**JEE Physics Quiz for Wave on a String MCQ:** Wave motion is a phenomenon that can be observed almost everywhere around us, and it appears in almost every branch of physics. Surface waves on bodies of matter are commonly observed. Sound waves and light waves are essential to our perception of the environment. All waves have a similar mathematical description, which makes the study of one kind of wave useful for the study of other kinds of waves.

In this chapter, we will concentrate on string waves, which are types of mechanical waves. Mechanical waves require a medium to travel through. Sound waves and water waves are other examples of mechanical waves. Light waves are not mechanical waves, these are electromagnetic waves that do not require a medium to propagate.

## Table of Contents

### Equation of travelling wave (Including Sine Wave)

For the wave shown in figure, the equation for the wave, travelling along +x axis with velocity
350 ms^{–1} when its position at t = 0 is as shown

- 0.05 sin (314/4 x –27475 t)
- 0.05 sin (379/5 x –27475 t)
- 1 sin (314/4 x –27475 t)
- 0.05 sin (289/5 x +27475 t)

The displacement of a wave disturbance propagating in the positive x-direction is given by
y = 1/(1 + x^{2}) at time t = 0 and y = 1/[1 + (x – 1)^{2}] at t = 2 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:

- 2.5 m/s
- 0.25 m/s
- 0.5 m/s
- 5 m/s

A transverse wave is described by the equation Y = Y_{0} sin 2π (ft – x/λ). The maximum particle velocity
is equal to four times the wave velocity if

- λ = πY
_{0}/4 - λ = πY
_{0}/2 - λ = πY
_{0} - λ = 2πY
_{0}

A travelling wave on a string is given by y = A sin [αx + βt + π/6]. If α = 0.56 /cm, β = 12/sec, A = 7.5 cm, then position and velocity of particle at x = 1 cm and t = 1s is

- 4.6 cm, 46.5 cm s
^{-1} - 3.75 cm, 77.94 cm s
^{-1} - 1.76 cm, 7.5 cm s
^{-1} - 7.5 cm, 75 cm s
^{-1}

A transverse wave of amplitude 0.50m, wavelength 1m and frequency 2 Hz is propagating on a string in the negative x-direction. The expression of the wave is

- y(x, t) = 0.5 sin (2πx – 4πt)
- y(x, t) = 0.5 cos (2πx + 4πt)
- y(x, t) = 0.5 sin (πx – 2πt)
- y(x, t) = 0.5 cos (2πx – 2πt)

Two small boats are 10m apart on a lake. Each pops up and down with a period of 4.0 seconds due to wave motion on the surface of water. When one boat is at its highest point, the other boat is at its lowest point. Both boats are always within a single cycle of the waves. The speed of the waves is

- 2.5 m/s
- 5.0 m/s
- 14 m/s
- 40 m/s

A wave pulse is generated in a string that lies along
x-axis. At the points A and B, as shown in figure, if R_{A}
and R_{B} are ratio of wave speed to the particle speed
respectively then:

- R
_{A}> R_{B} - R
_{B}> R_{A} - R
_{A}= R_{B} - Information is not sufficient to decide.

Wave pulse on a string shown in figure is moving to the
right without changing shape. Consider two particles at
positions x_{1} = 1.5 m and x_{2} = 2.5 m. Their transverse
velocities at the moment shown in figure are along
directions:

- positive y–axis and positive y–axis respectively
- negative y–axis and positive y–axis respectively
- positive y–axis and negative y–axis respectively
- negative y–axis and negative y–axis respectively

An observer standing at the sea-coast observes 54 waves reaching the coast per minute. If the wavelength of wave is 10m, The velocity of wave is

- 19 m/sec
- 29 m/sec
- 9 m/sec
- 39 m/sec

### Speed of a wave on a string

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 v_{1} and in CD it is v_{2}. The v_{1}/v_{2} is

- 1
- 2
- √2
- 1/√2

Three blocks I, II, & III having mass of 1.6 kg, 1.6 kg and 3.2 kg respectively are connected as shown
in the figure. The linear mass density of the wire AB, CD and DE are 10 g/m, 8 g/m and 10 g/m
respectively. The speed of a transverse wave pulse produced in AB, CD and DE are : (g = 10 m/sec^{2})

- 80 m/s, 20√10 m/s, 40 m/s
- 20√10 m/s, 40 m/s, 80 m/s
- 20√10 m/s in all
- 80 m/s in all

Three consecutive flash photographs of a travelling wave on a string are reproduced in the figure here. The following observations are made. Mark the one which is correct. (Mass per unit length of the string = 3 g/cm.)

- displacement amplitude of the wave is 0.25 m, wavelength is 1 m, wave speed is 2.5 m/s and the frequency of the driving force is 0.2/s.
- displacement amplitude of the wave is 2.0 m, wavelength is 2 m, wave speed is 0.4 m/s and the frequency of the driving force is 0.7/s.
- displacement amplitude of the wave is 0.25 m, wavelength is 2 m, wave speed is 5 m/s and the frequency of the driving force is 2.5 /s.
- displacement amplitude of the wave is 0.5 m, wavelength is 2 m, wave speed is 2.5 m/s and the frequency of the driving force is 0.2/s.

A heavy ball is suspended from the ceiling of a motor car through a light string. A transverse pulse
travels at a speed of 50 cm/s on the string when the car is at rest and 52 cm/s when the car accelerates
on a horizontal road. Then acceleration of the car is : (Take g = 10 m/s^{2})

- 2.7 m/s
^{2} - 3.7 m/s
^{2} - 2.4 m/s
^{2} - 4.1 m/s
^{2}

### Power transmitted along the string

A wave moving with constant speed on a uniform string passes the point x = 0 with amplitude A_{0},
angular frequency ω_{0} and average rate of energy transfer P_{0}. As the wave travels down the string it
gradually loses energy and at the point x = l, the average rate of energy transfer becomes P_{0}/2. At the
point x = l, angular frequency and amplitude are respectively :

- ω
_{0}and A_{0}/√2 - ω
_{0}/√2 and A_{0} - less than ω
_{0}and A_{0} - ω
_{0}/√2 and A_{0}/√2

A sinusoidal wave with amplitude y_{m}> is travelling with speed V on a string with linear density ρ. The
angular frequency of the wave is ω. The following conclusions are drawn. Mark the one which is
correct.

- doubling the frequency doubles the rate at which energy is carried along the string
- if the amplitude were doubled, the rate at which energy is carried would be halved
- if the amplitude were doubled, the rate at which energy is carried would be doubled
- the rate at which energy is carried is directly proportional to the velocity of the wave.

Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string having a linear mass density
equal to 4.00 × 10^{-2} kg/m. If the source can deliver a average power of 90 W and the string is under a
tension of 100 N, then the frequency at which the source can operate is (take π^{2} = 10):

- 45 Hz
- 50 Hz
- 30 Hz
- 62 Hz

The average power transmitted through a given point on a string supporting a sine wave is. 0.40 watt when the amplitude of wave is 2 mm. What average power will transmitted through this point its amplitude is increased to 4 mm.

- 0.40 watt
- 0.80 watt
- 1.2 watt
- 1.6 watt