Solutions for Waves Objective

1.Ans a

2.Ans b

3.Ans a

4. Ans a

5.Ans 3

6.Ans a and c

7.Ans a

8.Ans a,b,c,d

Solutions to waves conceptual

1.Ans. b,d

2. Ans c

3.Ans b,d

4.a,c

5.Ans A-Q,B-P,C-Q,D-Q,E-P


6. ans b,c

7. Ans c


9.Ans c


10. Ans b,c

Subjective questions of waves

1. If a wave form has the equation
y₁=Psun(wt-kx)
y₂=Qcos(wt-kx)

Find
a. The resulting equation on superimposition
b. Find the amplitude of the superimposed wave


2. A standing wave results from the sum of two transverse travelling waves given by
y₁=cos(πx-4πt)
y₂=cos(πx-4πt)
find
a. Equation of wave
b What is the amplitude of oscillation at an anti-node?
c. What is the smallest positive value of x that corresponds to a node
d.at what time during the interval 0 <= t <=.50 sec,will the particle at x=0 have zero velocity

3.You are driving 8 m/s on a straight road and sounding a horn which you hear at a frequency of 600Hz. The sound of the horn gets reflected from the high rise building ahead and you hear the echo.(a)What is the frequency of the echo you hear? (b) What beat frequency you hear?


4.Two trains A and B are moving with speeds 20 m/s and 30 m/s respectively in the same direction on the same straight track, with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle.

Assume that the sound of the whistle is composed of components varying in frequency from f1 = 800 Hz to f2 = 1120 Hz. The spread in the frequency (highest frequency - lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s.

1. The speed of sound of the whistle is
(A) 340 m/s for passengers in A and 310 m/s for passengers in B
(B) 360 m/s for passengers in A and 310 m/s for passengers in B
(C) 310 m/s for passengers in A and 360 m/s for passengers in B
(D) 340 m/s for passengers in both the trains


2. The spread of frequency as observed by the passengers in train B is
(A) 310 Hz
(B) 330 Hz
(C) 350 Hz
(D) 290 Hz


5.A car is being driven towards a cliff at 100km/h. The horn is sounded for a short time. The frequency of the horn is 440Hz.An echo from the cliff is heard by the driver of the car and also by a stationary observer. If the speed of sound is
340m/s, calculate the apparent frequency of the echo as perceived by
a) the stationary observer and
b) the driver of the car.

Objective questions of wave

1. The amplitude of a wave disturbance propagating in the positive direction is given by y=1/(1+x²) at time t=0 and by y=1/[1+(x-5)²] at t=5 seconds where x and y are in meters.The shape of the wave disturbance does not change during the propagation.The velocity of the wave is
a. 1 m/sec
b. 1.5 m/sec
c. .5 m/sec
d. 2 m/s


2.A transverse wave in a medium is described by the equation

y=Asin²(wt-kx).
The magnitude of the maximum velocity of particles in the medium is equal to that of the wave velocity.if the value of A is
a.λ/2π
b λ/4π
c. λ/π
d 2λ/π


3.A plane progressive wave is represented by the equation
y=cos(2πt-πx)

The equation of the wave with triple of the amplitude and double the frequency
a. y=3cos(4πt-πx)
b. y=3cos(5πt-πx)
c. y=3cos(4πt+πx)
d. y=3cos(3πt-πx)


4. In the above example ,The equation of wave with double of the amplitude and double the frequency but travelling in the opposite direction
a. y=2cos(4πt+πx)
b. y=2cos(5πt-πx)
c. y=2cos(4πt-πx)
d. y=2cos(3πt-πx)

5.The displacement of a particle having wave motion given by
y=cos²(t/4)sin(50t)
This expression may be considered to be a result of the superposition of how many wave motions
a. one
b Two
c. Three
d. Five


6.A wave is represented by the equation
y=(1mm)sin[(60 s^-1)t+(4 m^-1)x]

which one of the following is true
a. Frequency =30/π
b Amplitude=.001mm
c. Maximum Velocity of the Particle 60 mm/sec
d. wave velocity is 100m/s


7.the displacement of the particles in a string streched in the x-direction is represented by y.Among the following expressions for y,those describing wave motion ares
a. cospxsinqt
b p²x²-w²t²
c.cos²(px+wt)
d. cos(p²x²-w²t²)



8.A transverse wave on a string,the string displacement is described as
y(x,t)=1/1+(x-at)²

where a is negative constant
which of the following is true
a. The Shape of the string at t=0 is y=/1+x²
b. The shape of the waveform does not change as its move along the string
c Waveform moves in the -x direction
d. The speed of the waveform is |a|

Solutions

Please take a look at these related posts also
SHM concept Part 1
SHM concept Part 2
Conceptaul Question for SHM
Subjective questions for SHM
objective Question for SHM
Waves Concept part 1
Waves Concept part 2
Waves Concept part 3
Conceptaul Question for waves
Subjective Question for waves

Conceptual Questions of waves

1. Transverse wave velocity in a stretched string depends on
a. frequency of wave
b. tension
c. length of string
d. linear mass density string


2. A transverse wave travels along the x axis.The particles of the medium must move
a. Along the z-axis
b Along the x-axis
c. In the Y-Z plane
d. Along the y axis



3.What is true for a standing wave on the string
a.All the particles are never at rest simultaneously
b. In one complete cycle,all the particles cross their mean position simultaneously twice
c. In one complete cycle,all the particles cross their mean position simultaneously once
d. All the particles acquire their positive extreme positions simultaneously once in a cycle



4.Choose the incorrect one
a. When a ultrasonic wave travels from air into water.It bends towards the normal to the air-water interface
b.Any function of the form y(x,t)=f(vt+x) represents a travelling wave
c.the velocity ,wavelenght and frequency of wave undergo change when it is reflected from a surface
d. None of the above



5.Match the following with the types of the wave

A) Thermal radiation received from the sun
B)Sound waves produced by the vibrating string of guitar
C) Radio waves sent out from broadcasting station
D)X Rays
E) Waves produced in the air by the vibrating tuning fork

P) longitudinal
Q) Transverse





6. Which of the following functions represent a travelling wave
a. y=pcos(qx)sin(rt)
b. y=psin(qx+rt)
c. y=psin(qx-rt)
d. none of the above


7. Which of the following is not a standing wave
a. y=pcos(qx)sin(rt)
b. y=psin(qx+rt)+psin(qx-rt)
c. y=psin(qx+rt)
d Non of the above



8. When a wave is refracted into another medium which of the following will change
a. Velocity
b. Frequency
c. Phase
d. Amplitude

9.A pipe closed at one end and open at other will give
a. All even harmonics
b, All odd harmonics
c. All the harmonics
d. None of the harmonics



10. To raise the pitch of a stringed musical instrument ,the player can
a. Lossen the string
b. Tighten the string
c. Shorten the string
d. Lengthen the string

Solutions

Waves Concept

PART 3

Beats:-
-Interference of two harmonic waves of different frequencies and wavelength produces beats.
-In beats phenomenon interfering harmonic waves have slightely differing frequencies ν₁ and ν₂ such that
|ν₁-ν₂|<<(ν₁+ν₂)/2
-Thus beats arises when two waves having slightly differing frequencies ν₁ and ν₂ and comparable amplitude are superposed. The beat frequency is
ν_beat= ν₁ ∼ ν₂
-Musicians use beat phenomenon for tuning their instruments.
-For tuning an instrument for certain standard frequency it is sounded against a standard frequency and it is tuned untill the beats get disappeared.

Doppler effect:-
-Doppler effect is a change in the observed frequency of the wave when the source s and the observer o moves relative to the medium.
-There are three different ways where we can analyse this change in frequency.
(1) When observer is stationary and source is moving then change in frequency for source aproaching observer is
ν=ν₀(1+v_s/v)
where, v_s=velocity of source relative to the medium
v=velocity of wave relative to the medium
ν=observed frequency of sound waves in term of source frequency
ν₀=source frequency
-Change in frequencywhen source receeds from stationary observer is
ν=ν₀(1-v_s/v)
-Observer at rest measures higher frequency when source aproaches it and it measures lower frequency when source receeds from the observer.
(2)Doppler effect in frequency when observer is moving with a velocity v_o towards source and the source is at rest is
ν=ν₀(1+v_o/v)
(3) If both source and observer are moving then frequency observed by observer is
ν=ν₀(v+v_o)/(v+v_s)
and all the symbols have respective meanings as told earlier.

Waves Concept

PART2

Interference of waves:-
-From principle of superposition we know that overlaping waves algbrically add togather to produce a net wave without altering the way of each other or the individual waves.
-If two sinusoidal waves of the same amplitude and wavelength travell in the same direction they interfere to produce a resultant sinusoidal wave travelling in that direction.
-The resultant wave due to interference of two sinusoidal waves is given by the relation
y′(x,t)=[2A_mcos(υ/2)]sin(ωt-kx+υ/2)where υ is the phase difference between two waves.
-If υ=0n then there would be no phase difference between the travelling waves and the interference would be fully constructive.
-If υ=π then waves would be out of phase and there interference would be distructive.

Reflection of waves:-
-When a apulse or travelling wave encounters any boundary it gets reflected.
-If the boundary is not completely rigid then then a part of wave gets reflected and rest of it's part gets transmitted or refracted.
-A travelling wave at a rigid boundary is reflected with a phase reversal but the reflection at open boundary takes place without phase change.
-if an incident wave is represented by
y_i(x,t)=A sin(ωt-kx)then reflected wave at rigid boundary is
y_r(x,t)=A sin(ωt+kx+π)
=-Asin(ωt+kx)
and for reflections at open boundary reflected wave is given by
y_r(x,t)=Asin(ωt+kx)
Standing waves:-
-The interference of two identical waves moving in opposite directions produces standing waves.
-For a string with fixed ends standing wave is given by
y(x,t)=[2Acos(kx)]sin(ωt)above equation does not represent travelling wave since it does not have characterstic form involving (ωt-kx) or (ωt+kx) in the argument of trignometric function.
-In standing waves amplitude of waves is different at different points i.e., at nodes amplitude is zero and at antinodes amplitude is maximumwhich is equal to sum of amplitudes of constituting waves.
-At intermediate points amplitude of wave varies between these two limits of maxima and minima

Normal modes of stretched string:--Frequency of transverse motion of stretched string of length L fixed at both the ends is given by
f=nv/2L
where n=1,2,3,4,.......
-The set of frequencies given by above relation are called normal modes of oscillation of the system.
-The mode with n=1 is called the fundamental mode with frequancy
f₁=v/2L-Similarly second harmonic is the oscillation mode with n=2 and so on.
-Thus the string has infinite number of possible frequency of viberation which are harmonics of fundamental frequency f₁ such that f_n=nf₁

Waves concept

-Definition of wave:-
It is a disturbance which travels through the medium due to repeated periodic motion of particles of the medium about their equilibrium position.

-Example of wave motion are sound waves traveling through an intervening mediun, water waves, light waves and many more such examples are there.

-Waves requiring material medium for their propagation are called MECHANICAL WAVES. Mechanical waves are governed by Newton's law of motion.

-Sound waves are mechanical waves in atmosphere between source and the listner and require medium for their propagation.

-Other examples of mechanical waves are sesmic waves and water waves.

-Those waves which does not require material medium for their propagation are called NON MECHANICAL WAVES.

-One familiar example of NON MECHANICAL WAVES is waves associated with light or light waves. Another such examples are radio waves, X-rays, micro waves, UV light, visible light and many more.

-Transverse waves are such waves where the displacements or oscillations are perpandicular to the direction of propagation of wave.

-Longitudinal waves are those waves in which displacement or oscillations in medium are parallel to the direction of propagation of wave for example sound waves.

-At any time t , displacement y of the particle from it's equilibrium position as a function of the coordinate x of the particle is
y(x,t)=A sin(ωt-kx)
where,
A is the amplitude of the wave
k is the wave number
ω is angular frequency of the wave
and (ωt-kx) is the phase.

-Wavelength λ and wave number k are related by the relation
k=2π/λ
-Time period T and frequency f of the wave are related to ω by
ω/2π = f = 1/T
-speed of the wave is given by
v = ω/k = λ/T = λf

-Speed of a transverse wave on a stretched string depends on tension and the linear mass density of the string not on frequency of the wave
i.e,
v=√T/μ
T=Tension in the string
μ=Linear mass density of the string

-Sounds waves are longitudinal mechanical waves that can travel through solids,liquid and gases

-Speed of longitudinal waves in a medium is given by
v=√B/ρ
B=bulk modulus
ρ=Density of the medium

-Speed of longitudinal waves in ideal gas is
v=√γP/ρ

P=Pressure of the gas
ρ=Density of the gas
γ=C_p/C_V

Principle of superposition:
When two or more waves traverse thrugh the same medium,the displacement of any particle of the medium is the sum of the displacement that the individual waves would give it.

y=Σy_i(x,t)

Oscillations

PART 2

(1) Some system Executing SHM

a)Oscillations of a Spring mass system

-In this case particle of mass m oscillates under the influence of hooke's law restoring force given by F=-Kx where K is the spring constant

Angular Frequency ω=√(K/m)

Time period T=2π√(m/K)

And frequency is =(1/2π)√(K/m)

Time period of both horizontal ans vertical oscillation are same but spring constant have diffrent value for horizontal and vertical motion

b) Simple pendulum

-Motion of simple pendulum oscillating through small angles is a case of SHM with angular frequency given by
ω=√(g/L)
and Timeperiod
T=2π√(L/g)
Where L is the length of the string.

-Here we notice that period of oscillation is independent of the mass m of the pendulum

c) Compound Pendulum

- Compound pendulum is a rigid body of any shape,capable of oscillating about the horizontal axis passing through it.
-Such a pendulum swinging with small angle executes SHM with the timeperiod

T=2π√(I/mgL)

Where I =Moment of inertia of pendulum about the axis of suspension
L is the lenght of the pendulum

(2) Damped Oscillation

-SHM which continues indefinitely without the loss of the amplitude are called free oscillation or undamped and it is not a real case

- In real physical systems energy of the oscillator gradually decreases with time and oscillator will eventually come to rest.This happens because in acutal physical systems,friction(or damping ) is always present

-The reduction in amplitude or energy of the oscilaltor is called damping and oscillation are call damped

(3) Forced Oscillations and Resonance.

- Oscillations of a system under the influence of an external periodic force are called forced oscillations

- If frequency of externally applied driving force is equal to the natural frequency of the oscillator resonance is said to occur

Oscillations

PART I
-If a particle moves such that it retraces its path regularly after regular interval of time,its motion is said to be periodic Ex-Motion of earth around Sun

-If a body in periodic motion moves back and forth over the same path then the motion is said to be oscillatory motion

-Simple harmonic motion is simplest form of oscillatory motion

-SHM is a kind of motion in which the restoring force is propotional to the displacement from the mean position and opposes its increase.Mathematically restoring force is
F=-Kx
Where K=Force constant
x=displacement of the system from its mean or equilibrium position
Diffrential Equation of SHM is
d²x/dt² + ω²x=0
Solutions of this equation can both be sine or cosine functions .We conveniently choose
x=Acos(ωt+φ) where A,ω and φ all are constants

-Quantity A is known as amplitude of SHM which is the magnitude of maximum value of displacement on either sides from the equilibrium position

-Time period (T) of SHM the time during which oscillation repeats itself i.e, repeats its one cycle of motion and it is given by
T=2π/ω where ω is the angular frequency

-Frequency of the SHM is the number of the complete oscillation per unit time i.e, frequency is reciprocal of the time period
f=1/T
Thus angular frequncy
ω=2πf

-Velocity of a system executing SHM as a function of time is
v=-ωAsin(ωt+φ)

-Acceleration of particle executing SHM is
a=-ω²Acos(ωt+φ)

So a=-ω²x

This shows that acceleration is proportional to the displacement but in opposite direction

-At any time t KE of system in SHM is
KE=(1/2)mv²
=(1/2)mω²A²sin²(ωt+φ)
which is a function varying periodically in time

-PE of system in SHM at any time t is
PE=(1/2)Kx²
=(1/2)mω²A²cos²(ωt+φ)

-Total Energy in SHM
E=KE+PE
=(1/2)mω²A²
and it remain constant in absense of dissapative forces like frictional forces