Stationary Waves Presentation by Ms. S. S. Patil

Study of vibrations in finite medium When wave incident at the boundary of rigid medium ( i. e. finite medium) there is phase change of п rad. If the wave travelling through a denser medium arrive at the surface of a rarer medium, then wave is partly reflected & partly transmitted. The reflection of waves from rarer medium takes place without change of phase.

Stationary waves When two identical progressive waves ( i.e. waves having same amplitude, wavelength & speed) travelling through medium along same path in opposite direction, interfere with each other, by superposition of waves, the resultant wave obtained in the form of loops, is called a stationary wave or standing wave.

Equation of stationary wave Y1 = a sin 2п ( nt – x/λ ) & Y2 = a sin 2п ( nt + x/λ ) Y = Y2 + Y1 Using sin C + sin D = 2 sin ( C+D/2) cos(C-D/2) & cos (-ө) = cosө Y = 2a sin2пt ∕ T cos 2пx ∕ λ Y = 2a sinwt coskx Y = A sinwt Where A is resultant amplitude . A = 2acoskx

Condition for antinode - Amplitude at antinode is maximum, A=± 2a A = 2a cos 2пx/λ 2a = 2a cos 2пx/λ x = P(λ/2) where P= 0,1,2…… For x = 0, λ/2, λ, 3λ/2, ………. antinodes are produced. Condition for node – Amplitude at node is zero, A=0 But A = 2a cos 2пx/λ 2a cos 2пx/λ = 0 x = (2P – 1)λ/2 where P = 1,2,…… For x = λ/4, 3λ/4, 5λ/4, ………… nodes are produced.

Properties of Stationary waves In a stationary wave there are some points at which amplitude is zero, such points are called nodes & at some points amplitude is maximum such points are called antinodes. The distance between two successive nodes or antinodes is λ/2 & distance between node & adjacent antinode is λ/4.

In stationary waves all the particles except the nodes vibrate with same period as that of interfering waves. The amplitude of vibration is different for different particles & it increases from node to antinode. All the particles within the loop are in same phase of vibration & particles in the adjacent loop are vibrating out of phase. The resultant velocity of stationary wave is zero & thus there is no transfer of energy through the medium. Stationary waves show double periodic phenomenon.

Where V = √T/m Name Structure 1 Antinode 2 Nodes 2nd Harmonic or Modes of vibration of string Picture of Standing Wave Name Structure                                               L = λ/2 n = V/2L 1st Harmonic or Fundamental frequency 1 Antinode 2 Nodes L                                              L = λ1 n1 = V/L n1 = 2n 2nd Harmonic or 1st Overtone 2 Antinodes 3 Nodes                                              L = 3λ2/2 n2 = 3V/2L n2 = 3n 3rd Harmonic or 2nd Overtone 3 Antinodes 4 Nodes Where V = √T/m

Laws of vibrating string Fundamental frequency of vibrating string is n = since m = ƍпr2, ƍ- linear density of string. T – Tension in string. L – Length of string.

Vibrations in Air Column When a loudspeaker producing sound is placed near the end of a hollow tube, the tube resonates with sound at certain frequencies. Stationary waves are set up inside the tube because of the superposition of the incident wave and the reflected wave travelling in opposite directions.

The natural frequency of a wind instrument is dependent upon Factors that determine the fundamental frequency of a vibrating air column The natural frequency of a wind instrument is dependent upon The type of the air column, The length of the air column of the instrument. Open tube Closed tube

Picture of Standing Wave Name Structure Modes of vibration for an open tube Picture of Standing Wave Name Structure                                              L = λ/2 n = v/2L 1st Harmonic or Fundamental frequency 2 Antinodes 1 Node                                              2nd Harmonic or 1st Overtone 3 Antinodes 2 Nodes L = λ1 n1 = v/L                                               L = 3/2λ2 n2 = 3v/2L 3rd Harmonic or 2nd Overtone 4 Antinodes 3 Nodes

Picture of Standing Wave Name Structure Modes of vibration for a closed tube Picture of Standing Wave Name Structure                                               1st Harmonic or Fundamental frequency 1 Antinode 1 Node L = λ∕4 n = v/4L                                               3rd Harmonic or 1st Overtone 2 Antinodes 2 Nodes L = ¾λ1 n1 = 3v/4L                                              5th Harmonic or 2nd Overtone 3 Antinodes 3 Nodes L = 5/4λ2 n2 = 5v/4L

End Correction (e) The antinode is not formed exactly at the open end, a little distance beyond it. Distance between open end & antinode is called end correction. Formula : e = 0.3d where ‘d’ is inner diameter of tube. Hence corrected length of air column = length of air column in pipe + end correction L = l + 0.3d For pipe closed at one end ‘e’ is 0.3d or 0.6r ( r – inner radius of pipe) In a pipe closed at one end, vel. of sound in air at room temp. is : V = 4n(l+0.3d)

For pipe open at both the ends ‘e’ is 0.6d or 1.2r In a pipe open at both ends, vel. of sound in air at room temp. is : V = 2n(l+0.6d) ‘e’ in terms of first & second resonating length is : e = (l2 – 3l1)/2

Free vibrations Forced vibrations Free & Forced vibrations Free vibrations Forced vibrations If a body is displaced from its equilibrium position & released, the restoring forces acting on the body may set the body in vibrations . Such vibrations are called natural or free vibrations. Frequency of vibration is called natural frequency & it depend on mass, dimensions, elastic properties & modes of vibration of the vibrating body. e.g. oscillation of simple pendulum. The vibrations of body under action of external periodic force in which body vibrates with frequency equal to frequency of external periodic force such vibrations are called forced vibrations. Frequency of vibratoin is equal to that of external periodic force. e.g. vibrations of pendulum in clock, vibrating tuning fork produces forced vibrations in an air column in a pipe.

It is a special case of forced vibrations. Reasonance The phenomenon in which the body vibrates under action of external periodic force , whose frequency is equal to the natural frequency of driven body, so that amplitude becomes maximum is called Reasonance. It is a special case of forced vibrations. This phenomenon is used to find unknown frequency of tuning fork in Melde’s & sonometer expt. Soldiers are ordered to break their regular stepping in marching while crossing a suspension bridge. RESONANCE A N=N0 F

Where N – frequency of fork. Melde’s experiment Use – To demonstrate formation of transverse stationary wave & to find the frequency of tuning fork. Parallel position – The direction of vibrations of prongs of tuning fork is parallel to the direction of a string & during 1 vibration of tuning fork sting complete half vibration. N=2n = Where N – frequency of fork. P – no. of loop. n – frequency of string. l – vibrating length.

Perpendicular position The direction of vibration of prongs of tuning fork is perpendicular to direction of vibration of string.So frequency of fork is equal to frequency of string. N= n=

i.e. Pperpendicular = 2Pparallel Note When parallel position is turned into perpendicular position by keeping other setting const. no. of loops get doubled. i.e. Pperpendicular = 2Pparallel In both position : TP2 = const. if l is const. T1P12 = T2P22 (m1+m0)P12 = (m2+m0)P22   Where m0 = mass of pan & m1 & m2 = masses put in the pan. TP2 = const . P α 1/√T OR P α √T = constant. If tension increases no. of loops decreases.  

Thank You.