( ) Sine wave travelling to the right side

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Κλεισθένης Παππάς
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1 SOUND WAVES (1) Sound wave: Varia2on of density of air Change in density at posi2on x and 2me t: Δρ(x,t) = Δρ m sin kx ωt (2) Sound wave: Varia2on of pressure Bulk modulus B is defined as: B = V dp dv P = Pressure V = Volume ( ) Sine wave travelling to the right side As ρ= m ΔV, Δρ= ρ V V = ρ $ dp ' & ) % B ( If Δρ is small, ρ ρ 0 = undisturbed density then, Δρ=ΔP ρ 0 B

2 Therefore the sound wave expressed in the form of varia2on in air density can also be expressed in the varia2on of pressure: ρ 0 sin( kx ωt) ΔP(x,t) ρ 0 B = ΔP m B ΔP(x,t) = ΔP m sin kx ωt where ΔP m = B ρ 0 Δρ m ( )

(3) Sound wave: A displacement wave - Consider a volume of gas with boundaries at posi2ons x and x+δx. The mass of the gas is δm. - Now a density varia2on (i.e. a sound wave) comes and the displacements of the two boundaries are s(x,t) and s(x+δx,t).

3 (3) Sound wave: A displacement wave - Consider a volume of gas with boundaries at posi2ons x and x+δx. The mass of the gas is δm. - Now a density varia2on (i.e. a sound wave) comes and the displacements of the two boundaries are s(x,t) and s(x+δx,t). Positions of the two boundaries: x + s(x,t) x + δ x + s(x + δ x,t) Original density: ρ 0 = δ m Aδ x After the wave comes, the length of the gas volume: " δx + S(x +δx,t) s(x,t) = δx 1+ # $ S(x +δx,t) s(x,t) δx % ) S, & ' = δx+ 1+. * δx -

4 The new density thus becomes: δm ρ = Aδx 1+ S = ρ 0 " % $ ' 1+ S # x & x Change in density: Δρ = ρ ρ 0 = ρ 0 1+ S x As S x << 1, Δρ ρ 0 S x 1 S 1+ x ρ 0 = ρ 0 1 S x 1+ S x S x = Δρ = Δρ m sin( kx ωt) ρ 0 ρ 0 Integrate both sides with respect to x: 1 ρ 0 S(x,t) = Δρ m cos( kx ωt) = S m cos( kx ωt) kρ 0 where S m = Δρ m kρ 0

5 S(x,t) = Δρ m cos( kx ωt) = S m cos( kx ωt) kρ 0 Velocity of gas layer (i.e. rate of change of the displacement S(t)): u x = S t = ωs m sin kx ωt ( ) = ω 2 S m cos( kx ωt) = ω 2 S a x = u x t i.e. SHM in horizontal direction NOT the phase velocity

6 ( ) ( ) ( ) Δρ(x,t) = Δρ m sin kx ωt ΔP(x,t) = ΔP m sin kx ωt S(x,t) = S m cos kx ωt u x = S t = u m sin( kx ωt)

7 Speed of sound wave At t=0, a sound wave (i.e. air compression) arrives at the boundary of a column of undisturbed gas. Consider the length of the gas column L 0 is such that aqer the compression its length becomes L. AQer some 2me, the sound wave pulse go through the air column and the gas column is compressed (i.e. pressure increases, or volume decreases) Let t be the 2me required for the wave to travel through the whole gas column Δx CM = 1 2 a CMt 2 a CM = 2 ( x CM, f x CM,i ) = t 2 ) 2 L 2 # L 0 + % * $ 2 t 2 &, (. '- = L 0 L t 2 = ΔL t 2 ΔL=change in column length

8 As the mass of the column: M = ρ 0 AL 0 ( )& ΔL F ext = AΔP = Ma CM = ρ 0 AL 0 But t = L 0 v P, v P = phase velocity AΔP = ρ 0 AL 0 $ v 2 P = 1 & & ρ 0 & % v P = $ ( ) ΔLv P & B ρ 0 ΔP AΔL AL 0 % L ' ) ( $ % ' $ ' ) ) = 1 & ΔP ) & ) = B ρ ) 0 & ΔV ) ρ 0 ( % V ( t 2 ' ) ( In gas, B = γp 0 γ = specific heat ratio (~ ) P 0 = Undisturbed pressure

9 Power and intensity of sound waves ΔP(x,t) = ΔP m sin( kx ωt) Force ac2ng on the air by the sound waves: F x = AΔP m sin( kx ωt), A = Cross section area of the air column Work done by the force in a period of 2me Δt: ΔW = F x Δx, Power P = ΔW Δt = F x Δx Δt = F x u x Δx = displacement of the air column's CM N.B.: u x not the phase velocity v P As u x = u m sin( kx ωt) = v ΔP P m B P = Av P P ave = ( ΔP m) 2 B T 0 P dt T sin 2 ( kx ωt) = Av ΔP P ( m) 2 2B sin( kx ωt) = A ΔP m ( ) 2 2ρv P

10 Define sound level (SL): SL =10log I I 0, I 0 =10 12 Wm 2, unit=db Note: (1) Log scale is used because the ear response to intensity is logarithmic. (2) I 0 =10 12 Wm -2 is chosen because this is the minimum intensity that human ear commonly barely hears.

11 Interference of sound waves (2-dimensional) Consider two sources S 1 and S 2 having the same frequency, phase and amplitude. Observa2on is made at P. Path difference=δl The corresponding phase difference Δφ=2π ΔL λ If ΔL = mλ or Δφ = 2mπ, m=0, 1, 2,... ( ) y 1 = Asin kr 1 ωt y 2 = Asin kr 2 ωt = Asin( kr 1 ωt) ( ) = Asin k ( r 1 + mλ) ωt ( ) y P = y 1 + y 2 = 2Asin kr 1 ωt Constructive interference # #$ % & = Asin kr 1 ωt + 2π $ ' λ mλ % & ( Media at P in SHM with magnitude 2A, maximum amplitude

12 " If ΔL = m + 1 % $ # 2 & 'λ or Δφ = 2 " m + 1 % $ # 2 'π, m=0, 1, 2,... & ( ) y 1 = Asin kr 1 ωt y 2 = Asin kr 2 ωt = Asin( kr 1 ωt) y P = y 1 + y 2 = 0 ( ) = Asin + k $ r + $ m # 2 Destructive interference ) * " # " % %, 'λ' ωt & & -. = Asin ) kr ωt + 2π " 1 λ m + 1 %, + $ # 2 'λ. * & - Media at P does not oscillate.

open end free end fixed end Phase 0 o 180 o change 0 o 180 o

13 Standing longitudinal wave for tubes with one end closed Longitudinal wave reflec2on Transverse wave reflec2on closed end open end free end fixed end Phase 0 o 180 o change 0 o 180 o An2-node for closed end Node for open end An2-node for free end Node for fixed end

14 Standing longitudinal wave for tubes with both ends open λ = 2L, f = v 2L λ = L, f = v L λ = 2 3 L, f = 3v 2L λ n = 2L n, f n = n v 2L, n=1,2,3,4...

15 Standing longitudinal waves for tubes with one end closed and the other other open λ = 4L, f = v 4L λ = 4 3 L, f = 3v 4L λ = 4 5 L, f = 5v 4L λ n = 4L n, f n = n v 4L, n=1,3,5,...

16 Beats At a par2cular point when two waves are passing. The frequency of the two waves are closed. Let the varia2on in pressure for the two waves are: ΔP 1 (x,t) = ΔP m sinω 1 t ΔP 2 (x,t) = ΔP m sinω 2 t For simplicity, also assume their amplitudes are equal. Resultant pressure wave: # ΔP = ΔP 1 + ΔP 2 = ΔP m ( sinω 1 t + sinω 2 t) = 2ΔP m cos ω 1 ω 2 & t $ % 2 ' ( sin # ω 1 +ω 2 & t $ % 2 ' ( ω amp ω av - ω av ω 1 or ω 2 - ω av >> ω amp

17 The equa2on is oscilla2ng with the original frequency but with the amplitude varying with a much slower frequency. Beat Period = T beat = T amp 2 ω beat = 2ω amp = ω 1 ω 2 Phase velocity and group velocity Phase velocity: v P = f λ = ω k Group velocity: v G = dω dk For non-dispersive media, the two veloci2es are the same. For dispersive media, they are different.

18 ω = ck ( ) 2 group ω = ck 1+ k / k 0 phase

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