# Concept Of Physics Sound Waves

#### H C Verma

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 ?

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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.

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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.

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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 ?

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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 ?

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

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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 ?

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

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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 ?

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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.

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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 = $320\ m/s$.

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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.

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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 $P$?

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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 $O$.

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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.

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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 = $340\ m/s$.

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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 ?

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.
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 $340\ m/s$.