Kamal A.A. 1000 Solved Problems in Classical Physics: An Exercise Book

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338 7 Lagrangian and Hamiltonian Mechanics

∴ T (system) =

m( ˙x

+˙y

+ 2 ˙x ˙y cos α) +

(M + m) ˙x

(6)

V (system) = V (ball) =−mgy sin α (7)

L =

m( ˙x

+˙y

+2 ˙x ˙y cos α)+

(M + m) ˙x

+ mgy sin α (8)

Lagrange’s equations



∂ L

∂ ˙x



−

∂ L

∂x

= 0,



∂ L

∂ ˙y



−

∂ L

∂y

= 0(9)

become

¨x +

m ¨y cos α +(M + m) ¨x = 0 (10)

¨y +

m ¨x cos α − mg sin α = 0 (11)

Solving (10) and (11) and simplifying

¨x =−

5mg sin α cos α

5M + (5 +7sin

α)m

(for the wedge)

¨y =

5(5M + 12m)g sin α

7(5M + (5 +7sin

α)m)

(for the ball)

For M = m and α = π/4

¨x =

¨y =

√

189

Chapter 8

Waves

Abstract Chapter 8 deals with waves. The topics covered are wave equation, pro-

gressive and stationary waves, vibration of strings, wave velocity in solids, liquids

and gases, capillary waves and gravity waves, the Doppler effect, shock wave, rever-

beration in buildings, stationary waves in pipes and intensity level.

8.1 Basic Concepts and Formulae

The travelling wave: The simple harmonic progressive wave travelling in the posi-

tive x-direction can be variously written as

y = A sin

2π

(vt − x)

= A sin 2π



−



= A sin(ωt − kx)

= A sin 2π f



t −



(8.1)

where A is the amplitude, f the frequency and v the wave velocity, λ the wave-

length, ω = 2π f the angular frequency and k = 2π/λ, the wave number.

Similarly, the wave in the negative x-direction can be written as

y = A sin

2π

(vt + x) (8.2)

and so on.

The superposition principle states that when two or more waves traverse the same

region independently, the displacement of any particle at a given time is given by

the vector addition of the displacement due to the individual waves.

Interference of waves: Interference is the physical effect caused by the superposi-

tion of two or more wave trains crossing the same region simultaneously. The wave

trains must have a constant phase difference.

339

340 8Waves

Vibrating strings: Stationary waves are formed by the superposition of two

similar progressive waves travelling in the opposite direction over a taut string

clamped by rigid supports.

Wave equation:

∂

∂t

∂

∂x

(8.3)

Wave velocity:

v =



F/μ (8.4)

where F is the tension in the string and μ is the linear density, i.e. mass per unit

length.

The general solution of (8.3)is

y = f

(vt − x) + f

(vt + x) (8.5)

Harmonic solution:

y = 2A sin kx cos ωt (8.6)

When the displacement in the y-direction i s maximum (antinode) the amplitude

is 2A, the antinodes are located at x = λ/4, 3λ/4, 5λ/4 ... and are spaced half a

wavelength apart. The amplitude has a minimum value of zero (nodes). The nodes

are located at x = 0, λ/2, λ... and are also spaced half a wavelength apart. Ends

of the strings are always nodes. Neighbouring nodes and antinodes are spaced one-

quarter wavelength apart.

The frequency of vibration is given by

f =



ρ A

(8.7)

where ρ is the density and A the cross-sectional area of the string and N =

1, 2, 3,.... Vibration with N = 1 is called the fundamental or the ﬁrst harmonic,

N = 2 is called the ﬁrst overtone or the second harmonic, etc.

Power: The energy per unit length of the string is given by

E =

μV

(8.8)

where V

is the velocity amplitude of any particle on the string. Since the wave is

travelling with velocity v, the power (P) is given by

= Ev =

μV

v =



Fμ (8.9)

8.1 Basic Concepts and Formulae 341

Waves in Solids

In solids, compressional and shearing forces are readily transmitted.

(i) Transverse waves in wires/strings in which the elastic properties of the material

are disregarded:

v =



F/μ (8.4)

(ii) Transverse waves in bars/wires

V ∝



Y/ρ (8.10)

where Y is Young’s modulus of elasticity.

(iii) Longitudinal waves in wires and bars

V =



Y/ρ (8.11)

(iv) Torsional vibrations in wires/bars

V =



η/ρ (8.12)

where η is the shear modulus of elasticity.

In all t hese cases the material of restricted dimension is considered.

Waves in Liquids

The wave motion through liquids is inﬂuenced by the gravity and the characteristics

of the medium such as the depth and surface tension.

Canal Waves

If the wavelength is large compared with the wave amplitude, surface tension effect

is small. The controlling factors are then basically gravity (g) and the boundary con-

ditions. Furthermore, if the surface is sufﬁciently extensive so that the wall effects

are negligible then the depth (h) alone is the main boundary condition. The velocity

(v) of the canal waves is given by

V =



gh (8.13)

Surface Waves

These are the waves found on relatively deep water. The velocity of deep water

waves is given by

V =



gλ/2π (8.14)

342 8Waves

For long waves in shallow water

V =



gh (8.15)

Capillary Waves

Surface waves are modiﬁed by surface tension S.Ifh is large compared with λ

2πs

ρλ

gλ

2π

(8.16)

The minimum value of λ is given by minimizing (8.16)

min

= 2π



gρ

(8.17)

If λ is sufﬁciently large the second term dominates and the controlling factor being

mainly gravity. Thus, the velocity of the gravity waves is given by

V =



gλ/2π (8.18)

If λ is very small, the ﬁrst term in (8.16) dominates and the motion is mainly con-

trolled by capillarity and

V =



2πs

ρλ

(8.19)

Acoustic Waves

∂

∂t

= V

∂

∂x

(plane wave equation for displacement) (8.20)

∂

∂t

= V

∂

∂x

(plane wave equation for pressure) (8.21)

where

V =



B/ρ

(8.22)

B being the bulk modulus of elasticity.

Sound Velocity in a Gas

V =



γ P

(Laplace formula) (8.23)

8.1 Basic Concepts and Formulae 343

Sound Velocity in a Liquid

V =



γ B

(8.24)

where B

is the isothermal bulk modulus.

The energy in length λ is given by

λ (8.25)

The energy density

E = E

/λ =

(8.26)

The intensity, i.e. the time rate of ﬂow of energy per unit area of the wave front

I =

ρVA

(8.27)

Intensity Level (IL):Decibel Scale

IL = 10 log(I/I

) (8.28)

where log is logarithmic to base 10, I

is the reference intensity (the zero of the

scale) and IL is expressed in decibels.

Stationary Waves in Pipes

(i) Closed pipe (pipe closed at one end and opened at the other)

= V/4L, f

= 3V/4L, f

= 5V/4L ...

(ii) Open pipe_(pipe opened at both ends)

= V/2L, f

= V/L, f

= 3V/2L ...

Doppler effect is the apparent change in frequency of a wave motion when there

is relative motion between the source and the observer.

(a) Moving Source but Stationary Observer

If the source of waves of frequency f moves with velocity v and if v

is the sound

velocity in still air then the apparent frequency f would be

f v

v ± v

(8.29)

where the minus sign is for approach and plus sign for separation.

344 8Waves

(b) Source is At Rest, Observer in Motion

Let the observer be moving with speed v

. Then



= f

(v ± v

)

(8.30)

where the plus sign is for motion towards the source and the minus sign for motion

away from the source.



= f

(v ± v

)

(v ∓ v

)

(8.31)

(d) If the medium moves with velocity W relative to the ground along the line join-

ing source and observer,



= f

(v + W ± v

)

(v + W ∓ v

)

(8.32)

Shock waves are emitted when the observer’s velocity or the source velocity

exceeds the sound velocity and Doppler’s formulae break down. The wave front

assumes the shape of a cone with the moving body at the apex. The surface of the

cone makes an angle with the line of ﬂight of the source such that

sin θ = v/v

(8.33)

The ratio v

/v is called Mach number. An example of shock waves is the wave

resulting from a bow boat speeding on water, a second example is a jet-plane or

missile moving at the s upersonic speed, a third example is the emission of Cerenkov

radiation when a charged particle moves through a transparent medium with a speed

exceeding that of the phase velocity of light in that medium.

Echo is deﬁned as direct reﬂection of short duration sound from the surface of a

large area. If d is the distance of the reﬂector, V the speed of sound then the time

interval between the direct and reﬂected waves is

T = 2d/v (8.34)

Reverberation: A sound once produced in a room will get reﬂected repeatedly from

the walls and become so feeble that it will not be heard. The time t taken for the

steady intensity level to reach the inaudible level is called the time of reverberation:

= 0.16V/KS (Sabine law) (8.35)

8.2 Problems 345

where V is in cubic metres and S in square metre for the volume and surface area

of the room, respectively, and K is the absorption coefﬁcient of the material of the

ﬂoor, ceiling, walls, etc. summed over these components.

Beats: When two wave trains of slightly different frequencies travel through the

same region, a regular swelling and fading of the sound is heard, a phenomenon

called beats.

At a given point let the displacements produced by the two waves be

y = A sin ω

t (8.36)

y = A sin ω

t (8.37)

By the superposition principle, the resultant displacement is given by

y = y

+ y

=[2A cos 2π( f

− f

)t/2]sin 2π( f

+ f

)t/2 (8.38)

The resulting vibration has a frequency

f = ( f

+ f

)/2 (8.39)

and an amplitude given by the expression in the square bracket of (8.38). The beat

frequency is given by f

∼ f

8.2 Problems

8.2.1 Vibrating Strings

8.1 Show that the one-dimensional wave equation is satisﬁed by the function

y = A

√

(x + vt).

8.2 Show that the equation y = 2A sin(nπx/L) cos 2π ft for a standing wave is a

solution of the wave equation

∂

∂x

∂

∂t

where F is the tension and μ the mass/unit length.

8.3 A cord of length L ﬁxed at both ends is set in vibration by raising its centre a

distance h and let go. Obtain an expression for the displacement y at any point

x and time t as a series expansion assuming that initially the velocity is zero.

Also show that even harmonics are absent.

8.4 Show that the superposition of the waves y

= A sin(kx − ωt) and y

3A sin(kx + ωt) is a pure standing wave plus a travelling wave in the negative

direction along the x-axis. Find the amplitude of (a) the standing wave and (b)

the travelling wave.

346 8Waves

8.5 A sinusoidal wave on a string travelling in the +x direction at 8 m/s has a

wavelength 2 m. (a) Find its wave number, frequency and angular frequency.

(b) If the amplitude is 0.2 m, and the point x = 0 on the string is at its equi-

librium position (y = 0) at time t = 0, ﬁnd the equation for the wave.

8.6 A sinusoidal wave on a string travelling in the +x direction has wave number

3/m and angular frequency 20 rad/s. If the amplitude is 0.2 m and the point

at x = 0 is at its maximum displacement and t = 0, ﬁnd the equation of the

wave.

8.7 Show that when a standing wave is formed, each point on the string is

undergoing SHM transverse to the string.

8.8 The length of the longest string in a piano is 2.0 m and the wave velocity of

the string is 120 m/s. Find the frequencies of the ﬁrst three harmonics.

8.9 Two strings are tuned to fundamentals of f

= 4800 Hz and f



= 32 Hz. Their

lengths are 0.05 and 2.0 m, respectively. If the tension in these two strings is

the same, ﬁnd the ratio of the masses per unit length of the two strings.

8.10 The equation of a transverse wave travelling on a rope is given by y =

5sinπ(0.02x −4.00t), where y and x are expressed in centimetres and t is in

seconds. Find the amplitude, frequency, velocity and wavelength of the wave.

8.11 A string vibrates according to the equation y = 4sin

π x cos 20π t, where

x and y are in centimetres and t is in seconds. (a) What are the amplitudes

and velocity of the component waves whose superposition can give rise to this

vibration? (b) What is the distance between the nodes? (c) What is the velocity

of the particle in the transverse direction at x = 1.0 cm and when t = 9/4s?

8.12 A wave of frequency 250 cycles/s has a phase velocity 375 m/s. (a) How

far apart are two points 60

◦

out of phase? (b) What is the phase difference

between two displacements at a certain point at time 10

−3

s apart?

8.13 Two sinusoidal waves having the same frequency and travelling in the same

direction are combined. If their amplitudes are 6.0 and 8.0 cm and have a phase

difference of π/2 rad, determine the amplitude of the resultant motion.

8.14 Show that the one-dimensional wave equation is satisﬁed by the following

functions:

(a) y = A ln(x +vt) and (b) y = A cos(x +vt).

8.15

(a) A cord of length L is rigidly attached at both ends and is plucked to a

height h at a point 1/3 from one end and let it go. Show that the dis-

placement y at any distance x along the string at time t in the subsequent

motion is given by

y =

5/2

2π



sin

π x

cos

πvt

sin

2π x

cos

2πvt

−

sin

4π x

cos

4πvt

...



(b) and that the third, sixth and ninth harmonics are absent.

8.2 Problems 347

8.16 Given the amplitude A = 0.01 m, frequency f = 170 vibrations/s, the wave

velocity v = 340 m/s, write down the equation of the wave in the negative

x-direction.

8.17 (a) Show that the superposition of the waves y

= A sin(kx − ωt) and y

+A sin(kx + ωt) is a standing wave. (b) Where are its nodes and antinodes?

8.18 The wave function for a harmonic wave travelling in the positive x-direc-

tion with amplitude A, angular frequency ω and wave number k is y

= A sin

(kx − ωt).

The wave interferes with another harmonic wave travelling in the same direc-

tion with the same amplitude, frequency and wave number, but with a phase

difference δ. By using the principle of superposition, obtain an expression for

the wave function of the resultant wave and show its amplitude is |2A cos

δ|.

If each wave has an amplitude of 6 cm and they differ in phase by π/2, what

is the amplitude of the resultant wave?

For what phase differences would the resultant amplitude be equal to 6 cm?

Describe the effects that would be heard if the two waves were sound waves,

but with slightly different frequencies. How could you determine the differ-

ence between the frequencies of the two harmonic sound sources? [you may

use the result sin θ

+ sin θ

= 2 cos

(θ

− θ

) sin

(θ

+ θ

)].

[University of Durham]

8.19 Show that the average rate of energy transmission

P, of a travelling sine of

velocity v, angular frequency ω, amplitude A, along a stretched string of mass

per unit length, μ,is

P =

μvω

8.20 A fork and a monochord string of length 100 cm give 4 beats/s. The string is

made shorter, without any change of tension, until it is in unison with the fork.

If its new length is 99 cm, what is the frequency of the fork?

8.21 y(x, t) =

0.10

4+(2x−t)

represents a moving pulse, where x and y are in metres

and t in seconds. Find out the velocity of the pulse (magnitude and direction)

and point out whether it is symmetric or not.

[adapted from Hyderabad Central University 1995]

8.22

(a) A piano string of length 0.6 m is under a tension of 300 N and vibrates

with a fundamental frequency of 660 Hz. What is the mass density of the

string?

(b) What are the frequencies of the ﬁrst two harmonics?

What is the length of the pipe? (you may take the speed of sound as V =

340 m/s).

[University of Manchester 2006]

8.23

(a) Sketch the ﬁrst and second harmonic standing waves on a stretched string

of length L. Deduce an expression for the frequencies of the family of

standing waves that can be excited on the string.