Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com. PROPAGATION OF SOUND WAVES : Sound is a mechanical three dimensional and longitudinal wae that is created by a ibrating source such as a guitar string, the human ocal cords, the prongs of a tuning fork or the diaphragm of a loudspeaker. Being a mechanical wae, sound needs a medium haing properties of inertia and elasticity for its propagation. Sound waes propagate in any medium through a series of periodic compressions and rarefactions of pressure, which is produced by the ibrating source. Consider a tuning fork producing sound waes. A B Undisturbed tuning fork normal atmospheric pressure When Prong B moes outward towards right it compresses the air in front of it, causing the pressure to rise slightly. The region of increased pressure is called a compression pulse and it traels away from the prong with the speed of sound A B compression pulse normal atmospheric pressure After producing the compression pulse, the prong B reerses its motion and moes inward. This drags away some air from the region in front of it, causing the pressure to dip slightly below the normal pressure. This region of decreased pressure is called a rarefaction pulse. Following immediately behind the compression pulse, the rarefaction pulse also traels away from the prong with the speed of sound. A SOUND WAVES B......... rarefaction pulse } compression pulse If the prongs ibrate in SHM, the pressure ariations in the layer close to the prong also aries simple harmonically and hence increase in pressure aboe normal alue can be written as δp δp sin ωt where δp is the maximum increase in pressure aboe normal alue. As this disturbance trael towards right with wae elocity, the excess pressure at any position x at time t will be gien by δp δp sin ω(t x/ (. Using p δp, p δp, the aboe equation of sound wae can be written as : p p sin ω(t x/ (. Ex. The equation of a sound wae in air is gien by p (. sin [(3 t (9. x ], where all ariables are in S.I. units. (a Find the frequency, waelength and the speed of sound wae in air. (b If the equilibrium pressure of air is. 5 N/m, what are the maximum and minimum pressures at a point as the wae passes through that point? (a Comparing with the standard form of a traelling wae p p sin [w (t x/] we see that ω 3 s. The frequency is Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com (b ω 3 f Hz π π Also from the same comparison, ω/ 9. m. ω or, 9.m 3s 9.m m/s 3 / 3m / s π The waelength is λ m f 3 / πhz 9 The pressure amplitude is p. N/m. Hence, the maximum and minimum pressures at a point in the wae motion will be (. 5 ±. N/m.. FREQUENCY AND PITCH OF SOUND WAVES Each cycle of a sound wae includes one compression and one rarefaction, and frequency is the number of cycles per second that passes by a gien location. This is normally equal to the frequency of ibration of the (tuning fork source producing sound. If the source, ibrates in SHM of a single frequency, sound produced has a single frequency and it is called a pure tone.. Howeer a sound source may not always ibrate in SHM (this is the case with most of the common sound sources e.g. guitar string, human ocal card, surface of drum etc. and hence the pulse generated by it may not hae the shape of a sine wae. But een such a pulse can be obtained by superposition of a large number of sine waes of different frequency and amplitudes*. We say that the pulse contain all these frequencies. A normal person hears all frequencies between & KHz. This is a subjectie range (obtained experimentally which may ary slightly from person to person. The ability to hear the high frequencies decreases with age and a middle-age person can hear only upto to 4 KHz. Sound can be generated with frequency below Hz called infrasonic sound and aboe Hz called ultrasonic sound. Een through humans cannot hear these frequencies, other animals may. For instance Rhinos communicate through infrasonic frequencies as low as 5Hz, and bats use ultrasonic frequencies as high as KHz for naigating. Frequency as we hae discussed till now is an objectie property measured its units of Hz and which can be assigned a unique alue. Howeer a person s perception of frequency is subjectie. The brain interprets frequency primarily in terms of a subjectie quality called Pitch. A pure note of high frequency is interpreted as high-pitched sound and a pure note of low frequency as low-pitched sound * (This is possible for a pulse of any shape, according to an important theorem of mathematics, the Fourier theorem; the description of which is outside the scope of this work. Ex. A wae of waelength 4 mm is produced in air and it traels at a speed of 3 m/s. Will it be audible? From the relation νλ, the frequency of the wae is 3 m / s ν λ 3 75 Hz. 4 m This is much aboe the audible range. It is an ultrasonic wae and will not be audible to humans, but it will be audible to bats. 3. PRESSURE AMPLITUDE AND DISPLACEMENT AMPLITUDE : We can describe sound waes either in terms of excess pressure (equation. or in terms of the longitudinal displacement suffered by the particles of the medium. If s s sin ω(t x/ represents a sound wae where, s displacement of medium particle from its mean position at x, then it can be proed that P P sin {ω(t x/ + π/}... (3. represents that same sound wae where, P is excess pressure at position x, oer and aboe the aerage atmospheric pressure and the pressure amplitude P is gien by Bωs P BKs...(3.3 (B Bulk modulus of the medium, K angular wae number Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Note from equation (3. and (3. that the displacement of a medium particle and excess pressure Ex. 3 at any position are out of phase by π. Hence a displacement maxima corresponds to a pressure minima and ice-ersa. A sound wae of waelength 4 cm traels in air. If the difference between the maximum and minimum pressures at a gien point is. 3 N/m, find the amplitude of ibration of the particles of the medium. The bulk modulus of air is.4 5 N/m. The pressure amplitude is. N/m p 3 N/m. The displacement amplitude s is gien by p B k s or, s 3 p p Bk πλ B 3 N/m (4 4 π 4 6.6 Å N/m m Å 7π 4. SPEED OF SOUND WAVES : 4. Velocity of sound waes in a linear solid medium is gien by Y...(4. ρ where Y young s modulus of elasticity and ρ density. 4.. Velocity of sound waes in a fluid medium (liquid or gas is gien by B...(4. ρ where, ρ density of the medium and B Bulk modulus of the medium gien by, dp B V dv...(4.3 Newton s formula : Newton assumed propagation of sound through a gaseous medium to be an isothermal process. PV constant dp P dv V and hence B P and thus he obtained for elocity of sound in a gas, P RT where M molar mass ρ M Laplace s correction : Later Laplace established that propagation of sound in a gas is not an isothermal but an adiabatic process and hence dp dv γ P V where, B V dv dp gp and hence speed of sound in a gas, γ PV constant Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 3 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Ex. 4 γp γrt ρ M... (4.4 4.3 Factors affecting speed of sound in atmosphere. (a Effect of temperature : as temperature (T increases elocity ( increases. For small change in temperature aboe room temperature increases linearly by.6 m/s for eery C rise in temp. (b Effect of humidity : With increase in humidity density decreases. This is because the molar mass of water apour is less than the molar mass of air. (c Effect of pressure : The speed of sound in a gas is gien by γp γrt ρ M So at constant temperature, if P changes then ρ also changes in such a way that P/ρ remains constant. Hence pressure does not hae any effect on elocity of sound as long as temperature is constant. The constant γ for oxygen as well as for hydrogen is.4. If the speed of sound in oxygen is 45 m/s, what will be the speed of hydrogen at the same temperature and pressure? γrt M since temperature, T is constant, \ (H (O M M O H 3 4 Þ (H 4 45 8 m/s Ans. Aliter : The speed of sound in a gas is gien by u γp. At STP,.4 litres of oxygen has a mass of 3 g ρ whereas the same olume of hydrogen has a mass of g. Thus, the density of oxygen is 6 times the density of hydrogen at the same temperature and pressure. As γ is same for both the gases, or, f f (hydrogen (oxygen ρ ρ (oxygen (hydrogen f (hydrogen 4f (oxygen 4 45 m/s 8 m/s. Ans. 5. INTENSITY OF SOUND WAVES : Like any other progressie wae, sound waes also carry energy from one point of space to the other. This energy can be used to do work, for example, forcing the eardrums to ibrate or in the extreme case of a sonic boom created by a supersonic jet, can een cause glass panes of windows to crack. The amount of energy carried per unit by a wae is called its power and power per unit area held perpendicular to the direction of energy flow is called intensity. For a sound wae traelling along positie x-axis described by the equation. s s sin ω(t x/ P P cos ωt(t x/v the aerage intensity at position x is gien by ω s < > B P B Substituting B ρ, intensity can also be expressed as P ρ... (5.... (5. a If the source is a point source then I α and s α and s sin (ωt kr + θ r r r Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 4 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
If a sound source is a line source then I α r and s α s a sin (ωt kr + θ r and r For a Hz tone, the smallest sound intensity that a human ear can detect is watt./m. On the other hand, continuous exposure to intensities aboe W/m can result in permanent hearing loss. Ex. 5 Ex. 6 Ex. 7 π The pressure amplitude in a sound wae from a radio receier is. 3 N/m and the intensity at a point is 6 W/m. If by turning the Volume knob the pressure amplitude is increased to 3 3 N/m, ealuate the intensity. The intensity is proportional to the square of the pressure amplitude. Thus, p' p p' or p 3. 6 W/m.5 6 W/m. A microphone of cross-sectional area.4 cm is placed in front of a small speaker emitting p W of sound output. If the distance between the microphone and the speaker is. m, how much energy falls on the microphone in 5. s? The energy emitted by the speaker in one second is p J. Let us consider a sphere of radius. m centered at the speaker. The energy p J falls normally on the total surface of this sphere in one second. The energy falling on the area.4 cm of the microphone in one second.4 cm 4 π (.m p J 5 6 J. The energy falling on the microphone in 5. is 5 6 J 5 5 µj. Find the amplitude of ibration of the particles of air through which a sound wae of intensity 8. 6 W/m and frequency 5. khz is passing. Density of air. kg/m 3 and speed of sound in air 33 m/s. The relation between the intensity of sound and the displacement amplitude is ω sb, where B ρ and ω πν π s n ρ or, s (5. or, 8. 6 s 6 s 6.4 nm. W /m 3 (.kg/m (33m / s π ν ρ 6 DECIBEL SCALE : The decibel (db is a logarithmic scale used for comparing two sound intensities. The intensity leel β described in terms of decibels is defined as β log (db Here is the threshold intensity of hearing for human ear i.e. watt/m. Note that intensity leel β is a dimensionless quantity and is not same as intensity expressed in W/m. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 5 Ex. 8 If the intensity is increased by a factor of, by how many decibels is the intensity leel increased. Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com Let the initial intensity by I and the intensity leel be β and when the intensity is increased by times, the intensity leel increases to β. Then β log (I / I and β log ( / Thus, β β log ( / log 3 db. Ex. 9 A bird is singing on a tree. A person approaches the tree and perceies that the intensity has increased by db. Find the ratio of initial and final separation between the man and the bird. b log b log b b log r for point source r r or log Ans. Ex.. The sound leel at a point is increased by 4 db. By what factor is the pressure amplitude increased? The sound leel in db is β log. If β and β are the sound leels and and are the intensities in the two cases, β β or, 4 log or, log log 4. As the intensity is proportional to the square of the pressure amplitude, we hae p p. 6. INTERFERENCE OF SOUND WAVES : If p p m sin (ωt kx + θ and p p m sin (ωt kx + θ resultant excess pressure at point O is p p + p Þ p p sin (ωt kx + θ where, p m (i For constructie interference φ nπ p p m + p m (ii For destructie interference φ (n+ π p p m p m pm + pm + pm p cos φ, φ k (x x + (θ θ...(6. π If φ is only due to path difference, then φ x, and λ Condition for constructie interference : x nλ, n, ±, ± Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 6 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Condition for destructie interference : x (n + λ, n, ±, ± from equation (6. P m m m m P + P + P P cosφ Since intensity, (Pressure amplitude, we hae, for resultant intensity, + + cos φ...(6. If ( + cos φ 4 cos φ Hence in this case, for constructie interference : φ, π, 4π... and max 4 and for destructie interference :φ π, 3π... and min...(6.3 6. Coherence : Two sources are said to be coherent if the phase difference between them does not change with time. In this case their resultant intensity at any point in space remains constant with time. Two independent sources of sound are generally incoherent in nature, i.e. phase difference between them changes with time and hence the resultant intensity due to them at any point in space changes with time. Ex. Figure shows a tube structure in which a sound signal is sent from one end and is receied at the other end. The semicircular part has a radius of. cm. The frequency of the sound source can be aried from to khz. Find the frequencies at which the ear perceies maximum intensity. The speed of sound in air 34 m/s. The sound wae bifurcates at the junction of the straight and the semicircular parts. The wae through the straight part traels a distance p cm and the wae through the cured part traels a distance p π cm 3.4 cm before they meet again and trael to the receier. The path difference between the two waes receied is, therefore. p p p 3.4 cm cm.4 cm 33 m/ s The waelength of either wae is. For constructie interference, p nλ, where n is an υ υ integer. or, Dp n. ν Þ n Þ n. p n.34 (.4 3 n Thus, the frequencies within the specified range which cause maximum of intensity are 3, 3 and 3 3 Hz Ex. A source emitting sound of frequency 65 Hz is placed in front of a wall at a distance of m from it. A detector is also placed in front of the wall at the same distance from it. Find the minimum distance between the source and the detector for which the detector detects a maximum of sound. Speed of sound in air 33 m/s. The situation is shown in figure. Suppose the detector is placed at a distance of x meter from the source. The direct wae receied from the source traels a distance of x meter. The wae reaching the detector after reflection from the wall has traelled a distance of [( + x /4] / metre. The path difference between the two waes is ( / 4 x + x 4 metre. Constructie interference will take place when λ, λ,... The minimum distance x for a maximum Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 7 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com corresponds to λ...(i υ 33m / s The waelength is λ m. 65s Thus, by (i or, ( / x 4 + x / x 4 + x + 4 x or, 4 + + + x 4 4 or, 3 x. Thus, the detector should be placed at a distance of 3 m from the source. Note that there is no abrupt phase change. x 7. REFLECTION OF SOUND WAVES : Reflection of sound waes from a rigid boundary (e.g. closed end of an organ pipe is analogous to reflection of a string wae from rigid boundary; reflection accompanied by an inersion i.e. an abrupt phase change of p. This is consistent with the requirement of displacement amplitude to remain zero at the rigid end, since a medium particle at the rigid end can not ibrate. As the excess pressure and displacement corresponding to the same sound wae ary by π/ in term of phase, a displacement minima at the rigid end will be a point of pressure maxima. This implies that the reflected pressure wae from the rigid boundary will hae same phase as the incident wae, i.e., a compression pulse is reflected as a compression pulse and a rarefaction pulse is reflected as a rarefaction pulse. On the other hand, reflection of sound wae from a low pressure region (like open end of an organ pipe is analogies to reflection of string wae from a free end. This point corresponds to a displacement maxima, so that the incident & reflected displacement wae at this point must be in phase. This would imply that this point would be a minima for pressure wae (i.e. pressure at this point remains at its aerage alue, and hence the reflected pressure wae would be out of phase by π with respect to the incident wae. i.e. a compression pulse is reflected as a rarefaction pulse and ice-ersa. 8. LONGITUDINAL STANDING WAVES : Two longitudinal waes of same frequency and amplitude traelling in opposite directions interfere to produce a standing wae. If the two interfering waes are gien by p p sin (ωt kx and p p sin (ωt + kx + φ then the equation. of the resultant standing wae would be gien by p p + p p cos (kx + φ sin (ωt + φ p p sin (ωt + φ... (8. This is equation of SHM in which the amplitude p depends on position as p p cos (kx + φ...(8. Points were pressure remains permanently at its aerage alue; i.e. pressure amplitude is zero is called a pressure node, and the condition for a pressure node would be gien by p i.e. cos(kx + φ Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 8 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
i.e. kx + φ nπ ± π, n,,,...... (8.3 Similarly points where pressure amplitude is maximum is called a pressure antinode and condition for a pressure antinode would be gien by p ± p i.e. cos(kx + φ ± or (kx + φ nπ, n,,,......(8.4 Note that a pressure node in a standing wae would correspond to a displacement antinode; and a pressure anti-node would correspond to a displacement node. (when we label eqn (8. as SHM, what we mean is that excess pressure at any point aries simpleharmonically. If the sound waes were represented in terms of displacement waes, then the equation of standing wae corresponding to (8. would be s s cos(ωt + φ where s s sin (kx + φ This can be easily obsered to be an equation of SHM. It represents the medium particles moing simple harmonically about their mean position at x. Ex. 3. A certain organ pipe resonates in its fundamental mode at a frequency of khz in air. What will be the fundamental frequency if the air is replaced by hydrogen at the same temperature? (take molar mass of air 9 g Suppose the speed of sound in hydrogen is h and that in air is a. The fundamental frequency of an organ pipe is proportional to the speed of sound in the gas contained in it. If the fundamental frequency with hydrogen in the tube is n, we hae or, ν Hz ν khz 9 h a M M Air H Þ n (Since both air and H are diatomic, y is same for both 9 khz. Ans. Ex. 4 A tube open at only one end is cut into two tubes of non equal lengths. The piece open at both ends has of fundamental frequency of 45 Hz and of 675 Hz. What is the st oertone frequency of the original tube. 45 l 675 4l length of original tube (l + l \ its first obtained frequency, n 3 ( l + l 9 3 + 675 4 3(7 9 3 7 5 Hz. 7 + 9 4 Ex. 5 The range audible frequency for humans is Hz to, Hz. If speed of sound in air is 336 m/s. What can be the maximum and minimum length of a musical instrument, based on resonance pipe. For an open pipe, f n l Þ l Similarly for a closed pipe, l.n f (n + 4f Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 9 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com l min l max 4f max f min 336 (n + min 4. mm 4 n max n max 8.4 (m n max clearly there is no upper limit on the length of such an musical instrument. 9. VIBRATION OF AIR COLUMNS Standing waes can be set up in air-columns trapped inside cylindrical tubes if frequency of the tuning fork sounding the air column matches one of the natural frequency of air columns. In such a case the sound of the tuning fork becomes markedly louder, and we say there is resonance between the tuning fork and aircolumn. To determine the natural frequency of the air-column, notice that there is a displacement node (pressure antinode at each closed end of the tube as air molecules there are not free to moe, and a displacement antinode (pressure-node at each open end of the air-column. In reality antinodes do not occurs exactly at the open end but a little distance outside. Howeer if diameter of tube is small compared to its length, this end correction can be neglected. 9. Closed organ pipe (In the diagram, A p Pressure antinode, A s displacement antinode, pressure node, N s displacement node. p A P N P Fundamental mode : The smallest frequency (largest waelength that satisties the boundary condition for resonance (i.e. displacement node at left end and antinode at right end is λ 4l, where l length of closed pipe the corresponding frequency. p n λ 4L A p A p is called the fundamental frequency...(9. s s N S A S N s A s N s A s First Oertone : Here there is one node and one antinode apart from the nodes and antinodes at the ends. 4l λ l 3 3 and corresponding frequency, n λ 3n This frequency is 3 times the fundamental frequency and hence is called the 3rd harmonic. nth oertone : In general, the nth oertone will hae n nodes and n antinodes between the two nodes. The corresponding waelength is 4l λ l n n + n + and ν n ( n + n...(9. This corresponds to the (n + th harmonic. Clearly only odd harmonic are allowed in a closed pipe. 4. Open organ pipe : Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
S A s N s Fundamental mode : The smallest frequency (largest wae length that satisfies the boundary condition for resonance (i.e. displacement antinodes at both ends is, l l corresponding frequency, is called the fundamental frequency s N s n l N s A s A s A s p P...(9.3 A p A p A p st Oertone : Here there is one displacement antinode between the two antinodes at the ends. l λ l l and, corresponding frequency n λ n This frequency is times the fundamental frequency and is called the nd harmonic. nth oertone : The nth oertone has n displacement antinodes between the two antinode at the ends. l λ l n n + n + and n n (n + n...(9.4 This correspond to (n + th harmonic: clearly both een and odd harmonics are allowed in an open pipe. 9.3 End correction : As mentioned earlier the displacement antinode at an open end of an organ pipe lies slightly outside the open lend. The distance of the antinode from the open end is called end correction and its alue is gien by e.6 r P A p e.6r where r radius of the organ pipe. with end correction, the fundamental frequency of a closed pipe (f c and an open argon pipe (f will be gien by f c and 4( l +.6r f ( l +.r...(9.5 Ex. 6 A tuning fork is ibrating at frequency Hz. When another tuning fork is sounded simultaneously, 4 beats per second are heard. When some mass is added to the tuning fork of Hz, beat frequency decreases. Find the frequency of the other tuning fork. f 4 r Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com f 95 or 5 when st tuning fork is loaded its frequency decreases and so does beat frequency > f f 95 Hz.. INTERFERENCE IN TIME : BEATS When two sound waes of same amplitude and different frequency superimpose, then intensity at any point in space aries periodically with time. This effect is called beats. If the equation of the two interfering sound waes emitted by s and s at point O are, p p sin (πf t kx + θ p p sin (πf t kx + θ s By principle of superposition x p p + p p cos {p(f f t + θ + θ } sin {p(f + f t + + f i.e., the resultant sound at point O has frequency as (t p p cos π f f φ φ t + ( f θ + θ } s while pressure amplitude (t ariex with time p f f Hence pressure amplitude at point O aries with time with a frequency of. Hence sound intensity will ary with a frequency f f. This frequency is called beat frequency (f B and the time interal between two successie intensity maxima (or minima is called beat time period (T B f B f f T B f f (. IMPORTANT POINTS : (i The frequency f f should be less than 6 Hz, for it to be audible. (ii Beat phenomenon can be used for determining an unknown frequency by sounding it together with a source of known frequency. (iii If the arm of a tuning fork is waxed or loaded, then its frequency decreases. (i If arm of tuning fork is filed, then its frequency increases.. DOPPLER S EFFECT When there is relatie motion between the source of a sound/light wae & an obserer along the line joining them, the actual frequency obsered is different from the frequency of the source. This phenomenon is called Doppler s Effect. If the obserer and source are moing towards each other, the obsered frequency is greater than the frequency of the source. If the obserer and source moe away from each other, the obserered frequency is less than the frequency of source. ( elocity of sound wrt. ground., c elocity of sound with respect to medium, m elocity of medium, O elocity of obserer, s elocity of source. (a Sound source is moing and obserer is stationary : If the source emitting a sound of frequency f is traelling with elocity s along the line joining the source and obserer, obsered frequency, f f...(. s s and Apparent waelength λ λ...(. In the aboe expression, the positie direction is taken along the elocity of sound, i.e. from source to obserer. Hence s is positie if source is moing towards the obserer, and negatie if source is moing away x O Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
from the obserer. (b Sound source is stationary and obserer is moing with elocity along the line joining them : The source (at rest is emitting a sound of frequency f traelling with elocity so that waelength i s λ /f, i.e. there is no change in waelength. How eer since the obserer is moing with a elocity along the line joining the source and obserer, the obsered frequency is f f...(.3 In the aboe expression, the positie direction is taken along the elocity of sound, i.e. from source to obserer. Hence O is positie if obserer is moing away from the source, and negatie if obserer is moing towards the source. (c The source and obserer both are moing with elocities s and along the line joining them : The obsered frequency, f f s...(.4 s and Apparent waelength λ λ...(.5 In the aboe expression also, the positie direction is taken along the elocity of sound, i.e. from source to obserer. In all of the aboe expression from equation. to.5, stands for elocity of sound with respect to ground. If elocity of sound with respect to medium is c and the medium is moing in the direction of sound from source to obserer with speed m, c + m, and if the medium is moing opposite to the direction of sound from obserer to source with speed m, c m Ex. 7 A whistle of frequency 54 Hz is moing in a circle of radius ft at a constant angular speed of 5 rad/s. What are the lowest and height frequencies heard by a listener standing at rest, a long distance away from the centre of the circle? (elocity of sound in air is ft/sec. The whistle is moing along a circular path with constant angular elocity ω. The linear elocity of the whistle is gien by A S ωr O where, R is radius of the circle. P B At points A and B, the elocity S of whistle is parallel to line OP; i.e., with respect to obserer at P, whistle has maximum elocity s away from P at point A, and towards P at point B. (Since distance OP is large compared to radius R, whistle may be assumed to be moing along line OP. Obserer, therefore, listens maximum frequency when source is at B moing towards obserer: f max f s where, is speed of sound in air. Similarly, obserer listens minimum frequency when source is at A, moing away from obserer: f min f + For f 54 Hz, s ft 5 rad/s 3 ft/s, and ft/s, we get (approx. f max 555 Hz and, f min 55 Hz. s Ex. 8 A train approaching a hill at a speed of 4 km/hr sounds a whistle of frequency 6 Hz when it is at a distance of km from a hill. A wind with a speed of 4 km/hr is blowing in the direction of motion of the train. Find, (a the frequency of the whistle as heard by an obserer on the hill. (b the distance from the hill at which the echo from the hill is heard by the drier and its frequency. (Velocity of sound in air km/hr. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 3 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
Get Solution of These Packages & Learn by Video Tutorials on www.mathsbysuhag.com A train is moing towards a hill with speed s with respect to the ground. The speed of sound in air, i.e. the speed of sound with respect to medium (air is c, while air itself is blowing towards hill with elocity m (as obsered from ground. For an obserer standing on the ground, which is the inertial frame, the speed of sound towards hill is gien by c + m (a The obserer on the hill is stationary while source is approaching him. Hence, frequency of whistle heard by him is (b f f s for f 6 Hz, s 4 km/hr, and V ( + 4 km/hr, we get 4 f 6. 6 Hz. 4 4 The train sounds the whistle when it is at distance x from the hill. Sound moing with elocity with respect to ground, takes time t to reach the hill, such that, t x x...(i c + m After reflection from hill, sound waes moe backwards, towards the train. The sound is now moing opposite to the wind direction. Hence, its elocity with respect to the ground is c m Suppose when this reflected sound (or echo reaches the train, it is at distance x from hill. The time taken by echo to trael distance x is gien by t x' x' c...(ii m Thus, total time (t + t elapses between sounding the whistle and echo reaching back. In the same time, the train moes a distance (x x with constant speed s, as obsered from ground. That is, x x (t + t s. Substituting from (i and (ii, for t and t, we find or, x x c + m c + m s c + s m x + s c + s + c c For x km, c km/hr, s 4 km/hr, and m 4 km/hr, we get (c + 4 4 4 m m m x x 4 + 4 x 4 6 or, x.935 km. 4 Thus, the echo is heard when train is 935 m from the hill. Now, for the obserer moing along with train, echo is a sound produced by a stationary source, i.e., the hill. Hence as obsered from ground, source is stationary and obserer is moing towards source with speed 4 km/hr. Hence O 4 km/hr. On the other hand, reflected sound traels opposite to wind elocity. That is, elocity of echo with respect to ground is. Further, the source (hill is emmitting sound of frequency f which is the frequency obsered by the hill. Thus, frequency of echo as heard by obserer on train, is gien by f f + ( 6 ( 4 Þ f 6 6 O 86 Hz 9 9. A train approaching a railway crossing at a speed of 7 km/h sounds a short whistle at frequency 64 Hz when it is km away from the crossing. The speed of sound in air is 33 m/s. A road intersects the crossing perpendicularly. What is the frequency heard by a person standing on the road at the distance of 73 m from the crossing. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 4 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.
The obserer A is at rest with respect to the air and the source is traelling at a elocity of 7 km/h i.e., m/s. As is clear from the figure, the person receies the sound of the whistle in a direction BA making an 73 angle θ with the track where tan θ 3, i.e. θ 6º. The component of the elocity of the source (i.e., of the train along this direction is cos q m/s. As the source is approaching the person with this component, the frequency heard by the obserer is υ ucosθ υ 33 64 Hz 66 Hz. 33 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir, Bhopal Phone : 93 93 7779, 9893 5888. page 5 Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffectie People don't.