“The weak can
never forgive. Forgiveness is the attribute of the strong.”

– Mahatma Gandhi

You need to remember the following points to make you strong in answering multiple choice questions involving wave motion including sound:

**(1).**We come across three types of waves. They are:

(i) Mechanical waves

(ii) Electromagnetic waves and

(iii) Matter waves

Sound waves, surface waves in water and the waves in a stretched string are examples of mechanical waves. They require a material medium for their propagation.

Electromagnetic waves can propagate even through through empty space. They are propagated with different velocities through different media. Visible light, ultraviolet rays, X-rays, infrared rays, microwaves and radio waves are examples of electromagnetic waves. All electromagnetic waves propagate through free space with the same velocity equal to 3×10

^{8}ms^{–1}. This is the maximum velocity that can be attained by any particle.
The concept of matter waves is more abstract and is used in explaining the dual nature of matter (de Broglie's theory).

In this post we will concentrate in the study of

**only.***mechanical waves***(2).**In a

**transverse wave**the particles of the medium oscillate perpendicular to the direction of propagation of the wave. In a

**longitudinal wave**the particles of the medium oscillate parallel to the direction of propagation of the wave.

Ripples on the surface of water and the waves in a stretched wire when you pluck or strike it are examples of transverse waves. Sound waves are longitudinal waves.

**(3).**The velocity (

*v*) of a wave is related to its frequency (

*n*) and wave length (

*λ*) as

*v = nλ*

**(4). Speed (**

*v*)of transverse waves in a stretched string is given by

*v =***√(**where

*T/m*)*T*is the tension in the string and ‘

*m*’ is the linear density (mass per unit length) of the string

**(5) Frequency (**

*n*) of vibration of a stretched string is given by

*n*= (1/2

*ℓ*)**√(**where

*T/m*)*ℓ*is the length of the string.

Note that this is the frequency in the

*fundamental*mode. This frequency is called the*fundamental*frequency or the frequency of the first*harmonic*. A string can vibrate in various modes and generally, the frequency is given by

*n*= (*s*/2

*ℓ*)**√(**where s = 1,2,3,….etc. In the fundamental mode,

*T/m*)*s*= 1.

In the next mode

*s*= 2 and the frequency of vibration is*twice*the frequency in the fundamental mode. The note produced by the string in this case is called the*first overtone*or the*second harmonic*. When*s*= 3 we obtain the*second overtone*or the third harmonic.
The possible frequencies of vibration of a given length of a string with a given tension are integral multiples of the fundamental frequency.

**(6) Speed of sound (v) in a medium is generally given by**

*v*=**√(**

*E/***where E is the modulus of elasticity and**

*ρ*)*ρ*is the density of the medium.

**(i) Newton-Laplace equation**for the velocity (v) of sound in a gas is

*v*=**√(**

*γP/***where**

*ρ*)*γ*is the ratio of specific heats of the gas,

*P*is the pressure and

*ρ*is the density of the gas.

**(ii) Velocity of sound in a solid rod**is given by

*v*=**√(**

*Y/***where**

*ρ*)*Y*is the young’s modulus and

*ρ*is the density of the solid.

**(7) Equation of a progressive wave:**

The simplest equation for a wave is that which represents a wave proceeding in a particular direction (let us say, the x-direction) with the particles of the medium vibrating simple harmonically. Unlike in the case of the equation of a simple harmonic motion, the equation for a wave contains ‘

*x*’ in addition to ‘*t*’ since the equation basically shows the variation of the displacement ‘*y*’ of any particle of the medium with space and time.
You will encounter the wave equation in various forms. For a progressive wave proceeding along the

**, the wave equation is***positive X-direction*

*y***=**

*A*sin [

*ω***(**

*t**–x/v***) +**

*φ*]
where

*A*is the amplitude of the wave,*ω*is the angular frquency,*v*is the velocity (of the wave) and*φ*is the initial phase of the particle of the medium at the origin.
This has some similarity to the general equation of a simple harmonic motion [

*y*=*A*sin (*ωt*+*φ)*] with an initial phase*φ*. But the term*ωt*is replaced by*ω*(*t**–x/v*) since the state of vibration of the particle of the medium at the origin at any instant*t*is attained by a particle of the medium at distance*x*only after a time*x/v*. The wave equation is thus an equation that gives the state of vibration of the particle of the medium at any location*x*at any time*t*.
If the initial phase of the particle at the origin (

*φ*) is taken as zero, the above equation has the following forms:**(a)**

*y*=*A*sin

*ω***(**

*t**–x/v***)**

**(b)**Since

*ω =*2π/

*T*where

*T*is the period of the wave (which is the period of vibration of the particles of the medium), the wave equation can be written also as

*y =**A*sin [(2**π/**

*T*)(*t**– x/v***)]**

**(c)**Since

*T*=

*λ/v*, where

*λ*is the wave length, the wave equation can be written also as

*y*=*A*sin [(2**π/**

*λ*)(*vt**–x***)]**

**(d)**Another form of the wave equation obtained from the above is

**y =**

*A*sin [2**π(**

*t/T**– x/*

*λ***)]**

**(e)**The equation

*y =*

*A*sin (

*ωt – kx***)**also can be used to represent a progressive wave proceeding along the positive x-direction. This follows from the form shown at (a), where

*k = ω/v*which is equal to

*2π/*

*λ*.

It will be useful to remember that the

**velocity of the wave,***v*= Coefficient of*t*/Coefficient of*x***(8)**Equation of a progressive wave proceeding in the

**is**

*negative X-direction*

*y =*

*A*sin

*ω***(**

*t**+ x/v***)**

**Note that the negative sign of the term containing**

*x*in the case of the equation for a wave proceeding along the positive X-direction is replaced with positive sign. This has to be so in order to ensure the phase lead of the particle at distance

*x*compared to the particle at the origin at any instant

*t*. You can replace the positive sign of the

*x*term in all the wave equations given above to obtain the equation of a wave proceeding in the negative x-direction.

**(9) Equation of a stationary wave is**

*y =*2*A*cos(2**π**

*x/λ*) sin(2**π**if the stationary wave is formed by the superposition of a wave with the same wave reflected at a free boundary of the medium (such as the free end of a string or the open end of a pipe).

*vt/λ*)
If the reflection is at a rigid boundary (such as the fixed end of a string or the closed end of a pipe), the equation for the stationary wave formed is

*y*= – 2*A*sin(2**π**

*x/λ*) cos(2**π**.

*vt/λ*)

The negative sign and the inter change of the sine term and the cosine term in this expression (when compared to the expression for the stationary wave formed by reflection at a free boundary) occurred because of the phase change of π suffered due to the reflection at the rigid boundary. The important thing to note is that the amplitude has a space variation between the zero value (at nodes) and a maximum vlue 2

*A*(at the anti nodes). Further, the**.***distance between consecutive nodes or consecutive anti nodes is λ/2***(10)**When two sound waves of

*nearly equal*frequencies and amplitudes, traveling in the same direction, are superimposed, the frequency of the resultant sound heard by a listener is the

*average*of the two frequencies. The intensity of the sound will waver (waxing and waning) at a frequency equal to the

*difference*between the two frequencies (

**). If the beats are to be clearly heard, the difference between the frequencies of the**

*beat frequency*
If ind ividual frequencies (

*n*_{1}and*n*_{2}are the*n*_{1}>*n*_{2}),**beat frequency =***n*_{1}–*n*_{2}**(11)**A

**or**

*closed*pipe**in acoustics (branch of physics dealing with sound) means a tube**

*closed*organ pipe*closed at one end*. An

**or**

*open*pipe**means a tube**

*open*organ pipe*open at both ends*. When a standing wave (stationary wave) is formed in an organ pipe, the closed end will be a node and the open end will be an antinode. This is why the length of the pipe in the fundamental mode is equal to λ/4 (which is the distance between neighbouring node and antinode) in a closed pipe. In an open pipe, in the fundamental mode, the length of the pipe is equal to λ/2 since the consecutive antinodes are located at the ends of the tube.

Note that a

**can produce***closed pipe***only where as an***odd harmonics***can produce***open pipe***. In other words, the frequencies of vibration of the air column in a closed pipe are in the ratio 1: 3 : 5 : 7 : etc., while those in an open pipe are in the ratio 1 : 2 : 3 : 4 : 5 : etc.***all harmonics***(12)**The phenomenon by which the frequency of a wave as measured by an observer is changed because of the motion of the source, observer and the medium is called

*Doppler Effect**.*

Let the source S of sound (fig.) move with velocity

*v*_{S}, the listener move with velocity*v*_{L}and the wind blow with velocity_{w}_{}, all in the*same*direction as shown in the figure.
Wind

**→**w S●**→***v*_{S}L●**→***v*_{L}
The apparent frequency (

*n’*) of sound is then given by*n’ = n*(

*v + w*–

*v*

_{L})/ (

*v + w*–

*v*

_{S})

where

*n*is the real frequency of the sound and*v*is the velocity of sound. It may be noted that the above relation has been derived on the assumption that the source is moving towards the listener, the listener is moving*away from*the source and the wind is blowing*from the source to the listener*.
Even if you don’t remember the above expression, you will be able to answer many questions if you note that the apparent frequency increases if the source moves towards the listener or the listener moves towards the source. If they move away from each other, the apparent frequency decreases.

If the wind blows from source to the listener the wind velocity

*w*gets added to the velocity*v*of sound. If the wind is blowing from the listener to the source, the wind velocity*w*gets subtracted from the velocity*v*of sound. If the air is stll (*w =*0), the apparent frequency (*n’*) of sound is given by*n’ = n*(

*v*–

*v*

_{L})/ (

*v*–

*v*

_{S})

Light and other electromagnetic radiations do not require a medium for their propagation. Hence the velocity Furth er, the speed of sound

*w*will be absent in the expression for the apparent frequency.*v*is to be replaced by the speed of light*c*and the apparent frequency in the case of**is given by***Doppler Effect in light**n’ = n*(

*c*–

*v*

_{L})/ (

*c*–

*v*

_{S})

Doppler effect in light is used to estimate the recessional velocities of stars and galaxies by measuring the

**of the spectral lines. When a star moves away from the observer, the frequency of light emitted by the star is decreased and the the wave length is increased. The expansion of the universe is proved from this red shift. The shift in frequency is (***red shift**n*–*n*^{’}) which is related to the recessional velocity*v*_{S}as*v*

_{S}=

*c*(

*n*–

*n*

^{’})/

*n*

^{’}

This can be written in terms of the wave length shift

*λ’*–*λ*as*v*

_{S}=

*c*(

*λ’*–

*λ*)/

*λ*

Or,

**where***v*_{S}=*cz**z*= (*λ’*–*λ*)/*λ*

*z*= (*λ’***–**

*λ***)/**is called

*λ**Doppler shift*or

*spectral shift*.

In the case of a star moving towards the observer, the apparent frequency is increased and the wave length is decreased giving rise to a

**as in the case of one member of a binary star system. The velocity of approach of the star is given by***blue shift*

*v*_{S}=*c*(*n’***–**

*n***)/**

*n*^{’}
Or,

*v*_{S}=*c*(*λ***–***λ’***)/***λ*
[Doppler shift of the microwaves reflected by aircraft and automobiles is used for measuring their velocities].

[It won’t be possible for you to remember all the expressions you came across in this post. Many questions at your level can be answered even if you do not remember them. But you will definitely get many useful ideas by examining the expressions given above].

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