**1.** A steel tube of length $1.00\ m$ is struck at one end. A person with his ear close to the other end hears the sound of the blow twice, one travelling through the body of the tube and the other through the air in the tube. Find the time gap between the two hearings. Use the table in the text for speeds of sound in various substances.

$V_{air} = 230\ m/s. V_{5} = 5200\ m/s$. Here $s = 7\ m$

So, $ t = t_{1}\ -\ t_{2} = \big(\frac{1}{330}\ -\ \frac{1}{5200}\big) = 2.75 \times 10^{-1}\ sec = 2.75\ ms$

**2.** At a prayer meeting, the disciples sing $JAI-RAM\ JAI-RAM$.
The sound amplified by a loudspeaker comes back after
reflection from a building at a distance of $80\ m$ from the
meeting. What maximum time interval can be kept
between one $JAI-RAM$ and the next $JAI-RAM$ so that the
echo does not disturb a listener sitting in the meeting.
Speed of sound in air is $320\ m/s$.

Here given $S = 80 m \times 2 = 160\ m$

$v = 320 m/s$

So minimum time interval will be $\\$ $t = \frac{5}{v} = \frac{160}{320} = 0.5$ seconds.

**3.** A man stands before a large wall at a distance of $50\cdot0\ m$
and claps his hands at regular intervals. Initially, the
interval is large. He gradually reduces the interval and
fixes it at a value when the echo of a clap merges with
the next clap. If he has to clap $10$ times during every
$3$ seconds, find the velocity of sound in air.

He ha to clap $10$ times in $3$ seconds. $\\$ So time interval between two clap = ($\frac{3}{10}$ second). $\\$ So the taken go the wall = $\big(\frac{3}{2} \times 10\big)$ = $\frac{3}{20}$ seconds. $\\$ = $333\ m/s$.

**4.** A person can hear sound waves in the frequency range
$20\ Hz$ to $20\ kHz$. Find the minimum and the maximum
wavelengths of sound that is audible to the person. The
speed of sound is $360\ m/s$.

a) For minimum wavelength $n = 20\ KHz$ $\\$ as $\big(\eta\ \infty\ \frac{1}{\lambda}\big)$

b) For minimum wavelength, $n = 20 KHz$ $\\$ $\therefore \lambda = \frac{360}{(20 \times 10^3)} = 18 \times 10^{-3}\ m = 18\ mm$ $\\$ $\Rightarrow x = (\frac{v}{n}) = \frac{360}{20} = 18\ m$

**5.** Sound waves from a loudspeaker spread nearly
uniformly in all directions if the wavelength of the sound
is much larger than the diameter of the loudspeaker.
(a) Calculate the frequency for which the wavelength of
sound in air is ten times the diameter of the speaker if
the diameter is $20\ cm$. (b) Sound is essentially
transmitted in the forward direction if the wavelength
is much shorter than the diameter of the speaker.
Calculate the frequency at which the wavelength of the
sound is one tenth of the diameter of the speaker
described above. Take the speed of sound to be $340\ m/s$.

a) For minimum wavelength $n = 20\ KHz$ $\\$ $\Rightarrow v = n\lambda \Rightarrow \lambda = \big(\frac{1450}{20 \times 10^3}\big) = 7.25\ cm$

b) for minimum wavelength $n$ should be minimum $\\$ $\Rightarrow v = n\lambda \Rightarrow \lambda = \frac{v}{n} \Rightarrow \frac{1450}{20} = 72.5\ m$

**6.** Ultrasonic waves of frequency $4.5\ MHz$ are used to
detect tumour in soft tissues. The speed of sound in
tissue is $1\cdot5\ km/s$ and that in air is $340\ m/s$. Find the
wavelength of this ultrasonic wave in air and in tissue.

According to the question, $\\$ a) $\lambda = 20\ cm \times 10 = 200\ cm = 2\ m$ $\\$ $v = 340\ m/s$ $\\$ So, $n = \frac{340}{2} = 170\ Hz$.

$N = \frac{v}{\lambda} \Rightarrow \frac{340}{2 \times 10^{-2}} = 17.00\ Hz = 17\ KH_{2}$ (Because $\lambda = 2 cm = 2 \times 10_{-2}\ m$)

**7.** The equation of a travelling sound wave is $y = 6\cdot0\ sin\ (600\ t - 1\cdot8 x)$ where $y$ is measured in $10^{-5}\ m$, $t$ in second and $x$ in metre. (a) Find the ratio of
the displacement amplitude of the particles to the
wavelength of the wave. (b) Find the ratio of the velocity
amplitude of the particles to the wave speed.

a) Given $V_{air} = 340\ m/s, n = 4.5 \times 10^{6}\ Hz$ $\\$ $\Rightarrow \lambda_{air} = \big(\frac{340}{4.5}\big) \times 10^{-6} = 7.36 \times 10^{-5}\ m$.

b) $V_{tissue} = 1500\ m/s \Rightarrow \lambda = \big(\frac{1500}{4.5}\big)\times 10^{-6} = 3.3 \times 10^{-4}\ m$.

**8.** A sound wave of frequency $100\ Hz$ is travelling in air.
The speed of sound in air is $350\ m/s$. (a) By how much
is the phase changed at a given point in $2\cdot5\ ms$ ?
(b) What is the phase difference at a given instant
between two points separated by a distance of $10\cdot0\ cm$
along the direction of propagation ?

Here given $r_{y} = 6.0 \times 10^{-5}\ m$ $\\$ a) Given $\frac{2\pi}{\lambda} = 1.8 \Rightarrow \lambda = (\frac{2\pi}{1.8})$

So, $\frac{r_{y}}{\lambda} = \frac {6.0 \times (1.8) \times 10^{-5}\ m/s}{2\pi} = 1.7 \times 10^{-5}\ m/s$

b) Let, velocity amplitude = $V_{y}$ $\\$ $V = \frac{dy}{dt} = 3600\ cos\ (600 t - 1.8) \times 10^{-5}\ m/s$

Here $V_{y} = 3600 \times 10^{-5}\ m/s$

Again, $\lambda = \frac{2\pi}{1.8}$ and $T = \frac{2\pi}{600} \Rightarrow$ wave speed $ = v = \frac{\lambda}{T} = \frac{600}{1.8} = \frac{1000}{3}\ m/s$

So the ratio of $\big(\frac{V_{y}}{v}\big) = \frac{3600 \times 3 \times 10^{-5}}{1000}$.

**9.** Two point sources of sound are kept at a separation of
$10\ cm$. They vibrate in phase to produce waves of
wavelength $5\cdot0\ cm$. What would be the phase difference
between the two waves arriving at a point $20\ cm$ from
one source (a) on the line joining the sources and (b) on
the perpendicular bisector of the line joining the sources ?

**10.** Calculate the speed of sound in oxygen from the
following data. The mass of $22\cdot4\ litre$ of oxygen at $STP$
$(T = 273\ K\ and\ p = 1\cdot0 \times 10^5 N/m^2)$ is $32\ g$, the molar
heat capacity of oxygen at constant volume is $C_{v} = 2\cdot5\ R$
and that at constant pressure is $C_{p} = 3\cdot5\ R$.

a) Here given $n = 100, v = 350\ m/s$

$\Rightarrow \lambda = \frac{v}{n} =\frac{350}{100} = 3.5\ m$.

In $2.5\ ms$, the distance travelled by the particle is given by

$\Delta{x} = 350 \times 2.5 \times 10^{-3}$

So, phase difference $\phi = \frac{2\pi}{\lambda} \times \Delta{x} \Rightarrow \frac{2\pi}{(350/100)} \times 350 \times 2.5 \times 10^{-3} = \big(\frac{\pi}{2}\big)$

b) In the second case, Given $\Delta{\eta} = 10\ cm = 10^{-1}\ m$

So, $\phi = \frac{2\pi}{x} \Delta{x} = \frac{2\pi \times 10^{-1}}{(350/100)} = \frac{2\pi}{35}$

**11.** The speed of sound as measured by a student in the
laboratory on a winter day is 340 m/s when the room
temperature is 17°C. What speed will be measured by
another student repeating the experiment on a day when
the room temperature is 32°C ?

**12.** At what temperature will the speed of sound be double
of its value at 0°C ?

12 None

Solutions**13.** The absolute temperature of air in a region linearly
increases from T, to T2 in a space of width d. Find the
time taken by a sound wave to go through the region in
terms of T„ T2, d and the speed v of sound at 273 K.
Evaluate this time for T 1 = 280 K, T2 = 310 K, d = 33 m
and v = 330 m/s.

13 None

Solutions**14.** Find the change in the volume of PO litre kerosene when
it is subjected to an extra pressure of 2.0 x 10 5 N/m 2
from the following data. Density of kerosene
= 800 kg/m 3 and speed of sound in kerosene = 1330 m/s.

**15.** Calculate the bulk modulus of air from the following
data about a sound wave of wavelength 35 cm travelling
in air. The pressure at a point varies between
(P0 x 10 5 ± 14) Pa and the particles of the air vibrate
in simple harmonic motion of amplitude 5'5 x 10 -6 m.

15 None

Solutions**16.** A source of sound operates at 2.0 kHz, 20 W emitting
sound uniformly in all directions. The speed of sound in
air is 340 m/s and the density of air is P2 kg/m 3.
(a) What is the intensity at a distance of 6.0 m from the
source ? (b) What will be the pressure amplitude at this
point ? (c) What will be the displacement amplitude at
this point ?

16 None

Solutions**17.** The intensity of sound from a point source is
PO x 10 -6 W/m 2 at a distance of 5.0 m from the source.
What will be the intensity at a distance of 25 m from
the source ?

**18.** The sound level at a point 5.0 m away from a point
source is 40 dB. What will be the level at a point 50 m
away from the source ?

18 None

Solutions**19.** If the intensity of sound is doubled, by how many
decibels does the sound level increase ?

19 None

Solutions**20.** Sound with intensity larger than 120 dB appears painful
to a person. A small speaker delivers 2.0 W of audio
output. How close can the person get to the speaker
without hurting his ears ?

20 None

Solutions**21.** If the sound level in a room is increased from 50 dB to
60 dB, by what factor is the pressure amplitude
increased ?

21 None

Solutions**22.** The noise level in a class-room in absence of the teacher
is 50 dB when 50 students are present. Assuming that
on the average each student outputs same sound energy
per second, what will be the noise level if the number
of students is increased to 100 ?

22 None

Solutions**23.** In a Quincke's experiment the sound detected is changed
from a maximum to a minimum when the sliding tube
is moved through a distance of 2'50 cm. Find the
frequency of sound if the speed of sound in air is
340 m/s.

**24.** In a Quincke's experiment, the sound intensity has a minimum value I at a particular position. As the sliding tube is pulled out by a distance of $16\cdot5\ mm$, the intensity increases to a maximum of $9I$. Take the speed of sound in air to be $330\ m/s$. (a) Find the frequency of the soundsource. (b) Find the ratio of the amplitudes of the two waves arriving at the detector assuming that it does not change much between the positions of minimum intensity and maximum intensity.

24 None

Solutions**25.** Two audio speakers are kept some distance apart and
are driven by the same amplifier system. A person is
sitting at a place $6\cdot0\ m$ from one of the speakers and
$6\cdot4\ m$ from the other. If the sound signal is continuously
varied from $500\ Hz$ to $5000\ Hz$, what are the frequencies
for which there is a destructive interference at the place
of the listener? Speed of sound in air = $320\ m/s$.

**26.** A source $S$ and a detector $D$ are placed at a distance $d$
apart. A big cardboard is placed at a distance $\sqrt{2d}$ from
the source and the detector as shown in figure $(16-E2)$.
The source emits a wave of wavelength = $d/2$ which is
received by the detector after reflection from the
cardboard. It is found to be in phase with the direct
wave received from the source. By what minimum
distance should the cardboard be shifted away so that
the reflected wave becomes out of phase with the direct
wave ?

**27.** Two stereo speakers are separated by a distance of
$2\cdot40\ m$. A person stands at a distance of $3\cdot20\ m$ directly
in front of one of the speakers as shown in figure $(16-E3)$.
Find the frequencies in the audible range $(20 - 2000 Hz)$
for which the listener will hear a minimum sound
intensity. Speed of sound in air = $320\ m/s$.

27 None

Solutions**28.** Two sources of sound, $S1$ and $S2$ , emitting waves of
equal wavelength $20\cdot0\ cm$, are placed with a separation
of $20\cdot0\ cm$ between them. A detector can be moved on
a line parallel to $S_{1}$ $S_{2}$ and at a distance of $20\cdot0\ cm$ from
it. Initially, the detector is equidistant from the two
sources. Assuming that the waves emitted by the sources
are in phase, find the minimum distance through which
the detector should be shifted to detect a minimum of
sound.

28 None

Solutions**29.** Two speakers $S_{1}$ and $S_{2}$ , driven by the same amplifier,
are placed at $y$ = $1\cdot0\ m$ and $y$ = $-1\cdot0\ m$ $(figure 16-E4)$. The
speakers vibrate in phase at $600\ Hz$. A man stands at
a point on the $X$-axis at a very large distance from the
origin and starts moving parallel to the $Y$-axis. The
speed of sound in air is $330\ m/s$. (a) At what angle $\theta$
will the intensity of sound drop to a minimum for the
first time ? (b) At what angle will he hear a maximum
of sound intensity for the first time ? (c) If he continues
to walk along the line, how many more maxima can he
hear ?

**30.** Three sources of sound $S_{1}\ S_{2}$ and $S_{3}$ of equal intensity
are placed in a straight line with $S_{1}S_{2}$ = $S_{2}S_{3}$, $(figure\
16-E5)$. At a point $P$, far away from the sources, the
wave coming from $S_{2}$ is $120°$ ahead in phase of that from
$S_{1}$. Also, the wave coming from $S_{3}$ is $120°$ ahead of that
from $S_{2}$. What would be the resultant intensity of sound
at $P$?

30 None

Solutions**31.** Two coherent narrow slits emitting sound of wavelength $\lambda$
in the same phase are placed parallel to each other at
a small separation of $2\lambda$. The sound is detected by
moving a detector on the screen $\sum$ at a distance
$D(\gg\lambda)$ from the slit $S_{1}$ as shown in $figure\ (16-E6)$. Find
the distance $x$ such that the intensity at $P$ is equal to
the intensity at $O$.

31 None

Solutions**32.** $Figure (16-E7)$ shows two coherent sources $S_{1}$ and $S_{2}$
which emit sound of wavelength $\lambda$ in phase. The
separation between the sources is $3\lambda$. A circular wire of
large radius is placed in such a way that $S_{1}S_{2}$ lies in its
plane and the middle point of $S_{1}S_{2}$ is at the centre of the wire. Find the angular positions $\theta$ on the wire for which constructive interference takes place.

32 None

Solutions**33.** Two sources of sound $S_{1}$ and $S_{2}$ vibrate at same
frequency and are in phase $(figure 16-E8)$. The intensity
of sound detected at a point P as shown in the figure is
$I_{0}$. (a) If $\theta$ equals $45°$, what will be the intensity of sound
detected at this point if one of the sources is switched
off ? (b) What will be the answer of the previous part if $\theta$
= $60°$ ?

**34.** Find the fundamental, first overtone and second
overtone frequencies of an open organ pipe of length
$20\ cm$. Speed of sound in air is $340\ m/s$.

**35.** A closed organ pipe can vibrate at a minimum frequency
of $500\ Hz$. Find the length of the tube. Speed of sound
in air = $340\ m/s$.

35 None

Solutions**36.** In a standing wave pattern in a vibrating air column,
nodes are formed at a distance of $4\cdot0\ cm$. If the speed
of sound in air is $328\ m/s$, what is the frequency of $73$
the source ?

36 None

Solutions**37.** The separation between a node and the next antinode
in a vibrating air column is $25\ cm$. If the speed of sound
in air is $340\ m/s$, find the frequency of vibration of the
air column.

37 None

Solutions**38.** A cylindrical metal tube has a length of $50\ cm$ and is
open at both ends. Find the frequencies between
$1000\ Hz$ and $2000\ Hz$ at which the air column in the
tube can resonate. Speed of sound in air is $340\ m/s$.