Wave Motion

Wave motion is how energy moves from one place to another without the actual movement of the material around it. Instead, the particles in the medium simply wiggle back and forth in place, passing the energy along. Some waves, called mechanical waves, need a medium like air, water, or solids to travel through. Others, like light waves, are electromagnetic and can even travel through empty space. Mechanical waves come in two types: longitudinal, where particles move in the same direction as the wave, and transverse, where particles move perpendicular to the wave. Wave motion is all around us—in sounds we hear, light we see, and the waves on water.

1.0Wave Definition

It is a disturbance which propagates in space, transporting energy from one place to another without the transport of matter.

Example - Ripples on a pond, sound we hear etc.

2.0Classification of Wave

3.0Difference Between Transverse and Longitudinal Wave

Transverse waves	Longitudinal waves
Particles vibrate perpendicular to the wave’s direction of propagation.	Particles of a medium vibrate parallel to the wave’s direction of motion.
It travels in the form of crests (C) and troughs (T).	It travels in the form of compression (C) and rarefaction (R).
Transverse waves can be transmitted through solids, they can be set up on the surface of liquids. But they can not be transmitted into liquids and gases.	These waves travel through solids, liquids, and gases because they require volume elasticity.

4.0Wave Function

It is a mathematical description of the disturbance created by a wave. For a string, the wave function is displacement, for sound waves, it is a pressure or density fluctuation whereas for light waves it is electric or magnetic field.

The wave function of a wave tells about the displacement of particles of the medium at given time. 

If the wave pulse is travelling along – x-axis then y = f (x + vt),the quantities x, t must appear in combinations (Ax + Bt), where A and B are the constants such that velocity of the wave is given by $v=-\frac{B}{A}$

$v=-\frac{\text{ Coefficient of }t}{\text{ Coefficient of }x}$

Wave function when equation of motion of particle is given as $y(x,t)=g(x-vt)$

The quantity x – vt is called phase of the wave function. The phase of the pulse has fixed value x – vt = const. By taking the derivative w.r.t time $\frac{dx}{dt}=v$ is the phase velocity although often called wave velocity.

NOTE: If the wave is travelling in –x direction, then wave equation is written as $y(x,t)=f\left(t+\frac{x}{v}\right)$

5.0Wave Equation

$y=A\mathrm{Sin}(\omega t\pm kx+\varphi )$

Relation between Phase Difference And Path Difference, $\Delta \varphi =\frac{2\pi }{\lambda }\times \Delta x$

$v=-\frac{\text{ Coefficient of }t}{\text{ Coefficient of }x}$

$v=\frac{\omega }{k}\Rightarrow v=f\lambda$

Velocity and acceleration of particle present on wave:

• Velocity of particle $v=\frac{\partial y}{\partial t}=A\omega \mathrm{Cos}(\omega t-kx)$
• Acceleration of particle $a=\frac{{\partial }^{2}y}{\partial t^2}=-A{\omega }^2\mathrm{Sin}(\omega t-kx)$

The Linear Wave Equation

$\begin{array}{rl}& \begin{array}{rl}& y=A\mathrm{Sin}(\omega t-kx+\varphi )\\ & v_y={\left[\frac{dy}{dt}\right]}_{x=\text{ constant }}\Rightarrow \frac{\partial y}{\partial t}=\omega A\mathrm{Cos}(\omega t-kx+\varphi )\\ & a_y={\left[\frac{d{v}_y}{dt}\right]}_{x=\text{ constant }}\Rightarrow \frac{\partial v_y}{\partial t}=\frac{{\partial }^{2}y}{\partial t^2}=-{\omega }^{2}A\mathrm{Sin}(\omega t-kx+\varphi )\\ & v_{y,\max }=\omega A\end{array}\\ & \begin{array}{rl}& a_{y,\max }={\omega }^{2}A\\ & {\left[\frac{dy}{dx}\right]}_{t=\text{ constant }}\Rightarrow \frac{\partial y}{\partial x}=-kA\mathrm{Cos}(\omega t-kx+\varphi )\\ & \frac{{\partial }^{2}y}{\partial x^2}=-k^{2}A\mathrm{Sin}(\omega t-kx+\varphi )\\ & \frac{\partial y}{\partial t}=-\frac{\omega }{k}\frac{\partial y}{\partial x}\Rightarrow v_p=-v_w\times \mathrm{Slope}\end{array}\end{array}$

Speed of Transverse waves on strings

This velocity depends on two properties of the string.

(a) Elasticity (measured by tension F in the string)

(b) Inertia (measured by mass per unit length )

$v=\sqrt{\frac{F}{\mu }}$

If A is the area of cross section and is the density $\mu =\rho$A than $v=\sqrt{\frac{F}{\rho A}}$

6.0Energy Density in Travelling Wave on a String

Kinetic Energy: $dk=\frac{1}{2}(\mu dx){\left(-\omega y_m\right)}^2{\mathrm{Cos}}^2(kx-\omega t)$

Potential energy: Potential energy is carried in the string when it is stretched.

$dU=dW=T\Delta s=\frac{T}{2}{k}^2{y}_m^2{\mathrm{Cos}}^2(kx-\omega t)dx$

• KE and PE of an element are always equal to each other. Mechanical energy (KE + PE) of an element is not constant.
• At an instant when an element is passing through the mean position its KE and PE are maximum.
• At an instant when an element is at an extreme position its KE and PE are minimum.

Power Transmitted Along The String by a Sine Wave: $P_{avg}=2{\pi }^2\mu v{A}^2{f}^2$

Intensity of Wave on String: $⟨I⟩=\frac{1}{2}\rho v{A}^2{\omega }^2$ where $\rho$ is the density of material of string

7.0Interference and The Principle of Superposition

When two or more than two waves pass through the same region simultaneously, we say that waves interfere in that region.

$\begin{array}{rl}{y}_1& =A_1\mathrm{Sin}\left(\omega t-k{r}_1\right)\text{ in absence of }{S}_2\\ y_2& =A_2\mathrm{Sin}\left(\omega t-k{r}_2\right)\text{ in absence of }{S}_1\\ y& =y_1+y_2\Rightarrow y=A\mathrm{Sin}(\omega t+\alpha )\\ A& =\sqrt{A_1^2+A_2^2+2A_1{A}_2\mathrm{Cos}\Delta \varphi }\end{array}$

For maximum resultant amplitude, $\Delta \varphi =2n\pi ,A=A_1+A_2\approx \text{ Constructive interference }$

For minimum  resultant amplitude, $\Delta \varphi =(2n+1)\pi ,A=\left|A_1-A_2\right|\approx \text{ Destructive interference }$

Velocity analysis in interference

$\begin{array}{rl}& y=y_1+y_2\\ & \frac{\partial y}{\partial t}=\frac{\partial y_1}{\partial t}+\frac{\partial y_2}{\partial t}\\ & v_{\text{resultant }}=v_1+v_2\end{array}$

Reflection & Transmission of Wave on Composite String

Consider a wave travelling in a string of linear mass density ${\mu }_i$ as shown in the figure. At x = 0, string is joint with another string of different material of linear mass density ${\mu }_t$                       

$\begin{array}{rl}& \begin{array}{rl}& y_i=A_i\mathrm{Sin}\left(\omega t-k_{i}x\right)\to \text{ Incident Wave }\\ & y_r=A_r\mathrm{Sin}\left(\omega t-k_{r}x\right)\to \text{ Reflected Wave }\end{array}\\ & \begin{array}{rl}& y_t=A_t\mathrm{Sin}\left(\omega t-k_{t}x\right)\to \text{ Transmitted Wave }\\ & A_t=\left(\frac{2\sqrt{{\mu }_i}}{\sqrt{{\mu }_i}+\sqrt{{\mu }_t}}\right)A_i\text{ and }{A}_r=\left(\frac{\sqrt{{\mu }_i}-\sqrt{{\mu }_t}}{\sqrt{{\mu }_i}+\sqrt{{\mu }_t}}\right)A_i\end{array}\\ & \text{ In terms of wave velocity, }\\ & A_t=\left(\frac{2v_t}{v_t+v_i}\right)A_i\text{ and }{A}_r=\left(\frac{v_t-v_i}{v_t+v_i}\right)A_i\end{array}$

Reflection of Wave on String

Reflection from fixed or rigid end	Reflection from free end
After reflection from the fixed end, the shape of the wave/pulse gets inverted as shown in the figure.	After reflection from the free end, the shape of the pulse remains the same as shown in the figure.
There is a phase difference of π between incident wave and reflected wave at point of reflection.	The incident and reflected waves have no phase difference at the reflection point.

8.0Standing Waves 

Two sine waves with equal amplitude and frequency traveling opposite directions create standing waves.

$y=(2A\mathrm{Cos}kx)\mathrm{Sin}\omega t$

• The wave is not travelling and so is called a standing or stationary wave.
• The wave amplitude $A_s=(2A\mathrm{Cos}kx)$ varies periodically with position, not with time like beats.

Note: Nodes, Antinodes are also equally spaced with spacing $\left(\frac{\lambda }{2}\right)$ and Nodes and Antinodes are alternate with spacing $\left(\frac{\lambda }{4}\right)$

9.0Vibration of String

Fixed at both ends	Fixed at one end
$y=(2A\mathrm{Sin}kx)\mathrm{Cos}\omega t$ The positions of zero amplitude are called the nodes. Note that a distance of $\frac{\lambda }{2}$ or half a wavelength separates two consecutive nodes. Nodes: $\mathrm{Sin}kx=0\Rightarrow kx=n\pi$ $x=n\frac{\lambda }{2}\text{ for }n=0,1,2,3\dots \dots ..$	$y=(2A\mathrm{Sin}kx)\mathrm{Cos}\omega t$ The fundamental frequency is obtained when n = 0, i.e. $f_0=\frac{v}{4l}$
The positions of maximum amplitude. These are called the antinodes. $\begin{array}{rl}& \mathrm{Sin}kx=1\\ & x=\left(n+\frac{1}{2}\right)\lambda \text{ for }n=0,1,2,3\dots \dots \dots ..\end{array}$	First Overtone $f_1=\frac{3v}{4l}=3f_0$ Second Overtone $f_2=\frac{5v}{4l}=5f_0$
The frequencies corresponding to these wavelengths follow $f=n\frac{v}{2L}\text{ for }n=0,1,2,3\dots \dots \dots ..$	Note: Only the odd harmonics are the overtones.

Laws of Transverse Vibrations of a String

1. Law of Length :$f\propto \frac{1}{l}\Rightarrow \frac{f_1}{f_0}=\frac{L_2}{L_1}\text{ If }T\text{ and }\mu \text{ are constant }$
2. Law of Length :$f\propto \sqrt{T}\Rightarrow \frac{f_1}{f_2}=\sqrt{\frac{T_1}{T_2}}\text{ If }L\text{ and }\mu \text{ are constant }$
3. Law of Mass :$f\propto \frac{1}{\sqrt{\mu }}\Rightarrow \frac{f_1}{f_2}=\sqrt{\frac{{\mu }_2}{{\mu }_1}}\text{ If }T\text{ and }L\text{ are constant }$

Energy Density of String Carrying Standing Wave

$\frac{dE}{dx}=\frac{1}{2}\mu A^2{\omega }^2{\mathrm{Sin}}^2(kx){\mathrm{Cos}}^2(\omega t)+\frac{1}{2}T{A}^2{k}^2{\mathrm{Cos}}^2(kx){\mathrm{Sin}}^2(\omega t)$

Power Analysis in Standing Wave on String

$P=-\frac{1}{4}T{A}^{2}k\omega \mathrm{Sin}(2kx)\mathrm{Sin}(2\omega t)$

10.0Sound Waves

• Sound is a mechanical longitudinal wave from vibrating sources like guitar strings or vocal cords.
• It requires a medium with inertia and elasticity to propagate.
• Sound travels as periodic compressions and rarefactions caused by the vibrations.

Displacement Wave and Pressure Wave in One Dimension (Plane Wave)

A longitudinal wave in a fluid is described by particle displacement or the pressure changes from compression and rarefaction.

• Waves can then be described by the equation, $S=S_0\mathrm{Sin}\left(\omega t-\frac{x}{v}\right)$
• Volume Strain $\frac{\Delta V}{V}=\frac{-A{S}_0\omega \mathrm{Cos}\omega \left(t-\frac{x}{v}\right)\Delta x}{vA\Delta x}=\frac{-S_0\omega }{v}\mathrm{Cos}\omega \left(t-\frac{x}{v}\right)$
• Corresponding Stress $P=B\left(\frac{-\Delta V}{V}\right)$,Where B is the bulk modulus of the material.$P=B\frac{S_0\omega }{v}\mathrm{Cos}\omega \left(t-\frac{x}{v}\right)$
• The pressure amplitude $P_0$ and the displacement amplitude $S_0$ are related as $P_0=\frac{B\omega }{v}{S}_0=Bk{S}_0$, where k is the wave number

Note: Pressure waves are out of phase by 180° with displacement waves; pressure peaks where displacement is zero, and displacement peaks where pressure is normal.

Speed of Longitudinal (Sound) Waves

• Solid medium, $v=\sqrt{\frac{k+\frac{4}{3}\eta }{\rho }}$, k== Bulk modulus, $\eta$== Modulus of rigidity, $\rho$=Density
• Solid is in the form of long bar, $v=\sqrt{\frac{Y}{\rho }}$
• Velocity of sound waves in a fluid medium (liquid or gas), $V=\sqrt{\frac{B}{\rho }}$, where $B=-V\frac{dP}{dV}$

11.0Newton’s Formula and Laplace’s Correction

Factors Affecting Speed of Sound in Atmosphere

1. Effect of temperature: As temperature (T) increases velocity (v) increases. $v\propto \sqrt{T}$
2. Effect of Pressure: $v=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}$. At constant temperature, if pressure changes, density changes too so that $P/\rho$ stays constant. Thus, pressure doesn't affect sound velocity when temperature is constant.
3. Effect of Humidity: Humidity increases cause air density to decrease because water vapor has a lower molar mass than air.
4. Effect of Wind Velocity: Since wind moves the air, the sound’s velocity in a direction is the sum of the sound’s speed and the wind’s velocity component that way. $SL=v+w\cos .$

Intensity of Sound Waves

Power is the energy a wave carries per unit time, and intensity is power per unit area perpendicular to energy flow.

$\begin{array}{rl}& \text{ Average Intensity }=\frac{\text{ Average Power }}{\text{ Area }}\\ & ⟨I⟩=\frac{1}{2}\frac{{\omega }^2{S}_0^{2}B}{v}=\frac{p_0^{2}v}{2B}\\ & B=\rho v^2\\ & ⟨I⟩=\frac{P_0^2}{2\rho v}\end{array}$

Pitch and Frequency

Decibel Scale: The logarithmic scale for comparing sound intensities is called the decibel scale.$\beta =10\log \left(\frac{I}{I_0}\right)dB$

Loudness and Intensity

Interference of Sound Waves

Interference is the combination of waves in the same space to form a resultant wave, explained by the superposition principle.

$\begin{array}{rl}& P_1=P_{01}\mathrm{Sin}\left(\omega t-k{x}_1+{\theta }_1\right)\\ & P_2=P_{02}\mathrm{Sin}\left(\omega t-k{x}_2+{\theta }_2\right)\\ & P=P_1+P_2\\ & P=P_0\mathrm{Sin}(\omega t-kx+\theta )\\ & P_0=\sqrt{P_{01}^2+P_{02}^2+2P_{01}{P}_{02}\mathrm{Cos}\Delta \varphi }\\ & \Delta \varphi =\left|k\left(x_1-x_2\right)+\left({\theta }_2-{\theta }_1\right)\right|\end{array}$

$\text{ Intensity }\propto (\text{ Pressure Amplitude }{)}^{2}ResultantIntensityI=I1+I2+2I1I2CosI=I_1+I_2+2\sqrt{I_1{I}_2}\mathrm{Cos}\varphi$

Interference in Time: Beats

• When two sound waves of same amplitude travelling in the same direction with different frequencies superimpose, then intensity varies periodically with time. This effect is called Beats.
• Consider two sound waves of frequency $f_1$ and $f_2$  propagating in the same direction.  

$\begin{array}{rl}& y_1=A\mathrm{Sin}\left(2\pi f_{1}t-kx\right)\\ & y_2=A\mathrm{Sin}\left(2\pi f_{2}t-kx\right)\text{ where }{f}_1-f_2=\Delta f\end{array}$

By principle of superposition, $y=y_1+y_2$

$y=[2A\mathrm{Cos}(\Delta f\pi t)]\mathrm{Sin}\left[2\pi f_{2}t+\pi \Delta ft-kx\right]$

Frequency of variation of amplitude= $\frac{\Delta f}{2}=\frac{|f_1-f_2|}{2}$

Beat Time Period: The time interval between two successive maxima or minima is called Beat time period (T).

Beat Frequency: The number of beats per second is called Beat frequency. If frequency of parent waves are $f_{1}and{f}_2$,then Beat Frequency = $\left|f_1-f_2\right|$

Beats Time Period= $\frac{1}{|f_1-f_2|}$

Longitudinal Standing Waves

• Two longitudinal waves of equal frequency and amplitude traveling opposite directions form a standing wave.

$\begin{array}{rl}& P_1=P_0\sin (\omega t-kx)\\ & P_2=P_0\sin (\omega t+kx+\varphi )\end{array}$

The equation of the Resultant Standing Wave

$\begin{array}{rl}& P=P_1+P_2=2P_0\mathrm{Cos}\left(kx+\frac{\varphi }{2}\right)\mathrm{Sin}\left(\omega t+\frac{\varphi }{2}\right)\\ & P=P_0^{\prime }\mathrm{Sin}\left(\omega t+\frac{\varphi }{2}\right)\end{array}$

This is equation of SHM in which the amplitude $P_0^{\prime }$ depends on position as $P_0^{\prime }=2P_0\mathrm{Cos}\left(kx+\frac{\varphi }{2}\right)$

• Pressure amplitude is zero is called a pressure node, $P_0^{\prime }=0$

 $\begin{array}{rl}& \mathrm{Cos}\left(kx+\frac{\varphi }{2}\right)=0\\ & \left(kx+\frac{\varphi }{2}\right)=2n\pi \mp \frac{\pi }{2},\mathrm{n}=0,1,2\end{array}$

•  Pressure amplitude maximum is called a pressure antinode, $P_0^{\prime }=\mp 2P_0$

$\begin{array}{rl}& \mathrm{Cos}\left(kx+\frac{\varphi }{2}\right)=\mp 1\\ & \left(kx+\frac{\varphi }{2}\right)=n\pi ,\mathrm{n}=0,1,2\end{array}$

• If sound waves are represented as displacement waves,

$\begin{array}{rl}& S=S_0^{\prime }\mathrm{Sin}\left(\omega t+\frac{\varphi }{2}\right)\\ & S_0^{\prime }=2S_0\mathrm{Cos}\left(kx+\frac{\varphi }{2}\right)\end{array}$

Note: A pressure node in a standing wave would correspond to a displacement antinode. Similarly, a pressure anti-node would correspond to a displacement node.

Vibration of Air Columns (Closed Organ Pipe and Open Organ Pipe)

Resonance occurs when the tuning fork’s frequency matches a natural frequency of the air column in a cylindrical tube, causing a noticeable increase in sound volume.

In the diagram, $A_p$= Pressure antinode, $A_s$= displacement antinode, $N_p$= pressure node, $N_s$= displacement node.

Closed Organ Pipe	Open Organ Pipe
Fundamental frequency, $f_0=\frac{v}{{\lambda }_0}=\frac{v}{4l}\left({\lambda }_0=4l\right)$ First Overtone, $f_1=\frac{v}{{\lambda }_1}=3f_0$ $n^{th}$ overtone $f_n=(2n+1)f_0$ NOTE: Clearly only odd harmonics are allowed in a closed pipe.	Fundamental frequency , $f_0=\frac{v}{{\lambda }_0}=\frac{v}{2l}\left({\lambda }_0=2l\right)$ First Overtone, $f_1=\frac{v}{{\lambda }_1}=2f_0$ $n^{th}$ overtone $f_n=(n+1)f_0$ NOTE: Both even and odd harmonics are allowed in an open pipe.

End Correction

The displacement antinode at an open organ pipe extends slightly beyond the open end. This extra length is called the end correction, given by: e = 0.6 r

• where r = radius of the organ pipe
• Effective length of closed organ pipe is $l^{\prime }=l+e$
• Effective length of Open organ pipe is $l^{\prime }=l+2e$

frequency of a closed pipe $f_c$ and an open organ pipe $f_0$ will be

$f_c=\frac{v}{4(l+0.6r)}and{f}_o=\frac{v}{2(l+1.2r)}$
 

12.0Apparatus for Determining Speed of Sound

1. Quinck’s tube

• Two U-shaped metal tubes are used to produce sound waves with a tuning fork at A.
• Waves travel through tubes B and C, and interfere at D where a flame is sensitive to changes.
• If waves are in phase, constructive interference makes the flame flare; if not, destructive interference keeps it steady.
• Moving tube C changes the interference pattern; for destructive interference, $2x=\lambda$
• The distance y between successive interference patterns satisfies $2y=\frac{\lambda }{2}$
• Using x or y, the speed of sound can be calculated. $v=f\lambda$

2. Kundt’s Tube

• Used to determine the speed of sound in gases.
• Consists of a glass tube with lycopodium powder spread inside.
• The tube is rotated, causing the powder to slip.
• Rod CD is rubbed at end D to produce stationary waves.
• Disc C vibrates, causing the air column to vibrate at the rod’s frequency.
• The piston P is adjusted to match the air column’s frequency with the rod, causing resonance.
• Stationary waves form, with powder settling at antinodes and forming heaps at nodes.

$v_r=v_a\cdot \frac{l_r}{l_a}$

3. Resonance Tube

• A closed organ pipe has an air column of variable length.
• When the tuning fork's frequency matches the air column's frequency, resonance occurs, and sound intensity reaches its maximum.
• The frequency of vibration is given by $f=\frac{(2n+1)v}{4l}$ where n is an integer, v is the speed of sound, and l is the length of the pipe.
• As the water level in the resonance tube is lowered, the resonance frequency occurs at different lengths.
• Neglecting end correction, the lengths for resonance are: $\text{ 3l, 5l, 7l........ }$
• Including the end correction, if ll is the minimum length and x is the end correction, the lengths for resonance are:
  $\left(l_1+x\right):\left(l_2+x\right):\left(l_3+x\right)\dots \dots \dots \dots \dots \dots .\dots .\dots .\dots ..=1:3:5\dots \dots \dots .$              

4. Doppler’s Effect

• Doppler Effect occurs when there is relative motion between a wave source and an observer along the line joining them.
• The observed frequency differs from the actual frequency of the source.
• If they move toward each other, the observed frequency increases.
• If they move apart, the observed frequency decreases.

When,

v = velocity of sound w.r.t. ground

c = velocity of sound with respect to medium

$v_m$= velocity of medium

$v_0$= velocity of observer,

$v_s$ = velocity of source

(a) Sound source is moving and observer is stationary

                  Observed frequency, $f^{\prime }=f\left(\frac{v}{v-v_s}\right)$

                  Apparent wavelength, ${\lambda }^{\prime }=\lambda \left(\frac{v-v_s}{v}\right)$

(b) Sound source is stationary and observer is moving with velocity $v_0$ along the line joining them

                      Observed frequency, $f^{\prime }=f\left(\frac{v+v_o}{v}\right)$

(c) The source and observer both are moving with velocities vs and vo along the line joining them

                        Observed frequency,  $f^{\prime }=f\left(\frac{v+v_o}{v-v_s}\right)$

                        Apparent wavelength ${\lambda }^{\prime }=\lambda \left(\frac{v-v_s}{v+v_o}\right)$

Keypoints

• $v_0$ ​is positive when the observer moves toward the source and negative when moving away.
• $v_s$ is positive if the source moves towards the observer, and negative if moving away.
• v is the speed of sound relative to the ground.

If the medium moves:

• With the sound (source to observer) at speed $v_m$, then $v=c+v_m$
• Against the sound (observer to source) at speed $v_m$, then $v=c-v_m$

Here, c is the speed of sound relative to the medium.