Chapter 14 - Waves (Final Chapter)

Pichle chapter me objects oscillating in isolation padhe. What happens in system which is collection of such objects? Material medium provides example elastic forces bind constituents to each other and therefore motion of one affects that of other. If you drop little pebble in pond of still water water surface gets disturbed disturbance does not remain confined to one place but propagates outward along a circle. If you continue dropping pebbles you see circles rapidly moving outward from point where water surface disturbed gives feeling as if water moving outward from point of disturbance. If you put some cork pieces on disturbed surface seen cork pieces move up and down but do not move away from centre of disturbance shows water mass does not flow outward with circles but rather moving disturbance created. Similarly when we speak sound moves outward from us without any flow of air from one part of medium to another. Disturbances produced in air much less obvious and only ears or microphone can detect them. These patterns which move without actual physical transfer or flow of matter as a whole are called waves.

Waves transport energy and pattern of disturbance has information that propagate from one point to another. All our communications essentially depend on transmission of signals through waves. Speech means production of sound waves in air and hearing amounts to their detection. Often communication involves different kinds of waves for example sound waves may be first converted into electric current signal which in turn may generate electromagnetic wave that may be transmitted by optical cable or via satellite detection of original signal will usually involve steps in reverse order.

Not all waves require medium for propagation light waves can travel through vacuum light emitted by stars hundreds of light years away reaches us through interstellar space which is practically vacuum.

Most familiar type waves such as waves on string, water waves, sound waves, seismic waves etc is so-called mechanical waves these waves require medium for propagation they cannot propagate through vacuum they involve oscillations of constituent particles and depend on elastic properties of medium. Electromagnetic waves that you will learn in Class XII are different type electromagnetic waves do not necessarily require medium they can travel through vacuum light radiowaves Xrays are all electromagnetic waves in vacuum all EM waves have same speed c = 299,792,458 m/s.

Third kind wave is so-called Matter waves associated with constituents of matter electrons protons neutrons atoms molecules arise in quantum mechanical description of nature that you will learn later though conceptually more abstract than mechanical or EM waves they have already found applications in several devices basic to modern technology matter waves associated with electrons employed in electron microscopes. In this chapter we will study mechanical waves which require material medium for propagation.

Consider collection of springs connected to one another as shown Fig. If spring at one end pulled suddenly and released disturbance travels to other end. What happened? First spring disturbed from equilibrium length since second spring connected to first it also stretched or compressed and so on disturbance moves from one end to other but each spring only executes small oscillations about equilibrium position. As practical example consider stationary train at railway station different bogies coupled through spring coupling when engine attached at one end it gives push to bogie next to it this push transmitted from one bogie to another without entire train being bodily displaced.

Now consider propagation of sound waves in air as wave passes through air it compresses or expands small region of air this causes change in density of that region say δρ this change induces change in pressure δp in that region pressure is force per unit area so there is restoring force proportional to disturbance just like in spring in this case quantity similar to extension or compression of spring is change in density if region compressed molecules in that region packed together and they tend to move out to adjoining region thereby increasing density or creating compression in adjoining region consequently air in first region undergoes rarefaction if region comparatively rarefied surrounding air will rush in making rarefaction move to adjoining region thus compression or rarefaction moves from one region to another making propagation of disturbance possible in air. In solids similar arguments can be made in crystalline solid atoms or group atoms arranged in periodic lattice each atom or group atoms is in equilibrium due to forces from surrounding atoms displacing one atom keeping others fixed leads to restoring forces exactly as in spring so we can think of atoms in lattice as end points with springs between pairs of them.

We have seen motion of mechanical waves involves oscillations of constituents of medium. If constituents oscillate perpendicular to direction of wave propagation we call wave transverse wave. If they oscillate along direction of wave propagation we call wave longitudinal wave. Fig shows propagation of single pulse along string resulting from single up and down jerk if string very long compared to size pulse pulse will damp out before reaches other end and reflection from that end may be ignored. Fig shows similar situation but this time external agent gives continuous periodic sinusoidal up and down jerk to one end of string resulting disturbance on string is then sinusoidal wave in either case elements of string oscillate about their equilibrium mean position as pulse or wave passes through them oscillations are normal to direction of wave motion along string so this is example of transverse wave. We can look at wave in two ways we can fix instant of time and picture wave in space this will give us shape of wave as whole in space at given instant another way is to fix location i.e. fix attention on particular element of string and see its oscillatory motion in time.

Fig describes situation for longitudinal waves in most familiar example propagation of sound waves long pipe filled with air has piston at one end single sudden push forward and pull back of piston will generate pulse of condensations (higher density) and rarefactions (lower density) in medium air if push pull of piston continuous and periodic sinusoidal wave will be generated propagating in air along length pipe this is clearly example of longitudinal waves.

Waves considered above transverse or longitudinal are travelling or progressive waves since they travel from one part of medium to another material medium as whole does not move as already noted stream for example constitutes motion of water as whole in water wave it is disturbance that moves not water as whole likewise wind (motion of air as whole) should not be confused with sound wave which is propagation of disturbance (in pressure density) in air without motion of air medium as whole.

In transverse waves particle motion is normal to direction propagation of wave therefore as wave propagates each element of medium undergoes shearing strain transverse waves can therefore be propagated only in those media which can sustain shearing stress such as solids and not in fluids fluids as well as solids can sustain compressive strain therefore longitudinal waves can be propagated in all elastic media for example in medium like steel both transverse and longitudinal waves can propagate while air can sustain only longitudinal waves. Waves on surface of water are of two kinds capillary waves and gravity waves former are ripples of fairly short wavelength not more than few centimetre and restoring force that produces them is surface tension of water gravity waves have wavelengths typically ranging from several metres to several hundred meters restoring force that produces these is pull of gravity which tends to keep water surface at lowest level oscillations of particles in these waves are not confined to surface only but extend with diminishing amplitude to very bottom particle motion in water waves involves complicated motion they not only move up and down but also back and forth waves in ocean are combination of both longitudinal and transverse waves it is found that generally transverse and longitudinal waves travel with different speed in same medium.

Example given below some examples of wave motion state in each case if wave motion is transverse longitudinal or combination of both: (a) Motion of kink in longitudinal spring produced by displacing one end of spring sideways transverse and longitudinal? Actually kink sideways transverse component but spring longitudinal? Answer transverse and longitudinal, (b) Waves produced in cylinder containing liquid by moving its piston back and forth longitudinal, (c) Waves produced by motorboat sailing in water transverse and longitudinal, (d) Ultrasonic waves in air produced by vibrating quartz crystal longitudinal.

For mathematical description of travelling wave we need function of both position x and time t such function at every instant should give shape of wave at that instant also at every given location it should describe motion of constituent of medium at that location if we wish to describe sinusoidal travelling wave (such as one shown in Fig) corresponding function must also be sinusoidal for convenience shall take wave to be transverse so that if position of constituents of medium is denoted by x displacement from equilibrium position may be denoted by y sinusoidal travelling wave is then described by y(x,t)=a sin(kx-ωt+φ). Term φ in argument of sine function means equivalently that we are considering linear combination of sine and cosine functions y(x,t)=A sin(kx-ωt)+B cos(kx-ωt) From equations a=√(A²+B²) and φ=tan⁻¹(B/A). To understand why equation represents sinusoidal travelling wave take fixed instant say t=t0 then argument of sine function in equation is simply kx+constant thus shape of wave (at any fixed instant) as function of x is sine wave similarly take fixed location say x=x0 then argument of sine function is constant-ωt displacement y at fixed location thus varies sinusoidally with time that is constituents of medium at different positions execute SHM finally as t increases x must increase in positive direction to keep kx-ωt+φ constant thus Eq represents sinusoidal harmonic wave travelling along positive direction of x-axis on other hand function y(x,t)=a sin(kx+ωt+φ) represents wave travelling in negative direction of x-axis Fig gives names of various physical quantities appearing in Eq that we now interpret.

y(x,t): displacement as function of position x and time t, a amplitude of wave, ω angular frequency of wave, k angular wave number, kx-ωt+φ initial phase angle (at x=0 t=0) Using plots of Fig we now define various quantities of Eq.

Amplitude and Phase: Since sine function varies between 1 and -1 displacement y(x,t) varies between a and -a we can take a to be positive constant without loss generality then a represents maximum displacement of constituents of medium from their equilibrium position note that displacement y may be positive or negative but a is positive it is called amplitude of wave. Quantity (kx-ωt+φ) appearing as argument of sine function is called phase of wave given amplitude a phase determines displacement of wave at any position and at any instant clearly φ is phase at x=0 and t=0 hence φ called initial phase angle by suitable choice of origin on x-axis and initial time it is possible to have φ=0 thus there is no loss of generality in dropping φ i.e. in taking Eq with φ=0.

Wavelength and Angular Wave Number: Minimum distance between two points having same phase is called wavelength of wave usually denoted by λ for simplicity we can choose points of same phase to be crests or troughs wavelength is then distance between two consecutive crests or troughs in wave taking φ=0 displacement at t=0 is given by y(x,0)=a sin kx Since sine function repeats its value after every 2π change in angle sin kx = sin(kx+2nπ)=sin k(x+2nπ/k) That is displacements at points x and at x+2nπ/k are same where n=1,2,3... The least distance between points with same displacement (at any given instant) is obtained by taking n=1 λ is then given by λ=2π/k or k=2π/λ k is angular wave number or propagation constant its SI unit is radian per metre or rad m⁻¹. Here again radian could be dropped and units could be written merely as m⁻¹ Thus k represents 2π times number of waves (or total phase difference) that can be accommodated per unit length with SI units m⁻¹.

Period, Angular Frequency and Frequency: Fig shows again sinusoidal plot it describes not shape of wave at certain instant but displacement of element (at any fixed location) of medium as function of time we may for simplicity take Eq with φ=0 and monitor motion of element say at x=0 we then get y(0,t)=a sin(-ωt)= -a sin ωt Now period of oscillation of wave is time it takes for element to complete one full oscillation that is y(0,t)=a sin(-ωt)=a sin(-ω(t+T))=a sin(-ωt-ωT) Since sine repeats after every 2π ωT=2π or T=2π/ω ω called angular frequency of wave its SI unit rad s⁻¹ frequency ν is number of oscillations per second therefore ν=1/T=ω/2π Usually measured in hertz.

In discussion above reference has always been made to wave travelling along string or transverse wave in longitudinal wave displacement of element of medium is parallel to direction propagation of wave in Eq displacement function for longitudinal wave is written as s(x,t)=a sin(kx-ωt+φ) where s(x,t) is displacement of element of medium in direction of propagation of wave at position x and time t in Eq a is displacement amplitude other quantities have same meaning as in case of transverse wave except that displacement function y(x,t) is to be replaced by function s(x,t).

Example wave travelling along string described by y(x,t)=0.005 sin(80.0 x -3.0 t) SI units 0.005 m 80.0 rad/m and 3.0 rad/s calculate amplitude wavelength period and frequency of wave also calculate displacement y at distance x=30.0 cm and time t=20 s Answer on comparing displacement equation with y=a sin(kx-ωt) we find (a) amplitude 0.005 m=5 mm (b) angular wave number k and angular frequency ω are k=80.0 m⁻¹ and ω=3.0 s⁻¹ we then relate wavelength λ to k through λ=2π/k=2π/80=7.85 cm (c) Now relate T to ω by relation T=2π/ω=2π/3.0=2.09 s and frequency v=1/T=0.48 Hz displacement y at x=30.0 cm and time t=20 s is given by y=0.005 sin(80*0.3-3*20)=0.005 sin(24-60)=0.005 sin(-36)=0.005 sin(-36+12π) after adding 12π? Calculation gives 0.005 sin(1.699)=0.005*0.991? Actually sin 97°? Approx 5 mm.

We can fix attention on crest and see how it progresses with time figure shows plots of Eq for different values of time differing by equal intervals of time in wave crest is point of maximum positive displacement trough point of maximum negative displacement to see how wave travels we can fix attention on crest and see how it progresses with time in figure this is shown by cross on crest in same manner we can see motion of particular constituent of medium at fixed location say at origin of x-axis this is shown by solid dot. Plots show that with time solid dot at origin moves periodically i.e. particle at origin oscillates about its mean position as wave progresses this is true for any other location also we also see that during time solid dot has completed one full oscillation crest has moved further by certain distance.

Motion of crest of wave Fig gives shape of wave at two instants of time which differ by small time interval Δt entire wave pattern is seen to shift to right (positive direction of x-axis) by distance Δx in particular crest shown by dot moves distance Δx in time Δt wave pattern as whole shifts to right crest of wave (or point with any fixed phase) moves right by distance Δx in time Δt. Any other phase point will move with same speed v otherwise wave pattern will not remain fixed. Motion of fixed phase point on wave is given by kx-ωt=constant Thus as time t changes position x of fixed phase point must change so that phase remains constant thus kx-ωt=k(x+Δx)-ω(t+Δt) or kΔx-ωΔt=0 Taking Δx,Δt vanishingly small gives dx/dt=v=ω/k. Relating ω to T and k to λ we get v=ω/k = (2πν)/(2π/λ)=νλ=λ/T Eq general relation for all progressive waves shows that in time required for one full oscillation by any element of string crest has moved distance λ which is wavelength. Wave pattern as whole shifts with speed v.

Speed of transverse wave on string determined by two factors (i) linear mass density or mass per unit length μ and (ii) tension T v=√(T/μ). Speed of longitudinal wave in medium depends on elastic properties Bulk modulus B and density ρ v=√(B/ρ) for fluid? Actually for extended solid also shear etc. For sound in air v=√(γP/ρ) where γ ratio specific heats. If wire under tension 60 N what is speed of transverse waves on wire depends μ.

What happens when two wave pulses or two waves moving in same or opposite directions overlap? It is found that overlapping waves add up algebraically producing resultant wave displacement principle of superposition when two or more waves overlapping in space at same time at some position resultant displacement at that position at that time equals sum of displacements produced by each wave independently y_total = y1 + y2. As example consider two harmonic travelling waves on stretched string both with same amplitude and frequency but different initial phase according to Eq two waves described by functions y1(x,t)=a sin(kx-ωt) and y2(x,t)=a sin(kx-ωt+φ) where φ phase difference between two waves. Resultant y=y1+y2 = 2a cos(φ/2) sin(kx-ωt+φ/2) amplitude 2a cos(φ/2) depends φ if φ=0 constructive interference amplitude 2a double, if φ=π destructive cos(π/2)=0 amplitude zero cancel. Fig shows two pulses having equal and opposite displacements moving in opposite directions overlap.

Fig shows travelling waves on stretched string both with same amplitude? Actually principle.

When wave pulse travelling along string reaches boundary where properties of medium change partially reflected partially transmitted. At boundary where medium ends reflection happens. Two cases: wave reflecting from rigid fixed end and free open end. When wave reflects from fixed rigid end pulse flips upside-down phase change π inverted. Reason fixed end displacement zero must remain zero incident and reflected must cancel. When wave reflects from free/open end pulse reflects right-side-up no phase inversion free end can move displacement maximum. For harmonic wave reflection at fixed end reflected wave has phase difference π with incident, at free end same phase.

String with fixed ends superposition governs overlapping pulses reflection depends on boundary condition.

Interference of two identical waves moving in opposite directions produces standing waves for string with fixed ends standing wave is given by y(x,t)=[2a sin kx] cos ωt standing waves characterised by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. String fixed at both ends nodes at ends. Condition L = n λ/2 n=1,2,3... with n=1 λ=2L fundamental frequency ν=n v/2L =1*v/2L fundamental, n=2 λ=L ν=2*v/2L=v/L 2nd harmonic, n=3 λ=2L/3 ν=3*v/2L 3rd harmonic etc all harmonics present.

Organ pipes: Open both ends same as string L=n λ/2 λ=2L/n ν=n v/2L all harmonics n=1,2,3... Closed one end only odd harmonics L=(2n-1) λ/4 λ=4L/(2n-1) ν=(2n-1) v/4L n=1,2,3... only odd harmonics closed end is node open end antinode.

Guitar string modes example fundamental first harmonic one loop two nodes at ends one antinode middle, second harmonic two loops three nodes, third harmonic three loops etc nodes shown as antinodes at peaks with max.

Beats: Two close frequencies f1 & f2 interfere produce beats resulting wave superposition amplitude modulates over time waxing waning loud soft loud... beat frequency f_beat=|f1-f2| beat period T_beat=1/f_beat Example 440 Hz and 444 Hz f_beat=|444-440|=4 Hz 4 beats per second. Tuning instruments using beats, when beats disappear frequencies matched.

Doppler effect chapter in old book but deleted in rationalised 15.8, for completeness: apparent change in frequency due to relative motion between source and observer. Not required for current syllabus but good to know f' = f (v±vo)/(v∓vs) but deleted.