Two speakers separated by some distance emit sound of the same frequency. At some point P the intensity due to each speaker separately is I 0 . The path difference from P to one of the speakers is 1 2 λ greater than that from P to the other speaker. What is the intensity at P if (a) the speakers are coherent and in phase; (b) the speakers are incoherent; and (c) the speakers are coherent but have a phase difference of 180 ° ?

Consider two speakers S 1 and S 2 are separated by the distance d . The intensity of the speaker is I o (given).Let the observation point is P . According to the question, the path difference ∆ x is S 2 P - S 1 P . S 2 P - S 1 P = λ 2 (given). The phase difference, ∆ ϕ = k ∆ x , here k = 2 π λ is the angular wave number. ⇒ ∆ ϕ = 2 π λ × λ 2 = π (a) If the sources are coherent and there is no initial phase difference (as both the sources are in the same phase). Then the resultant intensity I R = 4 I 0 cos 2 ∆ ϕ 2 = 4 I o cos 2 π 2 = 0 So at the point P , destructive interference will be obtained or minima point is obtained at this point. (b) If the sources are incoherent, then the resultant intensity is I R = I o + I o = 2 I o (c) The speakers are coherent and the initial phase difference is 180 ° = π . So the total path difference ∆ x at the point P is, π + π = 2 π or π - π = 0 (as the path difference is also S 2 P - S 1 P = λ 2 , which is the equivalent phase difference of ∆ ϕ = π ). So the net intensity at the point P is now, I R = 4 I 0 cos 2 ∆ ϕ 2 = 4 I 0 cos 2 2 π 2 or = 4 I 0 cos 2 0 2 = 4 I 0 So in this condition, this point is a constructive interference point and maxima will be obtained at this point.

The transverse displacement of a string (clamped at its both ends) is given by y x , t = 0 .06 s i n ⁡ 2 π x 3 c o s 120 π t where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3 .0 × 10 - 2 k g . Answer the following: Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?

A wave pulse travelling in the positive direction of the x-axis is given by, y 1 ( x , t ) = a s i n ⁡ ( k x - ω t ) Similarly, a wave pulse travelling in the negative direction of the x-axis is given by, y 2 ( x , t ) = a s i n ⁡ ( k x + ω t ) . The superposition of these two waves gives: y = y 1 + y 2 = a s i n ⁡ ( k x - ω t ) + a s i n ⁡ ( k x + ω t ) , y = 2 a s i n ⁡ ( k x ) c o s ⁡ ( ω t ) , S i n c e , s i n s i n A + B + s i n s i n A - B = 2 s i n A c o s B . So, like wise the given standing wave can interpreted as superposition of two waves as, y = y 1 + y 2 = 0 . 03 s i n ⁡ ( 2 π 3 x - 120 π t ) + 0 . 03 s i n ⁡ ( 2 π 3 x + 120 π t ) . y = 2 a s i n ⁡ ( 2 π λ x ) c o s ⁡ ( 2 π ν t ) … (i) The transverse displacement is given by: y x , t = 0 .06 s i n ⁡ 2 π x 3 c o s 120 π t … (ii) Comparing equations (i) and (i), we get 2 π λ = 2 π 3 Therefore, the wavelength is 3 m . By comparing, we also get 120 π = 2 π ν So, the frequency of the wave is 60 H z Speed of the wave, v = ν λ v = ν λ = 60 × 3 = 180 m s - 1

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Important Questions on Waves

MEDIUM

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The number of natural numbers less than

7000

which can be formed by using the digits

0, 1, 3, 7, 9

(repetition of digits allowed) is equal to:

HARD

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

A string of length 0.4 m and mass 10^-2 kg is tightly clamped at its ends. The tension in the string is 1.6 N. Identical wave pulses are produced at one end at equal intervals of time Δt. The minimum value of Δt, which allows constructive interference between successive pulses, is

HARD

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Two speakers separated by some distance emit sound of the same frequency. At some point $P$ the intensity due to each speaker separately is $I_{0}$ . The path difference from $P$ to one of the speakers is $\frac{1}{2} λ$ greater than that from $P$ to the other speaker. What is the intensity at $P$ if
(a) the speakers are coherent and in phase;
(b) the speakers are incoherent; and
(c) the speakers are coherent but have a phase difference of $180 ° ?$

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

When a string fixed at its both ends vibrates in

1

loop,

2

loops,

3

loops and

4

loops, the frequencies are in the ratio

HARD

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

A soap film of refractive index

4 / 3

and thickness

1.5 \times 10^{- 4} cm

is illuminated by white light incident at angle of

45 °

. The reflected light is examined by a spectroscope in which a dark band corresponding to the wavelength

5 \times 10^{- 5} cm

is found. Find the order of the interference band.

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Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Find the ratio of the length of a closed pipe to that of an open pipe in order that the second overtone of the former is in unison with the fourth overtone of the latter.

MEDIUM

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The displacement of a particle is given by

\vec{r} = A [\cos (ω t) \hat{i} + \sin (ω t) \hat{j}]

. The motion of the particle is

MEDIUM

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Two periodic waves of intensities

I_{1}

and

I_{2}

pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is

MEDIUM

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Figure shows a Young’s double slit experiment setup. The source

S

of wavelength

4000 Å

oscillates along

y -

axis according to the equation

y = \sin π t

, where

y

is in millimeters and

t

is in seconds. The distance between two slits

S_{1}

and

S_{2}

0.5 m m

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The stationary wave

y = 2 a (\sin k x) (\cos ω t)

in a stretched string is the result of the superposition of

y_{1} = a \sin (k x - ω t)

and

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Four different independent waves are represented by

(i) $y_{1} = a_{1} \sin ω t$

(ii) $y_{2} = a_{2} \sin 2 ω t$

(iii) $y_{3} = a_{3} \sin ω t$

(iv) $y_{4} = a_{4} \sin (ω t + \frac{π}{3})$

With which two waves, interference is possible?

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The transverse displacement of a string (clamped at its both ends) is given by $y (x, t) = 0 .06 s i n (\frac{2 π x}{3}) c o s (120 π t)$ where x and y are in m and t in s. The length of the string is 1.5 m and its mass is $3 .0 \times 10^{- 2} k g$ .

Answer the following:
Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?

MEDIUM

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The length of a sonometer wire between two fixed points is

1.10 m

. Where should the two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio

1 : 2 : 3

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The displacement of an elastic wave is given by the function

y = 3 \sin ω t + 4 \cos

wt, where

y

is in

cm

and

t

is in second. Calculate the resultant amplitude.

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Two periodic waves of intensities

I_{1}

and

I_{2}

pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

The maximum intensity of fringes in Young's experiment is

I

. If one of the slit is closed then, the intensity at that place becomes

I_{o}

. Which of the following relations is true?

HARD

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Two radio antennas are located at points $A$ and $B$ , $200 m$ part as shown in the figure. The sources are coherent and are producing electromagnetic waves in phase at a frequency of $6 MHz$ . A radio receiver is moved out from point $B$ along a line perpendicular to the line connecting $A$ and $B$ ( line $B C$ shown in figure). At what distances from $B$ will there be destructive interference?
Question Image

EASY

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

In the case of light waves from two coherent sources S₁ and S₂, there will be constructive interference at an arbitrary point P, if the path difference S₁P - S₂P is

HARD

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

A person standing at a distance of $6 m$ from a source of sound receives sound wave in two ways, one directly from the source and others after reflection from a rigid boundary as shown in the diagram. The maximum wavelength for which, the person will receive maximum sound intensity, is

Question Image

MEDIUM

Physics>Oscillations and Waves>Waves>Principle of Superposition of Waves

Two parts of sonometer wire divided by a movable knife edge differ by

2 \times 10^{- 3} m

and produce

1 beat per second

when sounded together. Find their frequencies if the whole length of the wire is

1 metre

Which principle of perceptual organization involves perceiving smooth, continuous patterns rather than discontinuous ones?

Important Questions on Waves

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