes place from rarer boundary, there is no phase reversal. In this case, reflection co-efficient and transmission coefficient have considerable values

Transmission of Waves¶

Practically amplitude of transmitted wave is less than that of incident wave because some of energy of incident wave is wasted at point of discontinuity
Frequency of transmitted wave is same as that of incident wave but its velocity depends on density of medium as given $$ v\propto \frac{1}{\sqrt{\rho}} $$
Since velocity of transmitted wave is different from original wave, therefore, wave length ($\lambda$) changes as frequencies of both waves are equal

Standing Waves¶

Super position of two identical waves traveling opposite to each other in the same medium simultaneously, gives rise to stationary or standing waves
Both constructive and destructive interference takes place in the formation of stationary waves.
Points of constructive interference are called antinodes while points of destructive interference are called nodes
Amplitude is maximum at antinodes and minimum (zero) at nodes.
Nodes are stationary points whereas antinodes are points that vibrate with maximum amplitude.
Two consecutive nodes or antinodes are separated by distance equal to $\lambda/2$ and an antinode and its consecutive node by $\lambda/4$.

Info

The energy stored in anti nodes at their extreme position is wholly potential while at equilibrium position the energy stored is wholly kinetic but total energy remains the same.

Standing Waves in Stretched String¶

At fixed end of string always node is formed while at free end of string, always antinode is formed.

Info

The speed of stationary waves is independent of the No. of loops.

If string is fixed at both ends, then number of nodes is one greater than no of antinodes $N = A+1$
If string is free at one to end, then no. of antinodes is equal to that of nodes. $A = N$
Speed of string wave is $$ v=\sqrt{\frac{T}{m}}=\sqrt{\frac{F}{m}} $$ $m$ is called linear mass density.
Only the following quantized frequencies of transverse stationary waves on stretched string can be produced. $f_n=nf_1$ where $n=1,2,3,...$, $$ f_1=\frac{1}{2l}\sqrt{\frac{T}{m}} $$ $f_1$ is the lowest frequency (fundamental or basic frequency) at which first Stationary wave is formed.
All other frequencies ($f_2$, $f_1$,...,$f_n$) which are integral multiple of fundamental frequencies are called overtones or harmonics,

Standing Waves in Air Columns¶

Closed pipes¶

A loudspeaker sends a sound into a long tube Dust in the tube can show nodes and antinodes Nodes are half a wavelength apart So are antinodes Maximum amplitude shows maximum pressure variation and minimum motion of air (pressure antinode) Minimum amplitude shows minimum pressure variation and maximum motion of air (pressure node)

The fundamentals: The lowest frequency which can form a standing wave has wavelength equal to twice the length of the tube.

Pipes open at both ends¶

Sound can be reflected from an open end as well as from a closed end. This is how open organ pipes and flutes work

Pipes closed at one end¶

Pipes closed at one end are shorter for the same note. A clarinet is like this An oboe is too, but with a tapered tube. Some organ pipes are stopped at one end.

Frequencies of Standing Waving¶

	pipes open or closed at both ends strings fixed at both ends	pipes open at one end
length $L$	$$ L=n\frac{\lambda}{2} $$	$$ L=(2n-1)\frac{\lambda}{4} $$
harmonics	$nf$	$(2n-1)f$

Fundamental frequency of open pipe $= 2\times$ fundamental frequency of closed pipe
No. of harmonics in open pipe $= 2\times$ No. of harmonics in closed pipe

Point to Ponder

The open end of the organ pipe behaves as antinode while the closed end behave as node in case of stationary waves through organ pipe.

Doppler Effect¶

Apparent change in pitch (frequency) of sound is due to realtive motion of source and observer.
Doppler's effect was discovered by Doppler, an Australian physicist, in $1845$.
Apparent frequency of sound heard by stationary listener due to source moving toward him at speed $u_s$ is given as:$$ f'=\left(\frac{v}{v-u_s}\right)f; \qquad f'>f$$
Apparent frequency of sound heard by stationary listener due to source moving away from him at speed $u_s$ is given as$$ f'=\left(\frac{v}{v+u_s}\right)f; \qquad f'<f$$
Apparent frequency of sound heard by a listener moving towards a stationary source with speed $u_o$ is given as $$ f'=\left(\frac{v+u_o}{v}\right)f; \qquad f'>f $$
Apparent frequency of sound heard by a listener moving away from a stationary source with speed $u+0$ is given as: $$ f'=\left(\frac{v-u_o}{v}\right)f; \qquad f'<f $$
when source and listener move in same direction than frequency of sound heard by listener is given as: $$ f'=\left(\frac{v+u_o}{v-u_s}\right)f $$
Light observer Doppler's effect too.

Applications¶

Ships and submarine (sonar devices)
Bats (for traveling)
Radar (for detection)
Determine velocity of a star wrt earth
To monitor blood flow in major arteries

Star Movement Detection¶

When a star is moving away from earth then wavelength of light increase and red shift of spectrum is observed.
When a star is moving towards the earth then wavelength of light decreases and blue shift of spectrum is observed.

Wave	Speed (\(m/s\))
EM Waves	\(300,000,000\)
Sound in Air	\(340\)
Sound in water	\(1500\)
Sound in steel	\(5000\)

	pipes open or closed at both ends strings fixed at both ends	pipes open at one end
length \(L\)	$$ L=n\frac{\lambda}{2} $$	$$ L=(2n-1)\frac{\lambda}{4} $$
harmonics	\(nf\)	\((2n-1)f\)

Waves¶

Classification of Waves¶

Classification on the Basis of Nature¶

Progressive / Traveling Waves¶

Periodic Waves¶

Transverse Waves¶

Longitudinal Waves¶

Transverse Periodic Waves¶

Sound¶

Sound Intensity¶

Velocity in Air¶

Effect of moisture¶

Dependence of velocity of sound¶

Principle of Superposition¶

Applications¶

Interference¶

Beats¶

Stationary Waves¶

Interference Of Sound¶

Conditions for Interference¶

Condition for Constructive Interference¶

Condition for Destructive Interference¶

Beats¶

Reflection of waves¶

Transmission of Waves¶

Standing Waves¶

Standing Waves in Stretched String¶

Standing Waves in Air Columns¶

Closed pipes¶

Pipes open at both ends¶

Pipes closed at one end¶

Frequencies of Standing Waving¶

Doppler Effect¶

Applications¶

Star Movement Detection¶

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