Q: A wave is represented by an equation y= c1 sin (c2x+c3t) In which direction is the wave going? Assume that c1,c2 c3 are all positive.

Equation of the wave isy=c1sin(c2x+c3t) When the variable of the equation is (c2x+c3t), then the wave must be moving in the negative x-axis with time t.

Q: Two wave pulses identical in shape but inverted with respect to each other are produced at the two ends of a stretched string. At an instant when the pulses reach the middle, the string becomes completely straight. What happens to the energy of the two pulses?

When two wave pulses identical in shape but inverted with respect to each other meet at any instant, they form a destructive interference. The complete energy of the system at that instant is stored in the form of potential energy within it. After passing each other, both the pulses regain their original shape.

Q: Show that for a wave travelling on a string νmaxymax=αmaxνmax,where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to writeνmax−νmaxymax+νmax=νmax−αmaxνmax+αmax?

vmaxymar=amaxvmaxy=Asin(ωt−kx)y=Av=dtdy=Aωcos(ωt−kx)vmax=Aωa=dtdv=−Aω2sin(ωt−kx)amax=ω2A To prove,vmaxymax=amaxvmax LHSvmaxymar=AωA=ω1 RHSamaxvmax=ω2AAω=ω1 No, componendo and dividendo is not applicable. We cannot add quantities of different dimensions.

Q: What is the smallest positive phase constant which is equivalent to 7⋅5 π?

Equation of the wave: y=Asin(kx−ωt+Φ) Here,A is the amplitude, k is the wave number, ω is the angular frequency and Φ is the initial phase.The argument of the sine is a phase, so the smallest positive phase constant should besin(7.5π)=sin(3×2π+1.5π)=sin(1.5π) Therefore, the smallest positive phase constant is 1.5π.

Q: A string clamped at both ends vibrates in its fundamental mode. Is there any position (except the ends) on the string which can be touched without disturbing the motion? What if the string vibrates in its first overtone?

Yes, at the centre. The centre position is a node. If the string vibrates in its first overtone, then there will be two positions, i.e., two nodes, one at x = 0 and the other at x = L.

Q: A sine wave is travelling in a medium. The minimum distance between the two particles, always having same speed, is

λ/2A sine wave has a maxima and a minima and the particle displacement has phase difference of π radians. The speeds at the maximum point and at the minimum point are same although the direction of motion are different. The difference between the positions of maxima and minima is equal to λ/2

Q: A sine wave is travelling in a medium. A particular particle has zero displacement at a certain instant. The particle closest to it having zero displacement is at a distance

λ/2A sine wave has a maxima and a minima and the particle displacement has phase difference of π radians. Therefore, applying similar argument we can say that if a particular particle has zero displacement at a certain instant, then the particle closest to it having zero displacement is at a distance is equal to λ/2.

Question 1

You are walking along a seashore and a mild wind is blowing. Is the motion of air a wave motion?

Accepted Answer

No, in wave motion there is no actual transfer of matter but transfer of energy between the points where as when wind blows air particles moves with it.

Question 2

The radio and TV programmes, telecast at the studio, reach our antenna by wave motion. Is it a mechanical wave or nonmechanical?

Accepted Answer

It is a non-mechanical wave because this type of wave does not require a material medium to travel.

Question 3

A wave is represented by an equation y= c1​ sin (c2​x+c3​t) In which direction is the wave going? Assume that c1​,c2​ c3​ are all positive.

Accepted Answer

Equation of the wave isy=c1​sin(c2​x+c3​t) When the variable of the equation is (c2​x+c3​t), then the wave must be moving in the negative x-axis with time t.

Question 4

Show that the particle speed can never be equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by 2π.

Accepted Answer

Equation of the wave is given byy=Asin(ωt−kx) whereA is the amplitudeω is the angular frequencyk is the wave numberVelocity of wave, v=kω​ Velocity of particle, vp​=dtdy​=Aωcos(ωt−kx) Max velocity of particle, vpmax​​=Aω As given​A<2πλ​vpmax​​=2πλω​vpmax​​<kω​[∵λ2π​=k]​

Question 5

Two wave pulses identical in shape but inverted with respect to each other are produced at the two ends of a stretched string. At an instant when the pulses reach the middle, the string becomes completely straight. What happens to the energy of the two pulses?

Accepted Answer

When two wave pulses identical in shape but inverted with respect to each other meet at any instant, they form a destructive interference. The complete energy of the system at that instant is stored in the form of potential energy within it. After passing each other, both the pulses regain their original shape.

Question 6

Show that for a wave travelling on a string νmax​ymax​​=αmax​νmax​​,where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to writeνmax​−νmax​ymax​+νmax​​=νmax​−αmax​νmax​+αmax​​?

Accepted Answer

​vmax​ymar​​=amax​vmax​​y=Asin(ωt−kx)y=Av=dtdy​=Aωcos(ωt−kx)vmax​=Aωa=dtdv​=−Aω2sin(ωt−kx)amax​=ω2A​ To prove,vmax​ymax​​=amax​vmax​​ LHSvmax​ymar​​=AωA​=ω1​ RHSamax​vmax​​=ω2AAω​=ω1​ No, componendo and dividendo is not applicable. We cannot add quantities of different dimensions.

Question 7

What is the smallest positive phase constant which is equivalent to 7⋅5 π?

Accepted Answer

Equation of the wave: y=Asin(kx−ωt+Φ) Here,A  is the amplitude, k is the wave number, ω is the angular frequency and Φ is the initial phase.The argument of the sine is a phase, so the smallest positive phase constant should besin(7.5π)​=sin(3×2π+1.5π)=sin(1.5π)​ Therefore, the smallest positive phase constant is 1.5π.

Question 8

A string clamped at both ends vibrates in its fundamental mode. Is there any position (except the ends) on the string which can be touched without disturbing the motion? What if the string vibrates in its first overtone?

Accepted Answer

Yes, at the centre. The centre position is a node. If the string vibrates in its first overtone, then there will be two positions, i.e., two nodes, one at x = 0 and the other at x = L.

Question 9

A sine wave is travelling in a medium. The minimum distance between the two particles, always having same speed, is

Accepted Answer

λ/2A sine wave has a maxima and a minima and the particle displacement has phase difference of π radians. The speeds at the maximum point and at the minimum point are same although the direction of motion are different. The difference between the positions of maxima and minima is equal to λ/2

Question 10

A sine wave is travelling in a medium. A particular particle has zero displacement at a certain instant. The particle closest to it having zero displacement is at a distance

Accepted Answer

λ/2A sine wave has a maxima and a minima and the particle displacement has phase difference of π radians. Therefore, applying similar argument we can say that if a particular particle has zero displacement at a certain instant, then the particle closest to it having zero displacement is at a distance is equal to λ/2.

Question 11

Choose the correct option:Which of the following equations represents a wave travelling along Y-axis?

Accepted Answer

x = A sin (ky - ωt)Here x is the particle displacement of the wave and the wave is travelling along the Y-axis because the particle displacement is perpendicular to the direction of wave motion.

Question 12

The equation y=A sin2 (kx−ωt) represents a wave motion with

Accepted Answer

Amplitude A/2, frequency ω/π ​y=Asin2(kx−ωt)[cos2θ=1−2sin2θsin2θ=21−cos2θ​]​ ​y=A[21−cos2(kx−ωt)​]y=2A​[1−cos2(kx−ωt)]​ Thus, we haveAmplitude =2A​ Frequency =2(2πω​)=πω​

Question 13

Which of the following is a mechanical wave?

Accepted Answer

Sound wavesThere are mainly two types of waves: first is electromagnetic wave, which does not require any medium to travel, and the second is the mechanical wave, which requires a medium to travel. Sound requires medium to travel, hence it is a mechanical wave.

Question 14

A cork floating in a calm pond executes simple harmonic motion of frequency ν when a wave generated by a boat passes by it. The frequency of the wave is

Accepted Answer

νThe boat transmits the same wave without any change of frequency to cause the cork to execute SHM with same frequency though amplitude may differ.

Question 15

Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double of the radius of B. A transverse wave travels on A with speed νA​ and on B with speed νB​. The ratio νA​/νB​

Accepted Answer

1 / 2Wave speed is given byν=μT​​ whereT is the tension in the stringv is the speed of the waveμ is the mass per unit leng μ=LM​=ρLV​=ρL(AL)​ M is the mass of the string, which can be written as ρV L is the length of the string​=ρ(πr2)=ρ(π4D2​)∴ν=ρπ4D2​T​​=D2​ρπT​​​ where D is the diameter of the string.Thus, v∝D1​ Since, rA​=2rB​ vA​∝2rΛ​1​∝2×2rB​1​ vB​∝2rB​1​ From Equations (1) and (2) we get vB​vA​​=21​

Question 16

Both the strings, shown in figure, are made of same material and have same cross section. The pulleys are light. The wave speed of a transverse wave in the string AB is ν1​  and in CD it is ν2​. Then ν1​/ν2​

Accepted Answer

1/2​TAB​=TTCD​=2T whereTAB​ is the tension in the string ABTCD​ is the tension in the string CD The relation between tension and the wave speed is given by​v=μT​​v∝T​​ where,v is the wave speed of the transverse wave μ is the mass per unit length of the stringv2​v1​​=2TT​​=2​1​

Question 17

Velocity of sound in air is 332 m s−1. Its velocity in vacuum will be

Accepted Answer

meaninglessSound wave is a mechanical wave; this means that it needs a medium to travel. Thus, its velocity in vacuum is meaningless.

Question 18

A wave pulse, travelling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength λ and the transmitted wave λ'

Accepted Answer

λ′<λAs  v=μf​​ A wave pulse travels faster in a thinner string.The wavelength of the transmitted wave is equal to the wavelength of the incident wave because the frequency remains constant.

Question 19

Two waves represented by y=asin(ωt−kx) and y=acos(ωt−kx) y=acos(ωt−kx) are superposed. The resultant wave will have an amplitude

Accepted Answer

2​aWe know that the resultant of the amplitude is given byRnet​ =A12​+A22​+2A1​A2​cosϕ​ For the particular case, we can write=a2+a2+2a2cos2π​​ =2​a

Question 20

Two wires A and B, having identical geometrical construction, are stretched from their natural length by small but equal amount. The Young modulus of the wires are YA​ and YB​ whereas the densities are ρA​ and ρB​. It is given that YA​>YB​ and ρA​>ρB​. A transverse signal started at one end takes a time t1​ to reach the other end for A and t2​ for B.

Accepted Answer

the information is insufficient to find the relation between t1​ and t2​.v=ρη​​But because the length of wires A and B is not known, the relation between A and B cannot be determined.

Chapter 15 : Wave Motion and Waves on a String