JEE Advanced PYQ Waves Miscellaneous Mixed Concept Problems

Q 1 :

A student is performing an experiment using a resonance column and a tuning fork of frequency $244 s^{- 1}$ . He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is $(0.350 \pm 0.005) m$ , the gas in the tube is

(Useful information: $\sqrt{167 R T} = 640 J^{1 / 2} {mole}^{- 1 / 2}; \sqrt{140 R T} = 590 J^{1 / 2} {mole}^{- 1 / 2}$ . The molar masses M in grams are given in the options. Take the values of $\sqrt{\frac{10}{M}}$ for each gas as given there.) [2014]

$Neon (M = 20, \sqrt{\frac{10}{20}} = \frac{7}{10})$
$Nitrogen (M = 28, \sqrt{\frac{10}{28}} = \frac{3}{5})$
$Oxygen (M = 32, \sqrt{\frac{10}{32}} = \frac{9}{16})$
$Argon (M = 36, \sqrt{\frac{10}{36}} = \frac{17}{32})$

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(4)

$Here, ν = \frac{v}{4 l} = \sqrt{\frac{γ R T}{M \times 10^{- 3}}} \times \frac{1}{4 l} \Rightarrow v = ν λ = ν \times 4 l$

$\Rightarrow ν (244) \times 4 \times l = 336.7 m/s to 346.5 m/s$ $[∵ l = 0.350 \pm 0.005]$

$For monatomic gas γ = 1.67$

$∴$ $v = \sqrt{\frac{γ R T}{M \times 10^{- 3}}} = \sqrt{100 γ R T} \times \sqrt{\frac{10}{M}}$

$= \sqrt{167 R T} \times \sqrt{\frac{10}{M}} = 640 \sqrt{\frac{10}{M}}$

$For Neon M = 20$

$∴$ $v = 640 \times \frac{7}{10} = 448 {m s}^{- 1}$

$For Argon M = 36$

$∴$ $v = 640 \times \frac{17}{32} = 340 {m s}^{- 1}$

$For diatomic gas γ = 1.4$

$v = \sqrt{140 R T} \sqrt{\frac{10}{M}} = 590 \sqrt{\frac{10}{M}}$

$For Oxygen M = 32$

$∴$ $v = 590 \times \frac{9}{16} = 331.87 {m s}^{- 1}$

$For Nitrogen M = 28$

$∴$ $v = 590 \times \frac{3}{5} = 354 {m s}^{- 1}$

Q 2 :

In the experiment for the determination of the speed of sound in air using the resonance column method, the length of the air column that resonates in the fundamental mode, with a tuning fork is 0.1 m. When this length is changed to 0.35 m, the same tuning fork resonates with the first overtone. Calculate the end correction. [2003]

0.012 m
0.025 m
0.05 m
0.024 m

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(2)

$Let x be the end correction.$

$ℓ_{1} + x = \frac{λ}{4} or, λ = 4 (ℓ_{1} + x)$

$(ℓ_{2} + x) = \frac{3 λ}{4} or λ = \frac{4}{3} (ℓ_{2} + x)$

$∴$ $ν_{1} = \frac{v}{λ_{1}} = \frac{v}{4 (ℓ_{1} + x)}$

$∴$ $ν_{2} = \frac{v}{λ_{2}} = \frac{3 v}{4 (ℓ_{2} + x)}$

$Given ν_{1} = ν_{2}$

$∴$ $\frac{v}{4 (ℓ_{1} + x)} = \frac{3 v}{4 (ℓ_{2} + x)}$

$or, x = 0.025 m$

Q 3 :

The ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$ . In one experiment, the displacement of the wire is $y_{1} = A \sin (\frac{π x}{L}) \sin ω t$ and energy is $E_{1}$ and in another experiment its displacement is $y_{2} = A \sin (\frac{2 π x}{L}) \sin 2 ω t$ and energy is $E_{2}$ . Then [2001]

$E_{2} = E_{1}$
$E_{2} = 2 E_{1}$
$E_{2} = 4 E_{1}$
$E_{2} = 16 E_{1}$

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(3)

$Energy, E \propto A^{2} ν^{2} where A = amplitude and ν = frequency.$

$Also ω = 2 π ν \Rightarrow ω \propto ν$

$Experiment 1: Amplitude = A and ν_{1} = ν$

$Experiment 2: Amplitude = A and ν_{2} = 2 ν$

$\frac{E_{2}}{E_{1}} = \frac{A^{2} ν_{2}^{2}}{A^{2} ν_{1}^{2}} = 4$

$∴$ $E_{2} = 4 E_{1}$

Q 4 :

Two pulses in a stretched string whose centers are initially 8 cm apart are moving towards each other as shown in the figure. The speed of each pulse is 2 cm/s. After 2 seconds, the total energy of the pulses will be [2001]

zero
purely kinetic
purely potential
partly kinetic and partly potential

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(2)

The speed of each pulse is 2 cm/s and initially two pulses are 8 cm apart and moving towards each other.

After two seconds pulses will overlap each other.

According to the superposition principle the string will not have any distortion and will be straight.

Hence there will be no P.E. The total energy will be only kinetic.

Q 5 :

An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the X-axis. They are pulled from the equilibrium position in opposite directions along the X-axis by a small angular amplitude $θ_{0} = \cos^{- 1} (0.9)$ and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is 330 m/s, the maximum variation in the frequency (in Hz) as measured by the receiver (Take the acceleration due to gravity $g = 10 {m/s}^{2}$ ) is _______. [2025]

(32)

$From Doppler's effect,$

$f_{\max} = \frac{v + v'}{v - v'} f$

$f_{\min} = \frac{v - v'}{v + v'} f$

$Δ f_{\max} = f_{\max} - f_{\min} = \frac{v + v'}{v - v'} f - \frac{v - v'}{v + v'} f$

$= \frac{{(v + v')}^{2} - {(v - v')}^{2}}{v^{2} - v'^{2}} f$

$Δ f_{\max} = \frac{4 v v'}{v^{2} - v'^{2}} f$

$\cos θ_{0} = 1 - \frac{θ_{0}^{2}}{2} = 0.9$

$\frac{θ_{0}^{2}}{2} = 0.1 \Rightarrow θ_{0} = \sqrt{0.2} = \frac{1}{\sqrt{5}}$

$v' = ℓ θ_{0} ω$ $(ω = angular frequency)$

$= ℓ θ_{0} \sqrt{\frac{g}{ℓ}} = θ_{0} \sqrt{g ℓ}$

$v' = \frac{1}{\sqrt{5}} \sqrt{10 \times 8} = 4$

$Putting value of v = 330 m/s, v' = 4 m/s and f = 660 Hz in equation (i)$

$Δ f_{\max} = \frac{4 \times 330 \times 4 \times 660}{330^{2} - 4^{2}} = \frac{16 \times 330 \times 660}{334 \times 326} \approx 32$

Q 6 :

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$ , is emitting a sound of frequency $f_{s}$ . The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_{0}$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_{s}$ , will the sound reaching the pipe lead to a resonance? [2021]

$u = 0.8 v$ and $f_{s} = f_{0}$
$u = 0.8 v$ and $f_{s} = 2 f_{0}$
$u = 0.8 v$ and $f_{s} = 0.5 f_{0}$
$u = 0.5 v$ and $f_{s} = 1.5 f_{0}$

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Select one or more options

(1, 4)

$From Doppler's effect$

$f = f_{s} (\frac{v}{v - u})$

$As the pipe is closed at one end$ $∴$ $For resonance,$

$f = f_{0} (\frac{v}{v - u}) = n f_{0}$ where $n =$ odd integer

$(1) For, u = 0.8 v and f_{s} = f_{0},$

$f = f_{0} (\frac{v}{v - 0.8 v}) = 5 f_{0}$

$(2) For, u = 0.8 v and f_{s} = 2 f_{0}$ $\Rightarrow f = 2 f_{0} (\frac{v}{v - 0.8 v}) = 10 f_{0}$

$(3) For, u = 0.8 v and f_{s} = 0.5 f_{0}$

$f = 0.5 f_{0} (\frac{v}{v - 0.8 v}) = 2.5 f_{0}$

$(4) For u = 0.5 v and f_{s} = 1.5 f_{0}$

$f = 1.5 f_{0} (\frac{v}{v - 0.5 v}) = 3 f_{0}$

Q 7 :

Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at $60 km/h$ along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let $ν (t)$ represent the beat frequency measured by a person sitting in the car at time $t$ . Let $ν_{P}$ , $ν_{Q}$ and $ν_{R}$ be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is $330 {m s}^{- 1}$ . Which of the following statement(s) is(are) true regarding the sound heard by the person? [2016]

$ν_{P} + ν_{R} = 2 ν_{Q}$
The rate of change in beat frequency is maximum when the car passes through Q.
The plot below represents schematically the variation of beat frequency with time.
The plot below represents schematically the variation of beat frequency with time.

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Select one or more options

(1, 2, 3)

$(1) ν_{P} = (ν_{N} - ν_{M}) [\frac{v + v_{c} \cos θ}{v}] = 121 - 118 [\frac{v + v_{c} \cos θ}{v}]$

$ν_{Q} = (ν_{N} - ν_{M}) = 121 - 118 = 3$

$ν_{R} = (ν_{N} - ν_{M}) [\frac{v - v_{c} \cos θ}{v}] = (121 - 118) [\frac{v - v_{c} \cos θ}{v}]$

$∴$ $ν_{P} + ν_{R} = 2 ν_{Q}$

In general, when the car is passing through A,

$ν = 3 [\frac{v + v_{c} \cos α}{v}]$ ...(i)

$∴$ $\frac{d ν}{d α} = - 3 [\frac{v_{c} \sin α}{v}]$ $| \frac{d ν}{d α} |$ $is maximum when \sin α = 1$

i.e., $α = 90 °$ $(at Q)$

From Eq. (i), $\frac{d ν}{d t} = \frac{3 v_{c}}{v} (- \sin α) \frac{d α}{d t}$ ....(ii)

Also, $\tan α = \frac{10}{x}$ $∴$ $\sec^{2} α \frac{d α}{d t} = - \frac{10}{x^{2}} \frac{d x}{d t}$

$∴$ $\frac{d α}{d t} = \frac{- 10 v}{x^{2} \sec^{2} α}$ ...(iii)

From Eqs. (ii) and (iii),

$\frac{d ν}{d t} = - \frac{3 v_{c}}{v} \sin α (\frac{- 10 v}{x^{2} \sec^{2} α}) = \frac{30 v_{c} \sin α}{x^{2} \sec^{2} α}$

$∴$ $\frac{d ν}{d t} = \frac{30 v_{c} \sin α}{{(10 \cot α)}^{2} \sec^{2} α} = 0.3 v_{c} \sin^{3} α$

At $α = 90 °$

$\frac{d ν}{d t} = 0.3 v_{c}$

$∴$ ${(\frac{d ν}{d t})}_{\max} = 0.3 v_{c}$

$\frac{d ν}{d t} = \max$

Q 8 :

A person blows into the open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe. [2012]

a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is open.
a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is open.
a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed.
a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed.

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Select one or more options

(2, 4)

When a sound pulse is reflected from the open end of a pipe, its phase changes by 180°. A high-pressure pulse i.e., compression is reflected as a low-pressure pulse i.e., rarefaction.

When a sound pulse is reflected through a rigid boundary (closed end of a pipe), no phase change occurs. Therefore, a high-pressure pulse is reflected as a high-pressure pulse.

Q 9 :

A student performed the experiment to measure the speed of sound in air using the resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air-column is the second resonance. Then, [2009]

the intensity of the sound heard at the first resonance was more than that at the second resonance
the prongs of the tuning fork were kept in a horizontal plane above the resonance tube
the amplitude of vibration of the ends of the prongs is typically around 1 cm
the length of the air-column at the first resonance was somewhat shorter than 1/4 of the wavelength of the sound in air.

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Select one or more options

(1, 4)

The length of the air column at the first resonance is somewhat shorter than $\frac{1}{4}$ th of the wavelength of the sound in air due to end correction (e).

$ℓ + e = \frac{λ}{4}$ $\Rightarrow ℓ = \frac{λ}{4} - e$

Hence, at second resonance the length of the air column is more as compared to first resonance. Now, longer the length of air column, more is the absorption of energy and lesser is the intensity of sound heard.

Q 10 :

Two trains A and B 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 $f_{1}$ = 800 Hz to $f_{2}$ = 1120 Hz, as shown in the figure. The spread in the frequency (highest frequency − lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s. [2007]

Q. The speed of sound of the whistle is

340 m/s for passengers in A and 310 m/s for passengers in B
360 m/s for passengers in A and 310 m/s for passengers in B
310 m/s for passengers in A and 360 m/s for passengers in B
340 m/s for passengers in both the trains.

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(2)

The speed of sound depends on the frame of reference of the observer.

$V_{S A} = 340 + 20 = 360 m/s$

and $V_{S B} = 340 - 30 = 310 m/s$

Q 11 :

Two trains A and B 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 $f_{1}$ = 800 Hz to $f_{2}$ = 1120 Hz, as shown in the figure. The spread in the frequency (highest frequency − lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s. [2007]

Q. The distribution of the sound intensity of the whistle as observed by the passengers in train A is best represented by

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(1)

There is no relative motion between the source and the observer for the passengers in train A. Since all the passengers in train A are moving with a velocity of 20 m/s, therefore the distribution of sound intensity of the whistle by the passengers in train A is uniform.

Q 12 :

Two trains A and B 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 $f_{1}$ = 800 Hz to $f_{2}$ = 1120 Hz, as shown in the figure. The spread in the frequency (highest frequency − lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s. [2007]

Q. The spread of frequency as observed by the passengers in train B is

310 Hz
330 Hz
350 Hz
290 Hz

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(1)

For the passengers in train B, the source is approaching with velocity $20 m/s$ and the observer is receding with velocity $30 m/s$

$ν' = ν_{1} [\frac{v - v_{0}}{v - v_{s}}] = 800 [\frac{340 - 30}{340 - 20}] = 800 \times \frac{31}{32}$

$ν'' = ν_{2} [\frac{v - v_{0}}{v - v_{s}}] = 1120 \times \frac{31}{32}$

$∴$ $ν'' - ν' = (1120 - 800) \times \frac{31}{32} = 320 \times \frac{31}{32} = 310 Hz$

Q 13 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. Find the number of times intensity is maximum in time interval of 1 sec.

4
6
8
10

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(1)

Beat frequency $=$ $| f_{1} - f_{2} |$ $= (50 - 46) = 4 s^{- 1}$

One beat frequency consists of one maximum and one minimum.

So, the number of maxima is $f = 4 .$

Q 14 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. The wave velocity of louder sound is

100 m/s
192 m/s
200 m/s
96 m/s

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(3)

Wave velocity $= \frac{ω}{k} = \frac{100 π}{0.5 π} = 200 m/s$

Q 15 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. The number of times $y_{1} + y_{2} = 0$ at $x = 0$ in 1 sec is

100
46
192
96

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(4)

The given equations $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ represent two progressive waves travelling in the same direction along the $x$ -axis with a slight difference in frequency.

Comparing with the standard equation

$y = A \cos (k x - ω t),$

we get

$ω_{1} = 100 π$ $\Rightarrow 2 π f_{1} = 100 π$ $\Rightarrow f_{1} = 50 Hz$

and

$k_{1} = 0.5 π$ $\Rightarrow \frac{2 π}{λ_{1}} = 0.5 π$ $\Rightarrow λ_{1} = 4 m$

Wave velocity $= λ_{1} f_{1}$ $= 200 m/s$ $[Alternatively, use v = \frac{ω}{k}]$

$ω_{2} = 92 π$ $\Rightarrow 2 π f_{2} = 92 π$ $\Rightarrow f_{2} = 46 Hz$

Therefore, beat frequency $= f_{1} - f_{2}$ $= 4 Hz$ and

$k_{2} = 0.46 π$ $\Rightarrow \frac{2 π}{λ_{2}} = 0.46 π$ $\Rightarrow λ_{2} = \frac{200}{46}$

Wave velocity $= \frac{200}{46} \times 46$ $= 200 m/s$

At $x = 0$ , $y_{1} + y_{2} = (A \cos 100 π t) + (A \cos 92 π t) = 0$

$\Rightarrow \cos 100 π t = - \cos 92 π t = \cos (- 92 π t)$ $= \cos [(2 n + 1) π - 92 π t]$

$\Rightarrow t = \frac{2 n + 1}{192}$

when $t = 0$ , $n = - \frac{1}{2}$ and when $t = 1$ , $n = \frac{191}{2} = 95.5$

$∴$ $In 1 second, y_{1} + y_{2} = 0 at x = 0 for 96 times.$

Topic Question Set

Q 4 :

Two pulses in a stretched string whose centers are initially 8 cm apart are moving towards each other as shown in the figure. The speed of each pulse is 2 cm/s. After 2 seconds, the total energy of the pulses will be [2001]

Q 8 :

A person blows into the open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe. [2012]

Q 13 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. Find the number of times intensity is maximum in time interval of 1 sec.

Q 14 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. The wave velocity of louder sound is

Q 15 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. The number of times $y_{1} + y_{2} = 0$ at $x = 0$ in 1 sec is

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Topic Question Set

Q 3 : The ends of a stretched wire of length L are fixed at x=0 and x=L. In one experiment, the displacement of the wire is y1=Asin(πxL)sinωt and energy is E1 and in another experiment its displacement is y2=Asin(2πxL)sin2ωt and energy is E2. Then [2001]

Q 4 : Two pulses in a stretched string whose centers are initially 8 cm apart are moving towards each other as shown in the figure. The speed of each pulse is 2 cm/s. After 2 seconds, the total energy of the pulses will be [2001]

Q 8 : A person blows into the open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe. [2012]

Q 13 : Waves y1=Acos(0.5πx-100πt) and y2=Acos(0.46πx-92πt) are travelling along the x-axis. (Here x is in m and t is in second) [2006] Q. Find the number of times intensity is maximum in time interval of 1 sec.

Q 14 : Waves y1=Acos(0.5πx-100πt) and y2=Acos(0.46πx-92πt) are travelling along the x-axis. (Here x is in m and t is in second) [2006] Q. The wave velocity of louder sound is

Q 15 : Waves y1=Acos(0.5πx-100πt) and y2=Acos(0.46πx-92πt) are travelling along the x-axis. (Here x is in m and t is in second) [2006] Q. The number of times y1+y2=0 at x=0 in 1 sec is

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Q 4 :

Two pulses in a stretched string whose centers are initially 8 cm apart are moving towards each other as shown in the figure. The speed of each pulse is 2 cm/s. After 2 seconds, the total energy of the pulses will be [2001]

Q 8 :

A person blows into the open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe. [2012]

Q 13 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. Find the number of times intensity is maximum in time interval of 1 sec.

Q 14 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. The wave velocity of louder sound is

Q 15 :

Waves $y_{1} = A \cos (0.5 π x - 100 π t)$ and $y_{2} = A \cos (0.46 π x - 92 π t)$ are travelling along the $x$ -axis. (Here $x$ is in $m$ and $t$ is in second) [2006]

Q. The number of times $y_{1} + y_{2} = 0$ at $x = 0$ in 1 sec is