✤ WAVES :-> It is the mode of transfer of energy & momentum.

✤ TRANSVERSAL WAVE: When transversal wave propagates in a medium then particles of medium vibrate in ⊥ direction of motion of wave:

            ❆ TIME PERIOD: Time taken/by wave ions oscillation (T)

            ❆ FREQUENCY: No. of Oscillation per second. f = 1/T Slunit: Hz, S⁻¹

✤ Relation blw frequency & speed:

               speed = distance / time

                    v = 1/T

                      v = fλ

    Note: Transversal Wave is only seen in liquid & solid medium

✤ LONGITUDINAL WAVES :- when particle of medium oscillate in the parallel direction of propagation of wave. then it is called longitudinal waves.

           These wave travels in the terms of compression & rarefaction

                                

                      * It is seen in all the mediums.

             [NOTE] When sound travel in a medium then medium should have following properties:

✤ WAVE FUNCTION :-

           It is a function of sine or cosine in the term of angular displacement & initial phase.

                               y = asin(wt + Ф₀)

                            

         ∴ at any displacement x, equation of plane progressive wave

                          y= a sin (wt - kx+ Ф₀)

                 Here k = 8x/λ (Propagation constant)

                          ω = 2π / T (Angular frequency)

           [NOTE] speed of wave = f X λ speed of particle = dy/dt= ωa

                   Accelerationg particle= d²y/dt²= -ω² a

             NOTE:- From left:

                                 y = a sin(2πt - 2π/ λ .x)

                        From right:

                                 y = a sin(2π /T . t + 2π/ λ . x)

✤ SPEED OF TRANSVERSE WAVES:-

         a) Speed of a transverse wave on a stretched string.

                   It is determined by two factors :-

                           (I) Tension which is the measure of elasticity

                           (II) Mass per unit length or linear mass density m

                                  m = Mass / Length

                                   v = c√I/m , if c=1 then v=√ I/m

         b) Speed of transverse wave in a solid

                          (I) Elasticity or shape or modulus of rigidity

                           (II) Mass per unit volume or p, determines inertia

                                   v= C √ η/p , if c=1 then v= √ η/p

✤ SPEED OF LONGITUDINAL WAVES:-

         a) Speed of longitudinal wave in a liquid

                   (I) Volume elasticity or bulk modulus K of the fluid.

                   (II) Density of the fluid which determines inertia

                           v= C√ K/p , if c=1 then v= √ K/p

         b) Speed of a longitudinal wave in a solid: -

                   v = √(K + (4/3 * η)) k + (4/3 * η) = elongational elasticity

         c) Speed of a longitudinal wave in a solid rod :-

                   v = √(Y/ρ) Y= Young's modulus

                    P= density

✤ SPEED OF A SOUND :-

              * Newton's formula for the speed of sound in a gas. He assumed that sound waves travel through a gas under isothermal  conditions. He argued                 that the small amount of heat produced in a compression is rapidly conducted to the surroundings rarefactions where light cooling is produced.

                                    V= √(Kk0 / ρ)

                       For isothermal change:

                                        PV= Constant

                        differentiating both sides

                                        PdV + VdP

                                        PdV = -VdP

                                   P= -Vd P/dV = -dP/dV/N

                                      = Volumestress/ Volumestrain = Kiso

                                    :. V= √(P/ρ)

                          AT STP. P = 1.013x105 N/m2

                                         = 1.293 kg/m3

                           Speed of Sound in Air = √(1.013x105/1.893 )≈ 280 m/s

                              This value is about 15% less than the experimental Values

❆ LAPLACE'S CORRECTION:-

                  Laplace pointed that sounds travels through a gas under adiabatic conditions not under isothermal conditions:-

                             V= √Kadia/ρ

                  For an adiabatic Change, PV = Constantdifferentiating both sides

                         P(V) + VdP= 0

                         AV + VdP= 0

                         √P = -dP/dV = Kadia

                                   dV/V

                        Y= cp/cv, with ratio of two specific heats.

                               V= √Y.P/ρ

                            For air = 7/5 then

                                   V= √Y.P/ρ

                                   V= √7/5 * 880

                                   V= 331.3 m/s

                          Hence Laplace formula is justified...

✤ PRINCIPLE OF SUPERPOSITION OF WAVES:-

          1- Independent behavior of gases, when a no. of waves travel through region at the same time, each wave travels independently of the others.

                  For Example: Musical notes instruments Pave different notes

          2- Principle superposition waves: The resultant displacement of the particle is equal to the algebraic sum of the individual displacements given out                     by two waves. This is the principle of superposition of waves.

 ❈ Constructive Interference: When the crest of two identical pulse superposes, then the displacement of the resultant pulse is twice

                by the either pulses

                        

 ❈ Destructive Interference: When the crest & trough of two identical pulse superpose, then they cancel each other.

✤ STATIONARY WAVES :-

          When two identical waves of same amplitude & frequency travelling in opposite direction with the same speed along the same path superpose each             other, the resultant wave does not travel in the either direction & is called stationary or standing wave.

                              y = 2a sin kx cosωt

  → Nodes: Particles which have zero displacements

  → Antinodes: Particle which always suffer max. displacement

  → A stationary wave can be produced only in a finite medium which has its boundaries.

 ✤ PROGRESSIVE WAVES

            The disturbance travels forward with a definite velocity. Each particle of the medium executes SHM about it's mean position with the same             amplitude.

             There is a continuous change of phase from one particle to the next.

             Nobellium’s particles is at most permanently

             There is no instant when all the particles are at the mean positions together.

             There's an flow of energy across every plane along the direction of propagation of the wave.

             The average energy over a wavelength is half potential & half kinetic.

✤ STATIONARY WAVES

            The disturbance remains confined to the region where it is produced

            Except nodes, all particles of the medium execute SHM with varying amplitude.

            All the particles bet ween successive nodes vibrate in the same phase but the phase reverses for particles between next pair of nodes.

            The particles of the medium at nodes are permanently at rest.

            Twice during each cycle all particles pass through their mean positions simultaneously.

            Energy of one region remains confined in particular region

            Twice during each cycle the energy becomes alternately potential & kinetic

✤ FACTOR AFFECTING THE SPEED OF SOUND

         ❈ Effect of temperature :-

    we know by Laplace's law

      v = √γP/ ρ

    We know, PV= RT

      p = RT / V

      v=√RT/V ρ

      v=√RT/M

        v=C√T

         ❈ Effect of Pressure: Pressure => has no effect on the speed of sound.

         ❈ Effect of Humidity => With rise in humidity, density decreases so speed increases.

         ❈ Effect of Wind => In same direction, θ = 0°

                                                              cos 0⁰ = 1

                                                              V'= v+w

         In opposition, θ = 180°

          cos 180° = -1

          V' = v-w

         ✤ Stationary Wave in stretched string :-

           let y₁ = a sin(wt-kx) ----①

           & y₂ = a sin(wt+kx)

           y₂ = -a sin(wt+kx) ------②

       y = a[sin(wt-kx) - sin(wt+kx)]

         y = a 2.cos(wt-kx+wt+kx)/2

         y = 2a coswt . sin(-kx)

         y = -2a coswt sin kx

         y = -3a Sinkx.Cost wt

          at x = 0

          y = 0

         x=L at x=L, then amplitude becames

              zero.

       (-2a) Sink L = 0

         Sink L = 0

        ⇒KL=nx

        2piL/ λ = nx

        λ = 2L/n

   we know, frequency = velocity (v) / wavelength (λ)

             ❈ For n=2; λ = 2L / 2

          λ = L

      :. v₂ = v / λ

      v₂ = v/L

      :. v₂ = 2v₁

     [ freq. of 2nd harmonic or 1st Overtone ]

 ❈ for n=3, λ = 2L / 3

    :. v₃ = v / (2L / 3)

     3(V/2L)

    :. v₃ = 3v₁

   [ freq. of 3rd harmonic or 2nd overtone ]

 ❈ NOTE ===> v₁: v₂: v₃ = 1:2:3

        ❈ It is defined as the alternate variation in the intensity of sound whenever two turing fork same, nature level slightly difference in frequency

            superimpose on each other

                                                        

                                y₁= aSin(wt+kx)                                                                                      y2 = aSin(w₂t-kx)

                                                                                         W₁-W₂

                                        (y₁+y₂)        

❈ Beat Period: Time taken in one beat is called beat period

❈ Beat frequency: No. of Beats per second

                             m=v₁-v₂ v1=v2±m

                                Beat frequency

❈ NOTE: TUNNING FORK In lower weight               In extra weight v₁ = v₂-m

                            v₁ = v₂+m m = mass

                                m=mass

❈ ORGAN PIPE: These are musical instrument which are based on reflection of sound.

✤ EQUATION OF STANDING WAVE IN OPEN ORGANPIPE

                  Weknow, y₁ = asin(wt-kx)

                                  y₂= asin(wt+kx)

                 By superposition....

                                  y=y₁+y₂

      y=a[sin(wt-kx)+ sin(wt+kx)]

      y=2a sin((wt-kx+wt+kx)/2)cos((wt+kx-wt-kx)/2)

      y= 2a sin(wt).cos kx

      y = 2a cos kx sin wt

          at x=L

         amplitude = ±1

         cos kx = ±1

          kx = nπ

            2π/λ.l = nπ

            λ = 2L/n

 ❈ It states the change in frequency of a wave (like sound or light) due to the relative motion between the source of the wave and an observer. When the         source and observer move closer, the observes frequency increases, leading to a higher pitch for sound or a shift toward blue for light.that changing         the sound listens the frequency delta.

                    f'= ((V+ V_0)/(V-V_s ))f f’ = Freq. of source listen by listener.

                                f = Actual Freq.